Topics we discussed:
Approximating shapes by triangles:
In most modern computer graphics pipelines,
smooth shapes (like cylinders and spheres)
are created out of triangles.
If you want the edges of the shapes to
look smooth, you need to use more triangles.
For example, a cylinder is approximated by
an polygonal prism. If the approximating
prism shape has N sides, then you need 4*N
triangles to build it (N for the
top cap, N for the bottom cap, and 2 triangles
for each "quad" (four sided polygon)
that forms a side of the prism).
Similarly, to build an approximation
of a sphere out of triangles, you can use a
grid to create an approximating polyhedron.
Every quad (four sided polygonal face) of this
approximating polyhedron requires two triangles.
Such a shape is said to be parametric,
because we can use two parameters, u and v,
to create the shape, where 0 ≤ u ≤ 1
and 0 ≤ v ≤ 1.
We can think of u and v as lying on an
abstract parametric square, which acts as
a kind of rubber sheet.
We mathemetically distort this rubber
sheet to describe the sphere.
We do this as follows: For any point u,v on our
parametric square, we define a longitude
angle theta, and a latitude angle phi,
0 ≤ theta ≤ 2 π
Then any point (x,y,z) on the sphere
can be described by:
-π / 2 ≤ phi ≤ π / 2
x = cos(theta) * cos(phi)
By looping over values of u and v,
we can create the little quads
that constitute our approximating polyhedron.
If we do this in smaller steps,
we get a better approximation to the sphere.
y = sin(theta) * cos(phi)
z = sin(phi)
Vertex shaders and fragment shaders:
We will build our higher level shapes,
like cylinders, spheres, cubes and human
faces, on the computer's CPU.
But in the low level code, where those shapes are defined
as individual triangles, we will use the computer's GPU
(graphics processing unit).
The GPU lets us write two kinds of programs:
In the next few weeks, we are going to start off
programming entirely within a square in the fragment shader,
so that you will get used to doing that right away.
In order to make that easier for you, I am going
to provide you with a library that does much of
the heavy lifting for you of making sure that the
fragment shader code you write will be visible
on your web page.
- The Vertex shader uses the power of the
GPU to place the vertices of our triangles within
the final image.
Usually this is a very simple program, but
sometimes it can be very elaborate.
- The Fragment shader is where most of
the computational intensive work is done by the GPU.
There is where we do such computations as shading
and texturing, which need to be done on a per-pixel
The reason it is called a "fragment shader" and
not a "pixel shader" is that we sometimes have two
or more fragments of visible geometry within the same pixel
of the image (for example, at the edge where two shapes intersect).
The GPU needs to do a separate computation for each
of those fragments of pixels.
As I mentioned in class, this means that our
geometry will really consist of two triangles.
More about that next Tuesday!
We watched the following films in class: