Mtg 8/23: Wed-02-Oct-2024

Outline for Today

Geometric Objects and Transformations

Administration

Today

Happy 1st meeting of October
will be grading for thoughtfulness of responses from today forward *October 11 will have 50th anniversary events related to Computer Science
- Pierre Lemire (BSc ‘87), CEO of Kent Imaging will give a presentation at 3:30 in the Education Auditorium
- for a lunchtime event, would you prefer sandwiches and wraps or pizza?
Chapter 4 Code

For Next Meeting

Wiki

Link to the UR Courses wiki page for this meeting

Media

Transcript

Audio Transcript

Okay, Okay.
So Happy first meeting of October, they were
going to enjoy the two transition from 31 to
13 and
rain in the wind.
Error, he here.
So I want to talk about October, 11. I
so that's the day that we're going
to be having
an event for The theme of the 50th
anniversary of the university. I
thank you. I so We have our guest. I
So he was a contemporary of mine in The undergraduate program.
He's now CEO of Kent imaging. He
does that sound like somebody would be interesting to even
talk with? I
Okay,
there wasn't a correct answer there,
but I went to get your input,
because there's a
tragic set of circumstances here at university. I think there's
no peace on campus presently who
thinks us no age,
it's not clear, because there's so many food service places that
are boarded up,
but apparently they don't have pizza as an option to order for
Food Services.
So
we can order
pizza from off campus.
So would you like to would you I?
Do you think a lunchtime event would be something you could
you might take in on the 11th?
Okay? I
And if we have a choice between sandwiches and rats and pizza,
who votes for pizza? You don't have to commit to going to the
lunch and
and who votes for sandwiches and rats? Oh, strong showing for
sandwiches and rats. So I
Is that a true statement of the fact that you can get free food,
that's that's the biggest factor.
Okay,
I'll be sending a note out
tonight or tomorrow with some more particular details. This
will be
recorded and available
later. So this is in the regular seminar slot for The Department
i Okay,
down to Business here for
so
points are you can draw points on the screen and
we can approximate well,
they don't have any size or shape or
they are
Just a location space i
What else Do we need?
Time points to describe geometric objects that we'd like
to render on the screen measures.
I enough to get a space there,
because
nationals are important a triangular mesh is an important
way to represent objects, to model them.
So how do we get to think about the triangles?
So
if we start with points,
scalars. So What are they?
Can be real numbers, maybe complex numbers,
usually real and
so if you think about the distance between points, this
is between 00, and one, zero is one
as a scalar.
And vectors.
So what's going on with vectors, direction
and direction? So
so we can think of
directed line segments and
so if two vectors have the same direction and magnitude,
They're the same because they don't have a position what if
there
were 10 years? I factors. Yeah,
so.
I diving. So we can talk about it as a frame. So we have
vectors plus two. We have vectors
x, y and z. We could
have, we could think of the most unit vectors along the X, Y and
Z axes and
so we don't usually think of them being drawn that way, but I
so we can put them in
together that way.
So what hand?
What handed, which handed coordinate system Is that
in here? Yeah, I
so when we're thinking about computer graphics,
We might deal with a number of different frames and
so you might think, first of all, object frame.
So the thing, the primitives that
we are going to work with,
we define them on their own coordinate
system, in their own frame. That's convenient. We have Q
define it from the origin and
should I try that again? I
looks more parallel, but it's still not I
it. So you can take the cube and say, We want to make it. You can
scale it, rotate it, translate it into
our model.
And we might think of many cubes
from the models or making the transformation from the object,
spade, object frame
of the cube into
the world coordinates that, pardon me, into the Model
coordinates and
so then we're putting them into
this new frame
where the transform
objects
are
in one frame together.
There's one origin and so on
What other frames to deal with, I
any thoughts i
What about a camera frame? So
when we're taking pictures of the camera.
How do we
specify the three three vectors for the
of the frame and the origin and
so We can think of the eye or the camera, and
at the origin here.
So what if I want to take a picture of
the exit sign here
we described
The screen so
I have the camera here.
That's a position in
space, x, y and Z, and I'm looking At the sign I'm
okay,
still doesn't look like he's running, but so I have a
position for the camera. I'm looking at something like the
chair. So that's one direct I can have that direction. I have
the vector going from the camera to the chair or whatever I'm
looking at.
What else do I need to specify the coordinates i
What if I want To take a portrait picture
or an angled picture,
which way is up and
and then we can make the third vector by
the cross product of look at and up and
so that's the core camera frame. I
what other you mentioned one set of coordinates the other Day
that was I
can you Imagine this is A key again. So
so that's another frame
that we're getting
the world, which might be hundreds of 1000s of kilometers,
and we squish it into
the clip or
depth frame to the
together, ready to display. I
so we had a window
frame as well. And
that relates to the viewport
that we set up and
does that make sense?
So we can Do transformations between them. Do
so We Can You
so let's look at A matrix to do scaling I
so We can have so it starts with A Okay,
so the identity matrix so
so if We apply a negative scale factor, we get reflections and
So we want to preserve the object would just reflect around
y,
so we could apply a scale factor of y minus one. What if
we Want to translate the point? I
so we take the hard scale
and maintenance you
And then we have
So we apply
transformation and
so We multiply our point, p1 px,
py, pz, by our transformation matrix, and then we would add a
translation component. So that's not Very effective or efficient.
I so using the homogeneous coordinates,
we have the 3d affine
transformation.
Simple example is the scaling matrix. So we have
suspects and and
now we can do
to specify the translation vector in the same matrix, so
then we have Tx
equal I instead.
So points
are represented as
four tuple, so we have x, y, z and w.
So w is often one, but for points, we
convert them to three dimensions and
so we can think of lights w equal
to one and vectors W equal to zero. And
okay, so the idea is that
we can, in our rendering pipeline, we
can transform our world coordinates
into Window coordinates by doing a series of
transformations you
so when I look at the cube, rotating cube, this is Coda
sound from chapter four. So
So if we're looking in the z direction, which X are we
rotating on here to Begin with you
so if We're out here somewhere,
is It Z?
No, it's an X, Y, I think it's x.
Let's do rotating or y. I
Does that make sense? And
we can add rotations as well? I
if you stare the blind enough with false feeling, maybe go.
Just did hypnosis. You
so eventually you're getting a rotation around the center and
the axis of rotation is a combination of x, y and z
angles. So that's as finite controls are given in this
example. Let's check out the source code for this.
Okay, so the vertex shader, so we have uniform VEC 3u theta, so
the angle is a uniform and so we have X, Y and Z values as part
of the VEC three we're passing for u, theta. And then the first
thing is to compute the size of cosines of theta for each of the
three axis and one computation.
So it's converting the angles into radians and then computing
vectors of cosines and sines based on the rate the the angles
specified in radians. So then. So you'll notice, as you look
through examples of the code that this typo has repeated many
times over.
So what does it mean that matrices are in column major
order? I
Yes. So a column major means that we're putting the column,
putting them together in columns. So
the first formula is going to the first column. Next four
going to the second column, the next four go in the third
column, and the last four going to the last column, and the
fourth, fourth and last column. Does that make sense? So these
are the way. This is the way to define rotation matrices around
the X, Y and Z axes, so using the dot notation here to specify
the x component of the vector, the y component of the vector
and the z component of the vector. Vector is the cosine.
So the vertex color is taken from the attribute color that we
specify in the JavaScript program. And GL position is our
attribute position that we've started with, the our initial
cube that we specified in the JavaScript code. And then we're
applying
X, Y and Z rotations to determine new position.
And then the vertex, the private shirt is just passing through
the color and
so we're specifying a cat. This is 512 by 512 so let's look at
the pupa js to see what's going on there. We're
so why do we have none positions as 36 so
secured by three or six sides.
So How many faces?
Four,
six, I?
And so how many? So
we're not going to represent them as squares, because squares
might be non planar. So how many triangles make up a square? Two?
Two, yeah, two triangles.
So now we've got 12. How
do we roll? So what's what?
What do we need three of them for each triangle.
So we have three vertices. So
three vertices per triangle. So six times two times three is 36
so we're going to generate we're going to store 36 positions. We
can do better than that. You don't have to store because how
many vertices are there? Not Q? I?
So there's four on the top and four on the bottom.
So we can use a more efficient way to
you can look at that for yourselves, and we'll talk about
it some more as well. Okay, so 10 minutes left, let's see
what's going on here.
Okay, so we do the standard stuff. So we define so we have
the red GL to campus, get the context, and we set the set up
the viewport. And we enable depth tests so that we're using
the Z buffer loading shades,
and you're restoring colors and the colors is defined.
Okay, so color cube is called first, so that sets up the
colors. So we're taking that brain that We've set up and
flattening it.
So we put the colors there, and now we're giving the A color.
We're linking that to the attribute, so it's a four RGBA
that we're passing as the color locate, color value, Color
attribute,
and then we're
storing the vertex data, another array there, and then we're
associating with the variable in the shader. And then finally, we
do the uniform variable for the theta, the X, Y and Z angles. So
if we click on the on the X button in the interface that we
get when we set the axis to X, Y and Z and we go to Render and
so in a render, we set the color bit, and we're doing the clear
of the color bit and the theft buffer bit, so we're resetting
the Z buffer and resetting The screen to be the background
cover that we set. So the thing we do is we increment the angle
by two degrees, depending on which axis we've chosen, and
then we set it. We connect the value that we've just said in
theta to the uniform variable in the shader. We say, draw arrays,
draw triangles to start at zero. We go for 36 positions, and we
say, request animation frame for render. So that's the rendering.
Use
the color key. So we do quads. So there are six calls to quad.
And so they will be given the vertices. So 1032,
so we start with so the first quad is minus point five, point
5.5, and one. And zero is minus point five, minus point 5.5,
and one and
then then three minus or point five minus point 5.5, and one
and two is point five point 5.5, And one.
So we specify the vertices of the quad based on our input to
the function, and then we also have colors, vertex Colors and
so we're so we take the indices, so we're taking the quad face
and dividing the two triangles. That means the four vertices
have become six because we're duplicating vertices, so we have
three vertices for each triangle.
So you can see A, B, C, A, C, D, so A and C are repeated.
So that's the way to describe the two triangles that
correspond to the one face the cube.
They've commented on the
stuff that assigns different colors to each vertex. So just
using solid colors so for each i
So for each vertex, we're pushing the same color. So A is
C,
see. So A is one for the first squad. So we're going to color
the those two triangles red. So the face is going to be red for
that.
Okay?
So, what would happen if we had a bigger Q instead of minus
point five to plus point five in x, y and z? What if it was one
minus one to one? I?
Because we're not specifying the details of the transformation,
so we're relying on our coordinates after our from the
vertex shader to be in the minus one to plus one range. Does that
make sense? So if you experiment and make just take this naive
implementation of the code and would change the vertex
vertices, and you won't be able to see the whole cube anymore.
So we'll talk about transforming objects, we're going to apply
the same transformations to viewing that's in the next
chapter, and we'll see how we can how we
can work with the clipping frame, clip coordinate frame.
It's a feature, not a bug, that
we have to put everything careful about keeping things in
the minus one to plus one range, the pipe where we set the
pipeline will handle that for us. Anyway. Does that make
sense? Any questions or concerns?
So we'll talk a bit more about
viewing transformations. Next
day, we'll get into chapter five as well.
Okay, thank you for today.
Have a good weekend.
Take care, everyone. Thank you
for today.
You
You

Responses

What important concept or perspective did you encounter today?

The important concept I learned from last meeting was that we use matrices to represent vectors, and other types of coordinates in order to make a geometric objects/model. We use 4x4 matrix for homogenous coordinates and 3x3 matrix for vector scale. However, the matrice depends upon the object we are creating thus we will have to adjust or change representation by modifying the coordinates.
Transformation matrices! I cannot wait to talk about quaternions >:3
The scaling matrix, translation, and rotation matrices that make up the points of every object in a scene, and how we interact with them in order to be able to properly perform transformations to each matrix, cause that was the main aspect of Linear Algebra I
Geometric object + Transformation and Camera
The concept of producing reflections which is as easy as applying a negative scale factor to transform the matrix. It made a lot of sense to me.
Geometric objects and Transformation, Things needed to define/render an object on screen, different frames that deal with CG, coordinate system and 3d rotating cube's code
how we can showcase both a vector and translation matrices together in one 4d matrix
Today, I learned about geometric objects and transformations. Scalars represent quantities like distance. A point indicates a location in space without size, while a vector shows direction. Vectors require an origin and basis vectors to define a frame. I learned different frames in the graphics rendering pipeline: object, model, camera/eye, and clip coordinates. I also studied 4x4 matrices in homogeneous coordinates and a cube program.
The geometric objects including points, scalars, vectors, meshes and their definition

Was there anything today that was difficult to understand?

Sierpenski Gasket
No
Why for the homogeneous matrix we use 1 points and 0 for vectors

Was there anything today about which you would like to know more?

I would like to know more about how to easily identify which axis something is spinning on.
I would like to know more about how rotation is calculated. I couldn't quite see it in cube.html, unless there was something I missed. I've had a similar problem in this course's first lab because the variable for rotation was also used to calculate speed and I could not find a way to create them as separate variables.
I would like to know more about camera/eye frames and how exactly they work
I would like to know more about how we can represent points and transforms as vec4's and 4x4 matrices. What other kinds of transformations can we represent with a 4x4 matrix? Can we do rotations?
The different kinds of coordinate systems, when to use them, why they were originally developed. Especially why Blender decided to have Z as the up/down axis.
coordinate systems seem to be similar to the frames we learnt about? is that so? like object space = object, world space = Model, view space = camera/eye, clip = projection space
it was mentioned that a cube could be represented in 3d using only 8 points, a showcase of this would be very interesting
It was all clear and we understood most !

Site built: 2025-06-20 12:45 -0600 • Give feedback about page