Mtg 16/26: Thu-06-Mar-2025

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Shapes

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So we're going to talk about shapes today
and fractals, I guess So,
oh, okay. So who's familiar with fractals?
Okay, what are some properties of Practice? I
there's one I'm a property that is important for our discussion
today.
Okay, so?
Now this is the point where I see where there drew the right
Number of blanks. T I
P, no, s, thoughts, CS,
I said, M Right, yeah, yeah.
L, M, say, f Oh, wait,
self similarity, do
I don't have any prizes, sorry, maybe I'll have confidence right
now. Anyway, I
so the idea is they wouldn't heard of the SIP ski gasket, or
SIP in ski triangle. So Benoit Mandelbrot called the Sierpinski
gasket as a joke, because a gasket is supposed to seal
things, and the Sierpinski gasket is full of holes.
That's my interpretation, he told he said that it was a joke,
but he didn't I filled into the details or the rationale or the
reason, certainly the way we constructed is
contained. Take an E collateral triangle, and then we take the
whole triangle, and we make three smaller copies, scale by a
half and
and so we translate towards The vertices. So that's the first
stage, and then the
And then, 012,
that's a different shape And
so you can see the relationship here. So the third stage, we're
taking three copies of the second stage and putting them in
into the spots here.
So self similarity is a property of nature of many natural
objects. So when I generate fractals on my computer, after a
few minutes out, or maybe a few seconds out, comes up and puts a
leaf on the image and it says, Well, this is a maple or comes a
few different clients. Does that make sense in terms of
construction? In
fact, it's,
it's a kind of patterns and that repeat itself in a small and
small Yeah? So we can, we can keep going with this
until our graphical fidelity of our screen doesn't show anything
anymore. Yeah. I Yeah, as soon as you get smaller than a pixel,
yes.
So we can do this construction. Take the triangles you can Take
you can.
Yeah, So You
so we can go from the top down, or we can think of concatenating
transformation matrices, because we can have a matrix for the
each of these two transformations, so we scale by
half, and this one, if it's at zero and I don't need to
translate. But here we're translating to
zero, and this one, we're translating to point 5.866,
point 866,
that's, that's,
that's approximately e collateral triangle. Or we could
center it at the center of the triangle, and then this would be
point, 5433, 433,
it's not quite right, that'll be here anyway.
I so
we can have a matrix to describe the different transformations,
and then we can multiply them out, and
we can create strings of transformation so that the net
effect of them are going To be producing pixel size points.
That makes sense. I
so We Get the
so a way to think about the shape the fractal that's
generated is The collection of fixed points of all these
transformations and
CS. So when we create a transformation matrices, We can
we
so The weight, so we first want to Move. We're
so we think of if we're applying transformation this point,
translate point, the origin, apply the transformation, the
scaling, the rotation and the shear, and then translate back
from the origin to the center. So that gives us
so If we apply matrix zero and
I guess So,
by the matrix that's three times matrix zero, and we apply it to
the fixed point at 00, what?
What point am I going to are we going To get out? Does
that make sense?
Okay, so we're multiplying the transformations out. So this is
m zero applied once, this is m zero applied twice. And
this is m zero applied three times.
So if we, if we transform the point 00 with this, because it's
the fixed point of this transformation matrix, we're
going to get 00 out again, because we're applying The same
transformation. So for each of these transformations, we're
equations, they have the fixed point at zero at the origin. So
the m zero is
so we're just scaling by a Half.
So how do we know when to stop and
concatenated transformations.
So what happens, let's, let's look at This matrix. Let's do
some matrix multiplication. I'm
so let's do
let's do this one first so we get two levels. So point five
times, point five is point two five and
The rest
rests. The rest of them are zeros and
and then we apply point five, the matrix again, to this one,
which is a
So then point five times point two Five is the
Does that make sense? So what do we notice about the progression
of of the matrices? Yeah, so an important property I didn't
mention is that the transformations have to be
contractive, so they converge to
to a fixed set of points. So let's say we have, we're going
to go, we have.
You want to do a square image from Zero to One in x
and in y and
Okay, so,
what other information do we Need to know when to stop our
recursion or our concat composition of Tran of
transformation matrices.
That was the sentence. Did you mean, like, if you mean how to
stop the recursion. You just got to send in how many depth you
want?
Yeah. So how do we determine the depths of we want? Really
depends on how many generations you want. So at the beginning,
if I say I wanted one, I would send in the second image you
have there. So generation one. So then you send in one, and you
subtract that one, and when you send it in a final time, it'll
be like if it's greater than if depth is greater than zero,
we'll keep going, and then if not, we'll draw, draw whatever
we have currently.
Yeah, okay, but going with the idea that we do, we want to make
a pixel sized shape, or dealing with a pixel size shape from
each of these strings of transformation. So how so this
is the space, the world space we're dealing with.
But what about the resolution we're going to create in our
image. So if we say 1024 by 1024 I
so we want to have the number of levels such that this
contraction factor is Less than or is approximately equal to
This size and
so compose until I
so we're going to start with, say this is the size that we're
starting With, so we have a width of one, and then after we
apply one transformation, it's point Five, and apply another
transformation, It's point two, five and
so we take that the size of our rural space, and then we use a
way to estimate the size of the
so The transformation is going to end up to produce a a point
which is a pixel size. Does that make sense? Can you hear me in
the back? Okay, my tailing off, trailing off a little bit. I'll
try and keep the volume up. Okay, so then that brings us to
the assignment to reimagine the desktop tetrahedron. So let's
see if I Can.
Chapter is honest. I
I hope I'm curious why it's taking so long. Been quicker
before
minutes i entertainable.
I Okay, so Here are three spheres.
Corresponds
to gasket, example, So
just a little better. Okay, so I
so I'm using Object instantiation here. So I made a
base object. I
i So I called it the object base, and then inside the object
definition, I say attribute begin. So that gives me so
because I'm putting the attribute begin and ends in
here, that I'm doing a transformation, not to the
current one, but starting from a fresh
so another other languages might do push and pop, so it saves the
state anyway. So then I'm not if I, if I don't separate them with
the attribute begin and ends, then
the third, the third sphere, would be point five times point
five times point five. So be point 125, and not, if that
makes sense. So I'm, so I define this base object. Here's that,
there's the example of it. So I'm starting with the I'm just
letting it be unit sphere of diffuse material.
So I have three spheres. I have an object, then object end. So
then to use the object, I See object instance And then base I
so do You see what I've done here?
So I'm doing a translation of the base object, first scaling
by one half in all directions, and then doing a translation,
scaling. And then the third one is scale,
and translate and scaled as well. So what is, what do you
think the output of this, this description, C, description will
be gasket. I,
any ideas how many spheres are Going to see light?
Nine. Yes, I
now, how do I get to 27 Copy
when you did there?
Thinking, versus for me, part.
Let's make a new object. I
so it should be the same image that we just had.
Okay, can't Do that. I
see, how Do we Start? I
Okay, what's going to be The impact of that
change? I
i guess 27
let's See if I correct. Do
So I still have nine spheres, because I'm transforming the
three spheres. I transformed the three spheres three times. So
now I have to fill in The transformations to Make the
Other I
It seems like a lot of mass on the board and maybe should be
nice On the Part I
Well, I offline. So what? What do we want to do?
So you want
one three spheres here, three spheres there, three spheres
here, three spheres here. I
And in each of these places. So what? 123456789, times, Three is
27, So
I Skip there.
I and it's
making little stars on the insides, which is pretty
cool, that one section you completed, It makes a new shape
You saw
You. Yes, I
so There's only
a few spheres short here, yeah,
so We have 18, so I see what's happening. You shared the base
of the bottom triangles with the top triangles.
Yeah. And
that the middle one, too should not. So you have it so that the
middle like the bottom quadrant, if you look at those three
triangles, that is correct. The rest of it is not. So if you
just, if you move everything over and upward, and you have to
have a second copy of the bottom left three triangles, and then a
third copy on top, and that's what you're aiming for. And then
there'll be a big hole in the middle, as you would expect, on
a seconds. Keep Yes.
Anyway, I'm i I'll give you this file. You can work on it, if you
like. But this is the idea behind constructing a model of
tetrahedron. So this is one way to do it, maybe a little bit
tedious in a way to do it. So imagine a way is to compute
transformation, compute the final transformations, thinking
about what kind of resolution you want. And then you can
specify the transformation matrix as the input.
In the same description you can do it you can specify the 16
coefficients of the matrix, and doing a transformation That way
gives you it starts with a fresh current transformation matrix,
so you're not concatenating them.
So there are places online to find the coordinates of the
vertices of the tetrahedron, and you can construct transformation
matrices like this, like we've done Here. I'll fix that up. I
realize I haven't I'm
I have anyway. So this is for students, a little demo of how
to do things, fractal shapes in pbrt, fill similar shapes. And
I'll fix that up, and I'll send it to you. Do
so we're out and so time. But I wanted to ask, what are two ways
to describe common forms for describing shapes?
So there was implicit, yeah, everything was parabolic,
separate, parametric, parametric,
okay, good,
okay, so how about we end there. I'll fix this file up for you.
Maybe I'll do a version with transformation matrices in it.
Okay? Does that sound all right? Thank you for today. Have a good
weekend.
See you on Tuesday.

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