 Lesson Details
 Transcript
 References
 Instructor
 Iliya Mirochnik
 Subjects
 Drawing
 Topics
 Composition, Drawing Exercises, Still life
 Mediums
 Paper, Paper Stump, Pencil
 Duration
 1h 38m 15s
In week two, instructor Iliya Mirochnik will teach you the fundamentals of drawing form, using simple geometric shapes such as the cube, cylinder, and cone. You will learn how to describe threedimensional volume and construct forms from observation.
Throughout this course, you’ll have access to the NMA community for feedback and critiques to improve your work as you progress.
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Okay so now that you have
learned to take an angle and transfer it onto a piece of paper and
also establish your proportions, it's now time to turn our attention to
an actual object in front of us. And so here we have the cube.
And we've already worked a little bit
with the cube in learning to
take our angles and to establish our proportions so I think you should be
comfortable with it by now.
So what you'll need for this is your
pencil, your eraser, and your needles.
Now we are going to be
drawing the cube using
the techniques that we have covered. And
it'll be entirely based on what's in front of us
and our perception of
the object from where
we are in space.
learned to take an angle and transfer it onto a piece of paper and
also establish your proportions, it's now time to turn our attention to
an actual object in front of us. And so here we have the cube.
And we've already worked a little bit
with the cube in learning to
take our angles and to establish our proportions so I think you should be
comfortable with it by now.
So what you'll need for this is your
pencil, your eraser, and your needles.
Now we are going to be
drawing the cube using
the techniques that we have covered. And
it'll be entirely based on what's in front of us
and our perception of
the object from where
we are in space.
AUTO SCROLL
So begin by
establishing an angle. Let's
start with the bottom edge over there, it is the
same place we
started before.
So you can
find that angle, all you have to do is,
as we practiced, you put one of the needles up as a
vertical constant, line up the other one with the edge, and
put it against your paper.
And you find
a point, hold it up, and see
that it lines up with another point
that you are making, make sure that
you're correct, even go back to the cube,
move back to your paper. There's no point in speeding this up.
And now you connect them.
And
continue all the way.
As far as you can. Okay
so the next point that we need
is our corner and then
the other edge that
we can perceive from our angle, going off of
the line that we made. So we're going to be
doing the same thing.
Hold it up,
lock it in place,
bring it up,
and get that angle. Good.
I'm gonna move it a little bit.
Alright.
And now transfer it to your paper and make sure the two lines
intersect around the
middle of your paper. So one of the points that you need
will actually be on the line that you already drew.
(checking)
Let's see if that's right. I might need to lower this a bit.
Pull it back, make sure you're on the right track.
And move
it across.
Okay. So now we've established
two
parts of our cube.
So what we need to do from here
is to get this
line that we see, the closest 
the closest intersection of the two planes that are in front of us,
the closest corner, and
you don't actually need to take this angle because we are going to 
I mean you can, but we're going to assume, and for the most part
it is going
to be because of how large it is,
a complete vertical line. So from this point that we have
all you need to do is
get a vertical.
(drawing)
And that's what we have
here. Okay. So we're going to continue.
So we now need to establish
how high
the corner up here
is. So in order to
do this
from the cube in front of us, only using the
techniques that we learned, we're going to take the angle
from that
corner to that one.
(checking)
And
then you place the line anywhere along 
(checking)
you place a line anywhere along that
will intersect these two lines that we already
have at any point along the line depending on how large you want the cube
to be on the page. So what I'm going to do
at this point is  I think
that this seems alright 
and then we find the proportion and then you establish 
I'm gonna go back over and make sure
it's right. Get them in my hand, move
it over and find that point on this line,
step back and make sure it's right, it seems okay, we're good.
So now we have
those two points that are going to be quite important.
And so, since we've decided that this line
is completely vertical, we're going to continue with that same logic
and our  all of our
vertical edges are going to be completely vertical
and just  you can take it all the way up.
Now in order to establish the amount that we have
on this side of the cube, we're going to do the
exact same thing that we did here, going
from that point to that point except now we're going from that point
to the opposite one along the surface
of that plane.
Let's see  so let's
let
get that in place.
That seems right. Do the same. Now
you already have a point there, hold it up
and mark where the other point
is, line them up, see if you're good.
I actually will probably move this over a little bit, see if you're
correct. Take your time and make sure you get everything exactly as it should be,
move it over,
I think that's good. And so now we have that.
At this point 
also this is a vertical. So
before we move on, it's now time to see
if our angles were correct by
double checking them with a proportional
measurement. So we're gonna take a
simple measurement that is horizontally across, a lot like
I showed you in one of the previous
assignments. So I'm curious to
see the amount that this part of the
cube will go into that part of the cube.
So when I see it in life, I'm locking it in,
the point of the needle is on the outer edge, my thumb
is on the closer edge and then I move
it over and then I do it again and I'm getting slightly under one and half
times. So let's see.
Do this here, that's what we need there, we'll move
this over, I'll even do a mark, and then we even
move it over and I'm
in exactly one and a half times. So let's
try this one more time to see if it's right. One.
Yeah, I think one and
a half is good. It seems to correspond
with our
angle that we took between these two points. And I will
double check again.
Okay.
It actually seems to be slightly
closer than the line I have, which will pull it to right under
one and a half times.
So I think we're good.
At this point if there are any lines that
you made that you're not happy with, that are potentially inaccurate, just 
you can clean them up with the eraser
and we're almost there.
So what we need
now is to get the angles of the
plane that is on top. So since we have 
since we have that point, all we need
to do is continue
with our angles.
Line them up, keep that there.
(checking(
Okay. We'll just hold it in place.
Make sure that line is on the point and find
where the other
is. Line them up,
look again,
a little higher, and connect
them.
(drawing)
Okay.
Let's keep at it. Now we're going from here
into there. And now we continue. Now we're
going from that point to the other side of our cube.
(drawing)
Line them up,
(drawing)
hold it in place, align it and make sure your point is there,
and connect them.
(drawing)
And clean out
what you don't want.
Okay.
Let's keep at it.
This is the time to go
over and get your lines a little cleaner.
You can reinforce them a little bit. It's looking
okay for now. Okay so the next
order of business is now to get these lines in the back.
Now there are a few ways.
You can obviously work along
this edge or you can establish that angle. I would do all of them
actually to make sure you get them right. So I'm gonna start with
that line.
(checking)
Let's find it there.
(checking)
Can find it right there, get the point
get another point, and
pull them all the way.
(drawing)
Okay. So
what we  so right now we're gonna get the other side
but you can get that angle also
because it'll tell you where that point is and then you'll be able to
connect them. The order here won't matter. I'll start from the outside though.
(checking)
Then I can transfer 
got that point there,
that point right there, look back, make sure
you're right,
and connect them.
(drawing)
And what we need
to do is to make sure we have it right. And now it's time to
get to double check our point.
Here I'm using the constant as a horizontal and
finding where this line is.
(drawing)
And that seems about right.
(checking)
And it
matches up to this 
this
point. So
now it's 
now that we have our lines you can remove
some of the extras.
(drawing)
So this was a cube
that you drew
entirely from observation.
(drawing)
We are going to be talking about light and shadow
so we're just gonna keep this in line
at the moment.
(drawing)
Alright. So there you have it. Next time
I'm going to explain a little bit about the construction of cubes.
So now that you've worked on a cube
entirely from observation, let me tell you a little bit about the
construction of a cube or
a rectangular prism. Now if you've
noticed that when you're taking each angle,
moving it over to your piece of paper, taking the angle after that,
this takes quite a long time. And so I believe that an understanding of
the structure of a cube can
speed things up and also allow you to work from imagination.
So I'm going to demonstrate a few concepts
on the piece of paper in front of me. And you can just follow along.
So in order to begin we need
to establish something that's known as
the horizon line.
Which we will just call  this is our horizon line,
you can call it H.
And so the basic
idea that I want to get across
is that
as an object moves farther away
from you it becomes smaller. And there are
a few things that happen to the angles within
that object  and especially a cube  that I
would like to cover. So along 
on our horizon line, let's put a point on one end
and a point on the other end. This will be
our vanishing point
one or
V one and this is our vanishing point number two or
V two. So
in order to structure a cube
on the plane that's here
all you need to do is extend a line from
your vanishing point and then extend another
line onto the surface.
And then do
the same thing here
and there. And so
this right here gives us a plane.
Then the next thing you should do is pull up
from
the corners of this plane, completely vertically.
You pick a point along that
line
and you move it back towards your vanishing point
here and
back towards your vanishing point over there and
all around.
(drawing)
Okay. So the idea
behind the horizon line is that it is exactly
where your eyes are, whether you are
standing up or
in a chair if you're looking in front of you, your eyes
are the horizon line. So
this'll be also your
eye line. So the
important thing to understand is that if you have
plane of an
object, in this case a cube or a
rectangular prism, and it lines up exactly
with your horizon line
it becomes a line.
(drawing)
And so the other idea is that if you have  and so if we
extend
what will become the closest edge to us 
we mark that as the end, we mark that as the end  all things below
your horizon line are going 
are going
up towards it. And all things above your horizon line
are going back
down towards it. And the same on the other side.
And then you lock it in.
And so obviously the closer
a plane gets to your
horizon line, the
closer it appears as a line. So the most important
aspect here is that
in a cube, or
on a plane
that has
parallel sides,
as they're moving away from you they are slightly converging.
So
you can apply all this without the perspective if you're
aware of that. So if you begin with
an edge and you get that line 
you get
that line, you don't actually
need to understand where  you don't need to
observe where any of the other lines are. Because all they are 
they're practically parallel and slightly
converging away from you.
So that line is actually shorter than that one.
And the same applies to the other side.
(drawing)
And so this also allows you
to tilt the cube in any way you want.
So as long as you keep  as long
as you keep as  as long as you are
aware of the point that is the closest point to you.
So for example.
(drawing)
And
(drawing)
so obviously
the plane that we have here
is going to be a larger one than the one that we have on the other
side. Or at least  and obviously in reality
they are identical but
on paper they have to appear smaller due to perspective. And so
the interesting  the only thing to keep
in mind is that if you're working on a piece of paper
and you establish a horizon line, you have a vanishing point on one end
and a vanishing point on the other end, the perspective will be drastically increased.
And our eye does not perceive 
perceive exactly
in the way that I showed it here. So if you were to
imagine our eyes, the vanishing points would probably be actually quite  would be
off the page. So the kind of
angles that I have on the
lines that we have here would not really
be observed on a cube that you can essentially  that is as
small as the one that we have in front of us. So
the idea is that you have to keep them
practically parallel with just a minor amount of
conversion. So
now that we've covered that, let's learn to combine
our understanding of the cube with the cube that we actually have in front of us.
Okay so now that we have
practiced working from observation and understand a little bit about
the construction of a cube or a rectangular prism,
it's time to combine all of this
into working from observation
utilizing certain observational principles that we
spoke about as well as our understanding of the forms in front of us.
So what you're going to need for this is your needles again.
Okay. So
let's begin by doing the exact
same thing that we did when we were working entirely from observation.
So I am going to
find the angle of that edge, the angle
of that edge.
So as we've already practiced, by now
what seems like a million times and that's great,
keep one as a vertical constant, lock in the other one
as you align it with the edge and move it over.
(checking)
And then you find it on your paper.
(drawing)
Alright.
Obviously if you're unsure you can always
(checking)
go back and compare it to the cube.
Now I'm going for the other side.
Do the same thing, I have my needles.
My more horizontal needle
aligning to that angle. I bring it over, I mark
a point, mark another point
right there, and now I extend that
out here. So now
we have these two extremely important
(drawing)
lines because we're going to construct
upwards using them. But before that,
we know that we can just take
the point right here and extend it upwards, really completely vertical.
(drawing)
And now it's important to
find the end of this edge
and so in order to find this end
we need to establish from point to point that line
and then from point to point that line. So
a diagonal across the surface of each
square plane.
Okay so let's see where that
takes us. So now I'm going to keep my constant
horizontal as opposed to vertical in order to make it
easier to get the angle right and I'm going to go
from those two points.
So here, oh it's
quite vertical. Take your time,
get them aligned, move them over,
and place them where you
want them.
(drawing)
Actually
here. And
then let's see if that's right. That's right.
Okay. So now we need to do the same thing on
this other side to find that diagonal between the two opposite
corners of the square plane.
There it is.
You already have that point right there,
move it over, keep it there,
and find that point right there, and connect them.
Okay. So that's all
that you'll be using the needles for because everything else
you can do from your understanding of the cube.
So now that we have that point and that point,
we know also that they just need to be extended up vertically.
Up vertically. And there you have it.
And then from the top corner,
in order to find the angle of that edge
all you need to do is take the link that you've already observed,
move it up and tilt it
off of that  this
point so that this edge right here appears smaller
than the edge that's closest to you. So in essence
this edge and that edge are going to be
converging towards a vanishing point.
(drawing)
Bring it down, you can hold it
and see where you are.
Alright. Now you're going to be doing the same thing here.
The interesting thing to
keep in mind is that if the surface of
a plane  if the surface of a plane on
the cube is more open, then you don't
actually need to  you don't need to exaggerate
the conversion of these two opposite 
these two opposite edges.
You have to exaggerate them only when the 
only when the surface of the cube is turning a bit further away
from you. So it's more in perspective
in essence. So here we go to there. You still don't want
them perfectly parallel but just a slight tilt
off of this point. And
that's that right there.
And always go from one edge to the
other and tilt them down a little bit, and there
you have this.
And you already have that point
so it's pretty much
it.
Okay. So that's
what we have right there. So now let's try something a tiny
bit more complicated. So I'm going to
put something under this cube in order to get at a slightly more
complex angle. I'm also, in order to make this a little more
interesting, we're going to turn it the other way. So now that's
what we have. Okay. So
here you'll  the only
difference that you have is that you can't count
on this edge being completely vertical. So you will also have to
sort of find the angle using
your needles and transfer it over. Everything else
will be established using our understanding
of the structure of a cube. So I will begin with
that edge because it's the only one
that is on the ground. So as we
said, think like an architect and begin
to structure from the ground up.
There it is. Find it there.
Now I'm going for
the other side of the cube, the one that's now
lifted off of the
ground.
And now I'm going to go for the nearest
edge to us, let's see what we have.
And we're good and move it over.
We're here, we're here, and that's all
that you need to observe. Now you can put
your needles away for a moment. Ah, I was wrong.
We still  we still actually need to
establish where this point is and where that point is. So now you're going to be 
you're going to be taking the angle of the diagonal of our
cube. Of a surface of our
cube.
If you're holding this you can always put your pencil behind your ear, it's one of the more convenient
places.
Here's this angle,
go from there  oops sorry. Remember to keep
your constant
constant. So it has to be a vertical when it's in front of you
taking a measurement and a vertical as you move it
over to the paper. That's the point of it. And then you go
from there to there, you already have that point,
whether this actually is rather close, corresponds quite well. So now
we have it. Okay.
So what we
can  what we can
do from here is now we know that that's the point that's closest to us.
So now we begin to work off of this edge,
thinking of every parallel,
every part that is actually parallel on the cube. So
we take this line, we move it over, slightly tilt it
off here because the top, which is closest to us, more open,
has to be  has to be
larger than
the bottom because everything that's going away from
you is essentially converging into
a single point.
So here, move this up,
now we're here.
That's parallel
slightly converging, bring it back
here, here,
slightly out,
you connect them. See and even here I'm slightly
exaggerating the perspective.
In the future we will talk about exaggerating,
it mainly depends on you and understanding the structure of any
object allows you to play around with it when
you're  when you're working from imagination.
Or even
in part, even when you're working from observation. It gives
you more control and more
options.
Okay so I'm going to
erase  or actually what I'm going to do is
connect these diagonals
so that they don't
take our attention away from the cube itself.
You could also simply erase that line but
this accomplishes this.
There we go. So.
(drawing)
Okay.
And so I would now want to talk
a tiny bit. I'm going to move the cube,
perhaps we can get even more of an interesting angle if it can hold up
but it probably can. Okay so now the cube is standing up on
only a single point. But here
I'm not going to use the needles at all. So the
idea of all of the tools that you have is in order to
essentially build up intuition.
The idea is to do this as much by
eye as possible and for it to be as correct as possible
but it won't ever be as correct as
you hope it will and that's part of all of this.
So you keep these tools on you in order to
correct your own errors.
So here I'm going to
do this
entirely by eye but I'm actually combining these two approaches.
There are lines and angles that I'm actually,
I'm looking at, I'm taking them, I'm trying to
take them as they are and transfer them onto the paper and there are
other parts that I'm not observing at all and I'm adding from my
understanding of the cube. And
at a certain point you can,
at least for a moment, put your pencil up to the angle to get
a general idea of what that angle is. So like as you
as you become a bit more confident with observing
an angle and transferring it onto your piece of paper, you won't 
it won't be as  it won't be as important
to have a constant and move it over
and mark points along the line. All of this
will happen a lot quicker.
(drawing)
And if you are off
you have a number of options. If you're off and you feel that you're off
you can always go back and then
see the actual angle and then move it over and correct it
or you can just construct off of your error and
construct in a way that
makes the cube look like a cube. So
I made a slight correction over there, but this is all
that we have here. So 
(drawing)
and I don't have the room but that's fine.
(drawing)
And then you can apply
all of this to also establishing
the planes of
the table that the cube is on.
(drawing)
And in some ways it will be sort of impossible
to show that this cube is
at  is on a point without talking about 
without talking about
light and shadow. And we're going to arrive at that quite soon.
establishing an angle. Let's
start with the bottom edge over there, it is the
same place we
started before.
So you can
find that angle, all you have to do is,
as we practiced, you put one of the needles up as a
vertical constant, line up the other one with the edge, and
put it against your paper.
And you find
a point, hold it up, and see
that it lines up with another point
that you are making, make sure that
you're correct, even go back to the cube,
move back to your paper. There's no point in speeding this up.
And now you connect them.
And
continue all the way.
As far as you can. Okay
so the next point that we need
is our corner and then
the other edge that
we can perceive from our angle, going off of
the line that we made. So we're going to be
doing the same thing.
Hold it up,
lock it in place,
bring it up,
and get that angle. Good.
I'm gonna move it a little bit.
Alright.
And now transfer it to your paper and make sure the two lines
intersect around the
middle of your paper. So one of the points that you need
will actually be on the line that you already drew.
(checking)
Let's see if that's right. I might need to lower this a bit.
Pull it back, make sure you're on the right track.
And move
it across.
Okay. So now we've established
two
parts of our cube.
So what we need to do from here
is to get this
line that we see, the closest 
the closest intersection of the two planes that are in front of us,
the closest corner, and
you don't actually need to take this angle because we are going to 
I mean you can, but we're going to assume, and for the most part
it is going
to be because of how large it is,
a complete vertical line. So from this point that we have
all you need to do is
get a vertical.
(drawing)
And that's what we have
here. Okay. So we're going to continue.
So we now need to establish
how high
the corner up here
is. So in order to
do this
from the cube in front of us, only using the
techniques that we learned, we're going to take the angle
from that
corner to that one.
(checking)
And
then you place the line anywhere along 
(checking)
you place a line anywhere along that
will intersect these two lines that we already
have at any point along the line depending on how large you want the cube
to be on the page. So what I'm going to do
at this point is  I think
that this seems alright 
and then we find the proportion and then you establish 
I'm gonna go back over and make sure
it's right. Get them in my hand, move
it over and find that point on this line,
step back and make sure it's right, it seems okay, we're good.
So now we have
those two points that are going to be quite important.
And so, since we've decided that this line
is completely vertical, we're going to continue with that same logic
and our  all of our
vertical edges are going to be completely vertical
and just  you can take it all the way up.
Now in order to establish the amount that we have
on this side of the cube, we're going to do the
exact same thing that we did here, going
from that point to that point except now we're going from that point
to the opposite one along the surface
of that plane.
Let's see  so let's
let
get that in place.
That seems right. Do the same. Now
you already have a point there, hold it up
and mark where the other point
is, line them up, see if you're good.
I actually will probably move this over a little bit, see if you're
correct. Take your time and make sure you get everything exactly as it should be,
move it over,
I think that's good. And so now we have that.
At this point 
also this is a vertical. So
before we move on, it's now time to see
if our angles were correct by
double checking them with a proportional
measurement. So we're gonna take a
simple measurement that is horizontally across, a lot like
I showed you in one of the previous
assignments. So I'm curious to
see the amount that this part of the
cube will go into that part of the cube.
So when I see it in life, I'm locking it in,
the point of the needle is on the outer edge, my thumb
is on the closer edge and then I move
it over and then I do it again and I'm getting slightly under one and half
times. So let's see.
Do this here, that's what we need there, we'll move
this over, I'll even do a mark, and then we even
move it over and I'm
in exactly one and a half times. So let's
try this one more time to see if it's right. One.
Yeah, I think one and
a half is good. It seems to correspond
with our
angle that we took between these two points. And I will
double check again.
Okay.
It actually seems to be slightly
closer than the line I have, which will pull it to right under
one and a half times.
So I think we're good.
At this point if there are any lines that
you made that you're not happy with, that are potentially inaccurate, just 
you can clean them up with the eraser
and we're almost there.
So what we need
now is to get the angles of the
plane that is on top. So since we have 
since we have that point, all we need
to do is continue
with our angles.
Line them up, keep that there.
(checking(
Okay. We'll just hold it in place.
Make sure that line is on the point and find
where the other
is. Line them up,
look again,
a little higher, and connect
them.
(drawing)
Okay.
Let's keep at it. Now we're going from here
into there. And now we continue. Now we're
going from that point to the other side of our cube.
(drawing)
Line them up,
(drawing)
hold it in place, align it and make sure your point is there,
and connect them.
(drawing)
And clean out
what you don't want.
Okay.
Let's keep at it.
This is the time to go
over and get your lines a little cleaner.
You can reinforce them a little bit. It's looking
okay for now. Okay so the next
order of business is now to get these lines in the back.
Now there are a few ways.
You can obviously work along
this edge or you can establish that angle. I would do all of them
actually to make sure you get them right. So I'm gonna start with
that line.
(checking)
Let's find it there.
(checking)
Can find it right there, get the point
get another point, and
pull them all the way.
(drawing)
Okay. So
what we  so right now we're gonna get the other side
but you can get that angle also
because it'll tell you where that point is and then you'll be able to
connect them. The order here won't matter. I'll start from the outside though.
(checking)
Then I can transfer 
got that point there,
that point right there, look back, make sure
you're right,
and connect them.
(drawing)
And what we need
to do is to make sure we have it right. And now it's time to
get to double check our point.
Here I'm using the constant as a horizontal and
finding where this line is.
(drawing)
And that seems about right.
(checking)
And it
matches up to this 
this
point. So
now it's 
now that we have our lines you can remove
some of the extras.
(drawing)
So this was a cube
that you drew
entirely from observation.
(drawing)
We are going to be talking about light and shadow
so we're just gonna keep this in line
at the moment.
(drawing)
Alright. So there you have it. Next time
I'm going to explain a little bit about the construction of cubes.
So now that you've worked on a cube
entirely from observation, let me tell you a little bit about the
construction of a cube or
a rectangular prism. Now if you've
noticed that when you're taking each angle,
moving it over to your piece of paper, taking the angle after that,
this takes quite a long time. And so I believe that an understanding of
the structure of a cube can
speed things up and also allow you to work from imagination.
So I'm going to demonstrate a few concepts
on the piece of paper in front of me. And you can just follow along.
So in order to begin we need
to establish something that's known as
the horizon line.
Which we will just call  this is our horizon line,
you can call it H.
And so the basic
idea that I want to get across
is that
as an object moves farther away
from you it becomes smaller. And there are
a few things that happen to the angles within
that object  and especially a cube  that I
would like to cover. So along 
on our horizon line, let's put a point on one end
and a point on the other end. This will be
our vanishing point
one or
V one and this is our vanishing point number two or
V two. So
in order to structure a cube
on the plane that's here
all you need to do is extend a line from
your vanishing point and then extend another
line onto the surface.
And then do
the same thing here
and there. And so
this right here gives us a plane.
Then the next thing you should do is pull up
from
the corners of this plane, completely vertically.
You pick a point along that
line
and you move it back towards your vanishing point
here and
back towards your vanishing point over there and
all around.
(drawing)
Okay. So the idea
behind the horizon line is that it is exactly
where your eyes are, whether you are
standing up or
in a chair if you're looking in front of you, your eyes
are the horizon line. So
this'll be also your
eye line. So the
important thing to understand is that if you have
plane of an
object, in this case a cube or a
rectangular prism, and it lines up exactly
with your horizon line
it becomes a line.
(drawing)
And so the other idea is that if you have  and so if we
extend
what will become the closest edge to us 
we mark that as the end, we mark that as the end  all things below
your horizon line are going 
are going
up towards it. And all things above your horizon line
are going back
down towards it. And the same on the other side.
And then you lock it in.
And so obviously the closer
a plane gets to your
horizon line, the
closer it appears as a line. So the most important
aspect here is that
in a cube, or
on a plane
that has
parallel sides,
as they're moving away from you they are slightly converging.
So
you can apply all this without the perspective if you're
aware of that. So if you begin with
an edge and you get that line 
you get
that line, you don't actually
need to understand where  you don't need to
observe where any of the other lines are. Because all they are 
they're practically parallel and slightly
converging away from you.
So that line is actually shorter than that one.
And the same applies to the other side.
(drawing)
And so this also allows you
to tilt the cube in any way you want.
So as long as you keep  as long
as you keep as  as long as you are
aware of the point that is the closest point to you.
So for example.
(drawing)
And
(drawing)
so obviously
the plane that we have here
is going to be a larger one than the one that we have on the other
side. Or at least  and obviously in reality
they are identical but
on paper they have to appear smaller due to perspective. And so
the interesting  the only thing to keep
in mind is that if you're working on a piece of paper
and you establish a horizon line, you have a vanishing point on one end
and a vanishing point on the other end, the perspective will be drastically increased.
And our eye does not perceive 
perceive exactly
in the way that I showed it here. So if you were to
imagine our eyes, the vanishing points would probably be actually quite  would be
off the page. So the kind of
angles that I have on the
lines that we have here would not really
be observed on a cube that you can essentially  that is as
small as the one that we have in front of us. So
the idea is that you have to keep them
practically parallel with just a minor amount of
conversion. So
now that we've covered that, let's learn to combine
our understanding of the cube with the cube that we actually have in front of us.
Okay so now that we have
practiced working from observation and understand a little bit about
the construction of a cube or a rectangular prism,
it's time to combine all of this
into working from observation
utilizing certain observational principles that we
spoke about as well as our understanding of the forms in front of us.
So what you're going to need for this is your needles again.
Okay. So
let's begin by doing the exact
same thing that we did when we were working entirely from observation.
So I am going to
find the angle of that edge, the angle
of that edge.
So as we've already practiced, by now
what seems like a million times and that's great,
keep one as a vertical constant, lock in the other one
as you align it with the edge and move it over.
(checking)
And then you find it on your paper.
(drawing)
Alright.
Obviously if you're unsure you can always
(checking)
go back and compare it to the cube.
Now I'm going for the other side.
Do the same thing, I have my needles.
My more horizontal needle
aligning to that angle. I bring it over, I mark
a point, mark another point
right there, and now I extend that
out here. So now
we have these two extremely important
(drawing)
lines because we're going to construct
upwards using them. But before that,
we know that we can just take
the point right here and extend it upwards, really completely vertical.
(drawing)
And now it's important to
find the end of this edge
and so in order to find this end
we need to establish from point to point that line
and then from point to point that line. So
a diagonal across the surface of each
square plane.
Okay so let's see where that
takes us. So now I'm going to keep my constant
horizontal as opposed to vertical in order to make it
easier to get the angle right and I'm going to go
from those two points.
So here, oh it's
quite vertical. Take your time,
get them aligned, move them over,
and place them where you
want them.
(drawing)
Actually
here. And
then let's see if that's right. That's right.
Okay. So now we need to do the same thing on
this other side to find that diagonal between the two opposite
corners of the square plane.
There it is.
You already have that point right there,
move it over, keep it there,
and find that point right there, and connect them.
Okay. So that's all
that you'll be using the needles for because everything else
you can do from your understanding of the cube.
So now that we have that point and that point,
we know also that they just need to be extended up vertically.
Up vertically. And there you have it.
And then from the top corner,
in order to find the angle of that edge
all you need to do is take the link that you've already observed,
move it up and tilt it
off of that  this
point so that this edge right here appears smaller
than the edge that's closest to you. So in essence
this edge and that edge are going to be
converging towards a vanishing point.
(drawing)
Bring it down, you can hold it
and see where you are.
Alright. Now you're going to be doing the same thing here.
The interesting thing to
keep in mind is that if the surface of
a plane  if the surface of a plane on
the cube is more open, then you don't
actually need to  you don't need to exaggerate
the conversion of these two opposite 
these two opposite edges.
You have to exaggerate them only when the 
only when the surface of the cube is turning a bit further away
from you. So it's more in perspective
in essence. So here we go to there. You still don't want
them perfectly parallel but just a slight tilt
off of this point. And
that's that right there.
And always go from one edge to the
other and tilt them down a little bit, and there
you have this.
And you already have that point
so it's pretty much
it.
Okay. So that's
what we have right there. So now let's try something a tiny
bit more complicated. So I'm going to
put something under this cube in order to get at a slightly more
complex angle. I'm also, in order to make this a little more
interesting, we're going to turn it the other way. So now that's
what we have. Okay. So
here you'll  the only
difference that you have is that you can't count
on this edge being completely vertical. So you will also have to
sort of find the angle using
your needles and transfer it over. Everything else
will be established using our understanding
of the structure of a cube. So I will begin with
that edge because it's the only one
that is on the ground. So as we
said, think like an architect and begin
to structure from the ground up.
There it is. Find it there.
Now I'm going for
the other side of the cube, the one that's now
lifted off of the
ground.
And now I'm going to go for the nearest
edge to us, let's see what we have.
And we're good and move it over.
We're here, we're here, and that's all
that you need to observe. Now you can put
your needles away for a moment. Ah, I was wrong.
We still  we still actually need to
establish where this point is and where that point is. So now you're going to be 
you're going to be taking the angle of the diagonal of our
cube. Of a surface of our
cube.
If you're holding this you can always put your pencil behind your ear, it's one of the more convenient
places.
Here's this angle,
go from there  oops sorry. Remember to keep
your constant
constant. So it has to be a vertical when it's in front of you
taking a measurement and a vertical as you move it
over to the paper. That's the point of it. And then you go
from there to there, you already have that point,
whether this actually is rather close, corresponds quite well. So now
we have it. Okay.
So what we
can  what we can
do from here is now we know that that's the point that's closest to us.
So now we begin to work off of this edge,
thinking of every parallel,
every part that is actually parallel on the cube. So
we take this line, we move it over, slightly tilt it
off here because the top, which is closest to us, more open,
has to be  has to be
larger than
the bottom because everything that's going away from
you is essentially converging into
a single point.
So here, move this up,
now we're here.
That's parallel
slightly converging, bring it back
here, here,
slightly out,
you connect them. See and even here I'm slightly
exaggerating the perspective.
In the future we will talk about exaggerating,
it mainly depends on you and understanding the structure of any
object allows you to play around with it when
you're  when you're working from imagination.
Or even
in part, even when you're working from observation. It gives
you more control and more
options.
Okay so I'm going to
erase  or actually what I'm going to do is
connect these diagonals
so that they don't
take our attention away from the cube itself.
You could also simply erase that line but
this accomplishes this.
There we go. So.
(drawing)
Okay.
And so I would now want to talk
a tiny bit. I'm going to move the cube,
perhaps we can get even more of an interesting angle if it can hold up
but it probably can. Okay so now the cube is standing up on
only a single point. But here
I'm not going to use the needles at all. So the
idea of all of the tools that you have is in order to
essentially build up intuition.
The idea is to do this as much by
eye as possible and for it to be as correct as possible
but it won't ever be as correct as
you hope it will and that's part of all of this.
So you keep these tools on you in order to
correct your own errors.
So here I'm going to
do this
entirely by eye but I'm actually combining these two approaches.
There are lines and angles that I'm actually,
I'm looking at, I'm taking them, I'm trying to
take them as they are and transfer them onto the paper and there are
other parts that I'm not observing at all and I'm adding from my
understanding of the cube. And
at a certain point you can,
at least for a moment, put your pencil up to the angle to get
a general idea of what that angle is. So like as you
as you become a bit more confident with observing
an angle and transferring it onto your piece of paper, you won't 
it won't be as  it won't be as important
to have a constant and move it over
and mark points along the line. All of this
will happen a lot quicker.
(drawing)
And if you are off
you have a number of options. If you're off and you feel that you're off
you can always go back and then
see the actual angle and then move it over and correct it
or you can just construct off of your error and
construct in a way that
makes the cube look like a cube. So
I made a slight correction over there, but this is all
that we have here. So 
(drawing)
and I don't have the room but that's fine.
(drawing)
And then you can apply
all of this to also establishing
the planes of
the table that the cube is on.
(drawing)
And in some ways it will be sort of impossible
to show that this cube is
at  is on a point without talking about 
without talking about
light and shadow. And we're going to arrive at that quite soon.
AUTO SCROLL
Okay it's time to move on to other
geometric forms. Here we have a
number of cylindrical elements. And so in this
particular case, it is possible to observe
certain elements but there's
an important aspect here that needs to be constructed.
And so I think it'll be interesting for you to see how your understanding
of a cube will allow you to structure a cylindrical element
into it. So why don't we
get started on that and I will explain a few things.
Okay. So obviously
a major element here is the ellipse.
And the interesting thing about an ellipse is that
it is not an oval. An ellipse immediately implies
perspective.
And the easiest
way that I can give you an example of this
is by constructing an ellipse into
a square. So if we have a square  so an ellipse
is actually a circle in perspective. So in order
to achieve this we need to establish a
square.
We need to cross our
diagonals to find the 
to find
the center of our square
and
then establish where the
central point of each side is
to find where our circle is going to make a tangent
is going to make a
tangent
with the sides of our square. But as
you see, it's not as precise as
you might want it in order to explain what this
is. So
this is a somewhat circular form.
(drawing)
So
(drawing)
right now I will explain 
I will explain how to make this just slightly more
complex and get you a more accurate
circle within a square. And then we'll take that square,
put it into perspective, and construct a circle into it.
And you will see an ellipse. So
what you actually need to do  and I will have a ruler
for that purpose. So
here I will make a
square. So
let's see. This square
I'm making it 18 centimeters across or
around 7 inches.
18 centimeters.
We're good. That's there.
18 centimeters on this side.
(drawing)
And then here you just connect them.
I'll use the right angle here for extra
precision. The idea  it's helpful, it's precise, but it's not
fundamental. So okay, so now we have a square
Now we officially cross our diagonals.
(drawing)
Which allows us
to find the halfway points of our sides
like so.
Okay so this
is what we had initially. But as I said this isn't enough
because the only way that you can put a circle into this is by adhering
to the tangents on the corner 
on the halfway point of each side.
There isn't any way to find where our circle actually is
anywhere else. So what you need to do is actually
do the exact same thing you did for the whole square, however, you would
need to do it for each of these quadrants.
(drawing)
Okay.
And now you connect them.
(drawing)
Okay.
So now you've divided your square into 16 squares
into 16 parts. And so the next
step is you're going to take a corner and you're going to make
a line that hits the
at the point along the edge, one quadrant in across from
it.
(drawing)
Like so. And then you're going to do the same
from the other corner
to the opposite side. So  and so the important part here is to
go all around in this way.
(drawing)
And so this looks like
a lot
but I assure you, this will give you a much
better idea of how to inscribe a circle into
the square. So now we have a number of points available. So we still have the
center point of each side, which is a tangent, but then
we also have a certain intersection. So we have the intersection of this line that we made
with one of 
with a side of the square that's closest to it.
So this is the point.
And we have the exact same thing here.
(drawing)
So this
gives us a considerably larger amount of points into which
to construct a circle. Even as you follow
along, it's much easier to get the arcs that you need.
Now this is simply a circle,
a combination 
a continuous line of points equidistant from a central point.
(drawing)
Okay so let's now take our square with the circle inscribed
in it and place it in perspective. So
what we're going to need for this is our horizon line
(drawing)
And I will take it all the way.
(drawing)
And then
H if you remember. And then a
vanishing point on one side and a vanishing point
on the other. Vanishing point one, vanishing point two.
And from these two points we begin.
(drawing)
We simply take a line out of them into
the center of page. I'll go all the way so that we
have options. And then here
I wouldn't go all the way down to the bottom of
our page, mainly because you don't want the perspective
to look too exaggerated and for the
square that we have too open towards us. But also I wouldn't go too far to the
top, I would go somewhere in the middle of the page so we get
what looks like more or less convincing perspective.
It's still exaggerated from what our eye sees. And then
we go to the other point.
(drawing)
And then we lock in
a square.
Okay. So now we have
a square. I would kind of
clean off some of the other lines, especially 
and actually it works quite well because if you erase them you still have
some lines remaining to keep you
within the construct
however it'll
it won't be taking your attention away from the
what we have in front of us. So, according to what we
did with our square,
with our flat square, the next thing we have to do is cross
our diagonals. Alright.
To find the center of our square.
So here we go. Okay.
Now we have it. The next thing we do is we have to find
the center points of the sides of our square. For this
we use the center of the square and our vanishing points.
So line them up, don't go
all the way, and just connect them.
We're gonna be doing the same thing with the other side.
(drawing)
Excellent. So now you've divided your square into
quadrants and then next  the next step that
we have is
that we need to do the same thing with each one of our
smaller squares. Because if you remember
we need to divide our square up into
16 squares.
(drawing)
So the interesting thing here is that
actually if we continue, this is also
as you see it's going into perspective.
So there's actually a smaller square inside
of our larger square.
Now just to make sure, because there are errors that
can happen, I would line up
the center of each quadrant with the vanishing point, they do
line up, and just connect them. Do the same thing. If you do everything
properly they will line up, everything is within our
perspective construct here. They line up here. Do the same
to the opposite vanishing point.
And up here.
Okay so now we've successfully
divided our square into 16 squares.
Okay so the next step is to go from
each corner, across 
jumping across
the squares along the side. To be
precise, four of them, into the
this corner, like so.
And then from the other corner
to the other side of
the closest line.
Okay, that's good. Now we have to do this
everywhere. I'm going to switch to my smaller ruler, it's easier to hold
when working on this.
(drawing)
Alright. And now
the other way.
(drawing)
Good.
And now from here to here.
(drawing)
Alright. And the final
side.
(drawing)
If you have a mechanical pencil
it helps to get a straighter line but it's not
fundamental. Okay
so now we have everything that we need
to inscribe a circle within this square in perspective.
So here are our points. We have the midpoint
of the sides
and then we have the point where
one of these lines is crossing
one of the quarter lines.
So we're there. We're here,
that's a point, that's a point, that's a
point. Here this is getting a little close so you're just gonna
have to get as close as you can.
We're there, we're there, and then
we're here. And so this is enough
to work an ellipse into this.
So
now just take your time, you have enough
points along the way to guide you.
(drawing)
We take from the top,
round it off.
(drawing)
Let's see where we are right there.
Inscribing the ellipse actually might take a little bit
of practice. Okay. So
then make certain corrections, a couple of them if you need to.
(drawing)
And so now we have
a constructed ellipse within
a square. So  and what 
obviously I'm not asking you to go through
this entire operation every time you see an
ellipse in front of you. However, it's helpful
because it can give you an idea of
an ellipse so that you can  the point of which is to
recall it as much as possible in your head so when you're
simply making it by hand, you have
an idea of where corrections need to be made.
So the interesting thing that's happening here is that
it's very clear where an ellipse is
not an oval. So an oval
is 
(drawing)
if you cut an oval
or rather you can cut an oval
into equal parts. The top is the same
as the bottom, the side is the same as the opposite
side. Now in an ellipse it's important
to keep in mind that you can't cut the ellipse in half and have them be equal.
The side that is closer to you is
actually going to be a bit larger, but it's
also going to be  it's going to be rounded off
a bit more and the top side
is going to be smaller and flatter. Now obviously
this applies to everything below your horizon line.
Everything above your horizon line will be all of this but inverted.
So now that you understand the mechanics of an ellipse, let's try practicing
them by hand. So
the idea is not to get a perfect ellipse straightaway
and there are a few ways that you can approach this.So
on a clean sheet of 
a clean sheet of paper you're going to 
you're going to make an ellipse. And then you're going to
look at it and ask yourself a few questions. So
the first one is if you cut this
ellipse in half,
is the part that's underneath larger than the part
that's on top? And I'm not so sure
in my case. So I'm going to slightly
move down the top
and slightly enlarge the bottom.
And so the question
that follows is  or that's sort of asked simultaneously  is
is the
bottom side of the ellipse, the closer side, is it
larger than the top?
And so at that point you're going to make certain
corrections. Now the interesting this is that of course
when you're working
on anything, you actually could probably get away with
an oval. But I feel like this is helpful to understand.
So I would
just keep practicing these.
In some cases you analyze them, in other cases you just move on.
So
there is a tendency at times to get
the ellipse by just moving around the ellipse and
it sort of, it cleans up the edges. But I tend to think
that a smaller amount of lines is always
preferable to a large amount of lines. So be as clean as possible.
If you need to do a few of course you
can and even if you need to do a lot of course you can.
So I'm going to go back onto that one
and smooth it out a little bit  which is also
a way to approach these. After you've analyzed it,
after you have understood what's off about it, then you
can go over it
a few times. But the point is to begin to,
not so much to put a proper ellipse on paper, but to understand when an ellipse is wrong.
And so you can practice
a few that are open to a certain
degree, then you can practice a few others that are actually
closer to your
horizon. You can practice them in all different kinds of directions
and these had to be the easiest ones.
And then you can attempt ellipses that are more open
that are coming closer to just being
a circle. And so I
highly recommend that
you put as many on paper
as you need in order to get comfortable with it.
geometric forms. Here we have a
number of cylindrical elements. And so in this
particular case, it is possible to observe
certain elements but there's
an important aspect here that needs to be constructed.
And so I think it'll be interesting for you to see how your understanding
of a cube will allow you to structure a cylindrical element
into it. So why don't we
get started on that and I will explain a few things.
Okay. So obviously
a major element here is the ellipse.
And the interesting thing about an ellipse is that
it is not an oval. An ellipse immediately implies
perspective.
And the easiest
way that I can give you an example of this
is by constructing an ellipse into
a square. So if we have a square  so an ellipse
is actually a circle in perspective. So in order
to achieve this we need to establish a
square.
We need to cross our
diagonals to find the 
to find
the center of our square
and
then establish where the
central point of each side is
to find where our circle is going to make a tangent
is going to make a
tangent
with the sides of our square. But as
you see, it's not as precise as
you might want it in order to explain what this
is. So
this is a somewhat circular form.
(drawing)
So
(drawing)
right now I will explain 
I will explain how to make this just slightly more
complex and get you a more accurate
circle within a square. And then we'll take that square,
put it into perspective, and construct a circle into it.
And you will see an ellipse. So
what you actually need to do  and I will have a ruler
for that purpose. So
here I will make a
square. So
let's see. This square
I'm making it 18 centimeters across or
around 7 inches.
18 centimeters.
We're good. That's there.
18 centimeters on this side.
(drawing)
And then here you just connect them.
I'll use the right angle here for extra
precision. The idea  it's helpful, it's precise, but it's not
fundamental. So okay, so now we have a square
Now we officially cross our diagonals.
(drawing)
Which allows us
to find the halfway points of our sides
like so.
Okay so this
is what we had initially. But as I said this isn't enough
because the only way that you can put a circle into this is by adhering
to the tangents on the corner 
on the halfway point of each side.
There isn't any way to find where our circle actually is
anywhere else. So what you need to do is actually
do the exact same thing you did for the whole square, however, you would
need to do it for each of these quadrants.
(drawing)
Okay.
And now you connect them.
(drawing)
Okay.
So now you've divided your square into 16 squares
into 16 parts. And so the next
step is you're going to take a corner and you're going to make
a line that hits the
at the point along the edge, one quadrant in across from
it.
(drawing)
Like so. And then you're going to do the same
from the other corner
to the opposite side. So  and so the important part here is to
go all around in this way.
(drawing)
And so this looks like
a lot
but I assure you, this will give you a much
better idea of how to inscribe a circle into
the square. So now we have a number of points available. So we still have the
center point of each side, which is a tangent, but then
we also have a certain intersection. So we have the intersection of this line that we made
with one of 
with a side of the square that's closest to it.
So this is the point.
And we have the exact same thing here.
(drawing)
So this
gives us a considerably larger amount of points into which
to construct a circle. Even as you follow
along, it's much easier to get the arcs that you need.
Now this is simply a circle,
a combination 
a continuous line of points equidistant from a central point.
(drawing)
Okay so let's now take our square with the circle inscribed
in it and place it in perspective. So
what we're going to need for this is our horizon line
(drawing)
And I will take it all the way.
(drawing)
And then
H if you remember. And then a
vanishing point on one side and a vanishing point
on the other. Vanishing point one, vanishing point two.
And from these two points we begin.
(drawing)
We simply take a line out of them into
the center of page. I'll go all the way so that we
have options. And then here
I wouldn't go all the way down to the bottom of
our page, mainly because you don't want the perspective
to look too exaggerated and for the
square that we have too open towards us. But also I wouldn't go too far to the
top, I would go somewhere in the middle of the page so we get
what looks like more or less convincing perspective.
It's still exaggerated from what our eye sees. And then
we go to the other point.
(drawing)
And then we lock in
a square.
Okay. So now we have
a square. I would kind of
clean off some of the other lines, especially 
and actually it works quite well because if you erase them you still have
some lines remaining to keep you
within the construct
however it'll
it won't be taking your attention away from the
what we have in front of us. So, according to what we
did with our square,
with our flat square, the next thing we have to do is cross
our diagonals. Alright.
To find the center of our square.
So here we go. Okay.
Now we have it. The next thing we do is we have to find
the center points of the sides of our square. For this
we use the center of the square and our vanishing points.
So line them up, don't go
all the way, and just connect them.
We're gonna be doing the same thing with the other side.
(drawing)
Excellent. So now you've divided your square into
quadrants and then next  the next step that
we have is
that we need to do the same thing with each one of our
smaller squares. Because if you remember
we need to divide our square up into
16 squares.
(drawing)
So the interesting thing here is that
actually if we continue, this is also
as you see it's going into perspective.
So there's actually a smaller square inside
of our larger square.
Now just to make sure, because there are errors that
can happen, I would line up
the center of each quadrant with the vanishing point, they do
line up, and just connect them. Do the same thing. If you do everything
properly they will line up, everything is within our
perspective construct here. They line up here. Do the same
to the opposite vanishing point.
And up here.
Okay so now we've successfully
divided our square into 16 squares.
Okay so the next step is to go from
each corner, across 
jumping across
the squares along the side. To be
precise, four of them, into the
this corner, like so.
And then from the other corner
to the other side of
the closest line.
Okay, that's good. Now we have to do this
everywhere. I'm going to switch to my smaller ruler, it's easier to hold
when working on this.
(drawing)
Alright. And now
the other way.
(drawing)
Good.
And now from here to here.
(drawing)
Alright. And the final
side.
(drawing)
If you have a mechanical pencil
it helps to get a straighter line but it's not
fundamental. Okay
so now we have everything that we need
to inscribe a circle within this square in perspective.
So here are our points. We have the midpoint
of the sides
and then we have the point where
one of these lines is crossing
one of the quarter lines.
So we're there. We're here,
that's a point, that's a point, that's a
point. Here this is getting a little close so you're just gonna
have to get as close as you can.
We're there, we're there, and then
we're here. And so this is enough
to work an ellipse into this.
So
now just take your time, you have enough
points along the way to guide you.
(drawing)
We take from the top,
round it off.
(drawing)
Let's see where we are right there.
Inscribing the ellipse actually might take a little bit
of practice. Okay. So
then make certain corrections, a couple of them if you need to.
(drawing)
And so now we have
a constructed ellipse within
a square. So  and what 
obviously I'm not asking you to go through
this entire operation every time you see an
ellipse in front of you. However, it's helpful
because it can give you an idea of
an ellipse so that you can  the point of which is to
recall it as much as possible in your head so when you're
simply making it by hand, you have
an idea of where corrections need to be made.
So the interesting thing that's happening here is that
it's very clear where an ellipse is
not an oval. So an oval
is 
(drawing)
if you cut an oval
or rather you can cut an oval
into equal parts. The top is the same
as the bottom, the side is the same as the opposite
side. Now in an ellipse it's important
to keep in mind that you can't cut the ellipse in half and have them be equal.
The side that is closer to you is
actually going to be a bit larger, but it's
also going to be  it's going to be rounded off
a bit more and the top side
is going to be smaller and flatter. Now obviously
this applies to everything below your horizon line.
Everything above your horizon line will be all of this but inverted.
So now that you understand the mechanics of an ellipse, let's try practicing
them by hand. So
the idea is not to get a perfect ellipse straightaway
and there are a few ways that you can approach this.So
on a clean sheet of 
a clean sheet of paper you're going to 
you're going to make an ellipse. And then you're going to
look at it and ask yourself a few questions. So
the first one is if you cut this
ellipse in half,
is the part that's underneath larger than the part
that's on top? And I'm not so sure
in my case. So I'm going to slightly
move down the top
and slightly enlarge the bottom.
And so the question
that follows is  or that's sort of asked simultaneously  is
is the
bottom side of the ellipse, the closer side, is it
larger than the top?
And so at that point you're going to make certain
corrections. Now the interesting this is that of course
when you're working
on anything, you actually could probably get away with
an oval. But I feel like this is helpful to understand.
So I would
just keep practicing these.
In some cases you analyze them, in other cases you just move on.
So
there is a tendency at times to get
the ellipse by just moving around the ellipse and
it sort of, it cleans up the edges. But I tend to think
that a smaller amount of lines is always
preferable to a large amount of lines. So be as clean as possible.
If you need to do a few of course you
can and even if you need to do a lot of course you can.
So I'm going to go back onto that one
and smooth it out a little bit  which is also
a way to approach these. After you've analyzed it,
after you have understood what's off about it, then you
can go over it
a few times. But the point is to begin to,
not so much to put a proper ellipse on paper, but to understand when an ellipse is wrong.
And so you can practice
a few that are open to a certain
degree, then you can practice a few others that are actually
closer to your
horizon. You can practice them in all different kinds of directions
and these had to be the easiest ones.
And then you can attempt ellipses that are more open
that are coming closer to just being
a circle. And so I
highly recommend that
you put as many on paper
as you need in order to get comfortable with it.
AUTO SCROLL
So in front of us we have a vertical
cylinder. So what I'm interested
in finding out before we even begin
is what the proportions are here. So I'm
interested in seeing the amount of times the width of it will fit
into its height. All the way from
the bottom point here to the end,
the upper
most point that I can see at the other end of the ellipse.
So
all we need to do, as we've practiced, is to align the point of our
needle, or your pencil, with one side of the
cylinder, your hand locks in the other. And then
move your needle and place it vertically
so that your hand is now at the very bottom
and then move it up
and the proportions, as I seem to get
all this relatively simple proportions,
is that the width of
the cylinder fits into its full height one and a half
times. So
all you need to do is to establish
your
width and then
move it up a single time and then you
can find the halfway point on that line.
Move it up and move it up again. Okay.
So now we have a two
dimensional, rectangular shape that
will establish the entirety of the proportions of our
cylinder. The
next order of business
is to take the proportions of the full ellipse and see how many
times the ellipse, in terms of
the height of the ellipse, will fit into our 
the rest of the cylinder.
So line it up so that the point
of the needle is at the farther part of the cylinder and your hand is at the
closest part of the rim.
Then move it down at one  it seems like a lot from this angle  two
three, four,
five, six, and almost exactly 
actually it's exactly seven times. So
in total eight.
In my case, eight is fairly easy.
So if you find the half way point  you can either approximate it or, as we
know, you can find the halfway point by crossing your
diagonals and then we can find this again
so this is now divided into quarters. And now to find the eighth
you can divide the final top quarter into 
(drawing)
into an eighth. So this
essentially is all that we require in terms
of observing what our cylinder is. So
it also gives us an area into
which we can  we can place
our ellipse.
So into here. So  before
we begin, we have to find the center
line. And I will be talking about the center line a lot during
the entirety of the course, all the way to the end, because it's possibly
one of the most important  most important
elements of the
constructive analysis. And so
in our case it functions as a center line
in order  it functions as a vertical axis,
as well as a way to keep the left side symmetrical
to the right. And so and here
we can just begin to
(drawing)
place the ellipse. Now if you were really interested in
a proper construction, you can
establish what the perspective is on top, make it into a square
in perspective and place the
entire ellipse according to
the formula that I explained in the previous assighnmemts.
And so this is what we have
so far. You can cut it
in half horizontally and make the
corrections according to what we discussed so that the
upper part, the one farther away is a little flatter.
You don't actually need to look at what's in front of you any more.
This is all entirely 
this is all now entirely
from your imagination. Or from your understanding rather.
Which is possibly the same thing. So
before we connect anything, what we need
to put in this area is
the other ellipse. The ellipse that is holding
everything up. The base of it,
of this entire structure. So what we need to do for that since we
can't actually  we can't actually
see what's happening, you have to imagine
what's going on and put it on paper as though the cylinder
is made of glass.
And so the only thing to think about is that because our horizon
is higher that necessarily,
will mean that the
top cylinder will be not as
open as the one underneath it. Because if you recall, as
a plane comes closer to our horizon
line, it flattens and flattens and flattens out until it eventually becomes practically a line.
And so here
you can even
take its height, find it here, and enlarge it slightly.
And 
(drawing)
And see
where it is, all the way
into it. And at that point what you need to do is connect
the two cylinders, we find the sides
of them.
(drawing)
And
there
you have it.
Okay so now, let's take the cylinder and put it on
its side. Before
we begin, I recommend taking a pencil
and placing it right up against
the ellipse that you see, the top of the cylinder, its opening.
And what this establishes is it gives us
the
plane that is perpendicular
to the
central axis of our cylinder
on top of the  on top of our
pedestal here. So the angles
that you need to take right away are
the edge, the
bottom edge and the line of
the  line of
the pencil. So
(drawing)
let's give this a go.
You establish  keep the vertical constant, align the edge
and there you have it.
There you have that line.
(drawing)
And now I'm going
to get the line that is
given to me by the pencil.
(drawing)
Make sure it's correct, it's a little straighter
and a little lower.
(drawing)
Alright, so
in essence that's kind of all you need. At this point
you can actually remove the pencil in order
to see the entirety of the cylinder. So
what we need to establish at
the point that we have here is now we need to 
we need to
construct a square
for the opening, for the top of our cylinder.
And you are going to be putting all of the techniques
that you've learned into use here. So
here we have a line on top
that's slightly converging with the line on
the bottom.
Here we're going to cross our diagonals.
(drawing)
And
the 
yeah so we're going to cross our
diagonals and then we're going to
establish the tangents that we have
with the ellipse.
And then we're going to place the ellipse. Now I'm not 
now if you want you
can do the entirety of the construction that I explained
earlier or you
(drawing)
can kind of eyeball it.
(drawing)
Alright so
there you have it. So what you're interested in
putting in now is actually you might need to take
another angle. The angle that I
would take would be the outermost points of the ellipse. So where the side
of the cylinder is coming into contact
with the ellipse. So that entirety of it 
the angle of its 
essentially the angle of the 
the angle of the ellipse. And it's not so much that you are
trying to establish this line but you're actually
kind of correcting what you have.
And the key is
to see if the outermost points that you
placed are correct.
(drawing)
And once you have this
line you can make some slight corrections.
For example, making the part of the ellipse that's
a little  that's away from you slightly flatter.
(drawing)
and rounding off the part that is 
that is
closer to you.
Okay so once you have
the points here  so the angle
that comes after this is the angle of the side
of the cylinder.
You take it and you transfer it.
(drawing)
And then
you find it on the other side. All you do is get it to be
parallel and slightly converging.
And now it's important to establish
where our  what the proportions are. So
the question that you're asking is
the  so along our 
along our central axis
how many times does the
ellipse fit into the
cylinder as a whole? So you're going to
lock in one side with your hand, put the point of the needle up to
the closest part of the
ellipse and move that up 
move it up. And I get one and half times.
So from here that is
a single time, cut that
in half and add it on.
And so you're almost 
you almost have it. The only thing you have to recall is that the
ellipse at the farther end is more
open. And you can 
if you want to explain it,
then it's  if you were to
construct a square here
and
make sure these lines are parallel but then slightly converging.
(drawing)
The square that's
in the back is slightly more open than the one
in front of you.
And so  but you already have everything
that you need. You have
the point that is a tangent
to our square, you can take that all the way.
Ah well, you can take that
all the way, that would actually
be along that line. And
you can.
(drawing)
Okay.
Okay, so now that you understand
a little bit about cylindrical forms
and ellipses, you can start
applying this entirely from imagination.
So
all you need to do to construct a cylinder from imagination
is begin
by constructing a box. A cube
or a rectangular prism.
The important thing is to have 
to think of it as if it were
completely transparent.
Then obviously
since we already know how to construct an ellipse, all we need to do
is establish where the
center of
each of these sides is
(drawing)
and use
them  use them
to insert our axis.
And once all that's in place,
you can begin to place your ellipse,.
Now keep in mind the ellipse does not need to be perfect.
It just needs to be approximate enough so that you are able
to make the corrections that you need.
The important thing here is to get the ellipse at the front
and
then  as well as in the back 
and then once it's in
place you can make sure to make a line
that tells you the
parts of the ellipse that you're actually able to
see with your eye, or you would have been to see them with your eye if
you were working from
observation. Okay so after having
practiced a few of these, it's time to attempt a few
that are at angles that are a bit harder to observe.
It's possible but it would require
it would require you to hang
an ellipse or to prop it up in ways that aren't particularly
convenient.
So here
I go and I'm constructing a rectangular
prism. In this case,
the side that I have here is actually not a square, it's a rectangle.
So  but the principle applies. So this is not
even a proper ellipse any more, however it is 
(drawing)
all of the same rules apply to
its construction. Alright
there we have it. And then begin 
then begin to place your
ellipses.
And connect them at their outer most points.
You could
also not
open up the top of the
you could not open up the ellipses much and keep it a bit
and keep it smaller and keep it more
compressed. And you can even
start by constructing an ellipse and
then structuring a
a cylindrical form out of it  and see here I don't
even complete it. Okay!
So now
that we've practiced that, let's talk a little bit about
forms that we haven't actually
encountered. S
what if we think about a cone?
Ask yourself about the construction of it. What is a cone?
So why don't we begin
with a
square again, in perspective. And
once again we will
cross our diagonals to find the center of the square.
We will  and then we will cut the square
into equal
quadrants. Okay. It's
at that point it's time to place our ellipse.
(drawing)
And
up until the point here
we're quite comfortable. But the step after this
is also a simple one. So
extend your axis
up, establish a point
anywhere along this axis and then just connect it to the outer edges
of your ellipse. So
it could be here, it could be
here. And that is the construction of a cone.
Okay. So
it is a little bit harder to tilt a cone.
Mainly because the angle
of  the angle of the 
so if we have
a cone here and a
cylinder here, it is easier to
establish the angle of the ellipse to the side of the cylinder
because it is always at a right angle. Here however
we run into a few problems.
So the vertical axis is going to be
important here. So begin,
if you're going to be tilting,
begin with the angle
of the ellipse.
(drawing)
And afterwards
extend
your vertical axis and then
connect the outer edges of the ellipse
to a point along that axis.
And so
that's all for a cone.
So the interesting thing about a cone is that
you can also
cut a cone at any point
using your understanding of an ellipse. So
if we establish a cone 
(drawing)
and here I'm working the other way a little bit
but
that's also a possibility
once you get a bit more comfortable with this.
At that point you can
you can construct 
construct
a square at any point of the cone.
And do the exact 
and follow
with the procedure. And now you have a cone that is
truncated.
And
the final thing that I'm going to talk about
is that you not only are
able to truncate a cone
with an ellipse that is at the same angle as the base of the
cone, but you can essentially cut it at any angle.
So why don't we give that a shot.
So let's
begin with the base of our cone,
another square.
(drawing)
Here we have our ellipse.
Move your vertical axis up, make the connection.
But then here 
there we go 
but then here
why don't we truncate the cone
at that angle.
And then, once you establish the
rectangle at the proper angle that you need, the same
procedure applies.
(drawing)
And there it is.
So I
would practice this as much as possible because
we're going to be using our understanding of
these
structures when we're analyzing organic
objects from observation.
cylinder. So what I'm interested
in finding out before we even begin
is what the proportions are here. So I'm
interested in seeing the amount of times the width of it will fit
into its height. All the way from
the bottom point here to the end,
the upper
most point that I can see at the other end of the ellipse.
So
all we need to do, as we've practiced, is to align the point of our
needle, or your pencil, with one side of the
cylinder, your hand locks in the other. And then
move your needle and place it vertically
so that your hand is now at the very bottom
and then move it up
and the proportions, as I seem to get
all this relatively simple proportions,
is that the width of
the cylinder fits into its full height one and a half
times. So
all you need to do is to establish
your
width and then
move it up a single time and then you
can find the halfway point on that line.
Move it up and move it up again. Okay.
So now we have a two
dimensional, rectangular shape that
will establish the entirety of the proportions of our
cylinder. The
next order of business
is to take the proportions of the full ellipse and see how many
times the ellipse, in terms of
the height of the ellipse, will fit into our 
the rest of the cylinder.
So line it up so that the point
of the needle is at the farther part of the cylinder and your hand is at the
closest part of the rim.
Then move it down at one  it seems like a lot from this angle  two
three, four,
five, six, and almost exactly 
actually it's exactly seven times. So
in total eight.
In my case, eight is fairly easy.
So if you find the half way point  you can either approximate it or, as we
know, you can find the halfway point by crossing your
diagonals and then we can find this again
so this is now divided into quarters. And now to find the eighth
you can divide the final top quarter into 
(drawing)
into an eighth. So this
essentially is all that we require in terms
of observing what our cylinder is. So
it also gives us an area into
which we can  we can place
our ellipse.
So into here. So  before
we begin, we have to find the center
line. And I will be talking about the center line a lot during
the entirety of the course, all the way to the end, because it's possibly
one of the most important  most important
elements of the
constructive analysis. And so
in our case it functions as a center line
in order  it functions as a vertical axis,
as well as a way to keep the left side symmetrical
to the right. And so and here
we can just begin to
(drawing)
place the ellipse. Now if you were really interested in
a proper construction, you can
establish what the perspective is on top, make it into a square
in perspective and place the
entire ellipse according to
the formula that I explained in the previous assighnmemts.
And so this is what we have
so far. You can cut it
in half horizontally and make the
corrections according to what we discussed so that the
upper part, the one farther away is a little flatter.
You don't actually need to look at what's in front of you any more.
This is all entirely 
this is all now entirely
from your imagination. Or from your understanding rather.
Which is possibly the same thing. So
before we connect anything, what we need
to put in this area is
the other ellipse. The ellipse that is holding
everything up. The base of it,
of this entire structure. So what we need to do for that since we
can't actually  we can't actually
see what's happening, you have to imagine
what's going on and put it on paper as though the cylinder
is made of glass.
And so the only thing to think about is that because our horizon
is higher that necessarily,
will mean that the
top cylinder will be not as
open as the one underneath it. Because if you recall, as
a plane comes closer to our horizon
line, it flattens and flattens and flattens out until it eventually becomes practically a line.
And so here
you can even
take its height, find it here, and enlarge it slightly.
And 
(drawing)
And see
where it is, all the way
into it. And at that point what you need to do is connect
the two cylinders, we find the sides
of them.
(drawing)
And
there
you have it.
Okay so now, let's take the cylinder and put it on
its side. Before
we begin, I recommend taking a pencil
and placing it right up against
the ellipse that you see, the top of the cylinder, its opening.
And what this establishes is it gives us
the
plane that is perpendicular
to the
central axis of our cylinder
on top of the  on top of our
pedestal here. So the angles
that you need to take right away are
the edge, the
bottom edge and the line of
the  line of
the pencil. So
(drawing)
let's give this a go.
You establish  keep the vertical constant, align the edge
and there you have it.
There you have that line.
(drawing)
And now I'm going
to get the line that is
given to me by the pencil.
(drawing)
Make sure it's correct, it's a little straighter
and a little lower.
(drawing)
Alright, so
in essence that's kind of all you need. At this point
you can actually remove the pencil in order
to see the entirety of the cylinder. So
what we need to establish at
the point that we have here is now we need to 
we need to
construct a square
for the opening, for the top of our cylinder.
And you are going to be putting all of the techniques
that you've learned into use here. So
here we have a line on top
that's slightly converging with the line on
the bottom.
Here we're going to cross our diagonals.
(drawing)
And
the 
yeah so we're going to cross our
diagonals and then we're going to
establish the tangents that we have
with the ellipse.
And then we're going to place the ellipse. Now I'm not 
now if you want you
can do the entirety of the construction that I explained
earlier or you
(drawing)
can kind of eyeball it.
(drawing)
Alright so
there you have it. So what you're interested in
putting in now is actually you might need to take
another angle. The angle that I
would take would be the outermost points of the ellipse. So where the side
of the cylinder is coming into contact
with the ellipse. So that entirety of it 
the angle of its 
essentially the angle of the 
the angle of the ellipse. And it's not so much that you are
trying to establish this line but you're actually
kind of correcting what you have.
And the key is
to see if the outermost points that you
placed are correct.
(drawing)
And once you have this
line you can make some slight corrections.
For example, making the part of the ellipse that's
a little  that's away from you slightly flatter.
(drawing)
and rounding off the part that is 
that is
closer to you.
Okay so once you have
the points here  so the angle
that comes after this is the angle of the side
of the cylinder.
You take it and you transfer it.
(drawing)
And then
you find it on the other side. All you do is get it to be
parallel and slightly converging.
And now it's important to establish
where our  what the proportions are. So
the question that you're asking is
the  so along our 
along our central axis
how many times does the
ellipse fit into the
cylinder as a whole? So you're going to
lock in one side with your hand, put the point of the needle up to
the closest part of the
ellipse and move that up 
move it up. And I get one and half times.
So from here that is
a single time, cut that
in half and add it on.
And so you're almost 
you almost have it. The only thing you have to recall is that the
ellipse at the farther end is more
open. And you can 
if you want to explain it,
then it's  if you were to
construct a square here
and
make sure these lines are parallel but then slightly converging.
(drawing)
The square that's
in the back is slightly more open than the one
in front of you.
And so  but you already have everything
that you need. You have
the point that is a tangent
to our square, you can take that all the way.
Ah well, you can take that
all the way, that would actually
be along that line. And
you can.
(drawing)
Okay.
Okay, so now that you understand
a little bit about cylindrical forms
and ellipses, you can start
applying this entirely from imagination.
So
all you need to do to construct a cylinder from imagination
is begin
by constructing a box. A cube
or a rectangular prism.
The important thing is to have 
to think of it as if it were
completely transparent.
Then obviously
since we already know how to construct an ellipse, all we need to do
is establish where the
center of
each of these sides is
(drawing)
and use
them  use them
to insert our axis.
And once all that's in place,
you can begin to place your ellipse,.
Now keep in mind the ellipse does not need to be perfect.
It just needs to be approximate enough so that you are able
to make the corrections that you need.
The important thing here is to get the ellipse at the front
and
then  as well as in the back 
and then once it's in
place you can make sure to make a line
that tells you the
parts of the ellipse that you're actually able to
see with your eye, or you would have been to see them with your eye if
you were working from
observation. Okay so after having
practiced a few of these, it's time to attempt a few
that are at angles that are a bit harder to observe.
It's possible but it would require
it would require you to hang
an ellipse or to prop it up in ways that aren't particularly
convenient.
So here
I go and I'm constructing a rectangular
prism. In this case,
the side that I have here is actually not a square, it's a rectangle.
So  but the principle applies. So this is not
even a proper ellipse any more, however it is 
(drawing)
all of the same rules apply to
its construction. Alright
there we have it. And then begin 
then begin to place your
ellipses.
And connect them at their outer most points.
You could
also not
open up the top of the
you could not open up the ellipses much and keep it a bit
and keep it smaller and keep it more
compressed. And you can even
start by constructing an ellipse and
then structuring a
a cylindrical form out of it  and see here I don't
even complete it. Okay!
So now
that we've practiced that, let's talk a little bit about
forms that we haven't actually
encountered. S
what if we think about a cone?
Ask yourself about the construction of it. What is a cone?
So why don't we begin
with a
square again, in perspective. And
once again we will
cross our diagonals to find the center of the square.
We will  and then we will cut the square
into equal
quadrants. Okay. It's
at that point it's time to place our ellipse.
(drawing)
And
up until the point here
we're quite comfortable. But the step after this
is also a simple one. So
extend your axis
up, establish a point
anywhere along this axis and then just connect it to the outer edges
of your ellipse. So
it could be here, it could be
here. And that is the construction of a cone.
Okay. So
it is a little bit harder to tilt a cone.
Mainly because the angle
of  the angle of the 
so if we have
a cone here and a
cylinder here, it is easier to
establish the angle of the ellipse to the side of the cylinder
because it is always at a right angle. Here however
we run into a few problems.
So the vertical axis is going to be
important here. So begin,
if you're going to be tilting,
begin with the angle
of the ellipse.
(drawing)
And afterwards
extend
your vertical axis and then
connect the outer edges of the ellipse
to a point along that axis.
And so
that's all for a cone.
So the interesting thing about a cone is that
you can also
cut a cone at any point
using your understanding of an ellipse. So
if we establish a cone 
(drawing)
and here I'm working the other way a little bit
but
that's also a possibility
once you get a bit more comfortable with this.
At that point you can
you can construct 
construct
a square at any point of the cone.
And do the exact 
and follow
with the procedure. And now you have a cone that is
truncated.
And
the final thing that I'm going to talk about
is that you not only are
able to truncate a cone
with an ellipse that is at the same angle as the base of the
cone, but you can essentially cut it at any angle.
So why don't we give that a shot.
So let's
begin with the base of our cone,
another square.
(drawing)
Here we have our ellipse.
Move your vertical axis up, make the connection.
But then here 
there we go 
but then here
why don't we truncate the cone
at that angle.
And then, once you establish the
rectangle at the proper angle that you need, the same
procedure applies.
(drawing)
And there it is.
So I
would practice this as much as possible because
we're going to be using our understanding of
these
structures when we're analyzing organic
objects from observation.
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1. Drawing Forms Overview1m 7sNow playing...
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2. Constructing the Cube42m 14s

3. Constructing a Ellipse24m 11s

4. Constructing Cylinders and Cones29m 55s

5. Assignment24s
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