Course Notes

Adding Control of the View Volume

We continue with This demo
But Dalton Sevier, a student produced a Much better demo in 2024.
Right hand vs Left Hand
- We generally use coordinate systems that observe the right hand rule
- In other words, we were looking along the negative z axis
- The right hand rule tells us how to orient in three space.
- Your author says
  - "if you point the thumb of your right hand in the direction of the positive z-axis, then when you curl the fingers of that hand, they will curl in the direction from the positive x-axis towards the positive y-axis."
  - "Another way to think about it is that if you curl the figures of your right hand from the positive x to the positive y-axis, then your thumb will point in the direction of the positive z-axis."
  - "The default OpenGL coordinate system (which, again, is hardly ever used) is a left-handed system."
In OpenGL, everything is drawn from the standard view volume.
- This is a cube from (-1, -1, -1) to (1,1,1)
- Everything is clipped
- Then everything is projected to 2D Normalized Device Coordinates.
- Finally that is mapped to Device Coordinates.

Remember, with the camera, we had a "view cone"

That if we clipped in the front became the view frustum.
And we frequently add a rear clipping plane.

One way to form a projection matrix is make a matrix that maps coordinates in the eye space to the frustum.

The derivation of this is not too bad, but probably outside what we want to do today.
OpenGL Projection Matrix by Song Ho Ahn is a nice article on this.
The Perspecitve and Orthographic Projection Matrix by Scratchpixel

In general, to form the projection matrix for frustum you need

The distance to the near and far clipping planes.
The coordinates of the top and bottom edges of the near plane.
The coordinates of the left and right edges of the near plane.
(source unknown)

Shifting coordinate systems

Generally we assume
- The camera is at (0,0,0) looking down the negative z axis.
- The positive y axis is up.
- The positive x axis is to the right.
In 3D NDC this is backwards
- The camera is looking along the positive Z axis.
So part of the transformation flips the axis.

In the end, the transformation is

$ \begin{bmatrix} \frac{2*n}{r-l} & 0 & \frac{r+l}{r-l} & 0 \\ 0 & \frac{2*n}{t-b} & \frac{t+b}{t-b} & 0 \\ 0 & 0 & -\frac{f+n}{f-n} & \frac{-2fn}{f-n} \\ 0 & 0 & -1 & 0 \end{bmatrix}$

And relized in

    Frustum(l,r,b,t,n,f) {
       var m =  mat4(1);
       m[0][0] = 2 * n / (r - l);
       m[0][1] = 0;
       m[0][2] = (r + l) / (r - l);
       m[0][3] = 0;

       m[1][0] = 0;
       m[1][1] = 2 * n / (t - b);
       m[1][2] = (t + b) / (t - b);
       m[1][3] = 0;

       m[2][0] = 0;
       m[2][1] = 0;
       m[2][2] = -(f + n) / (f - n);
       m[2][3] = -2 * f * n / (f - n);

       m[3][0] = 0;
       m[3][1] = 0;
       m[3][2] = -1;
       m[3][3] = 0;

       return m;
    }

Viewing can also be accomplished via a field of view.

This is specified by the angle from top to bottom that is visible.
(source unknown)
You also specify an aspect ratio
And a near and far plane.
Again the derivation is not bad.
MV.js has code to produce this matrix.

The camera

It is often useful to place the camera in modeling coordinates.
Then transform everything from modeling coordinates to viewing coordinates.
The camera is specified by three vectors
- Eye is the origin for the camera
- Look at is a vector from the origin of the camera to where it eye is focused
- UP is a vector pointing up for the camera.
With the three of these, a transformation from modeling to viewing coordinates is possible.
This is what they call a change of basis in linear algebra.
And is accomplished with the lookAt function in MV.js.

So in the demo look at the canvas constructor.

Items in the model coordinate system are
- Transformed into the viewing coordinate system with the frustum transformation.
- The moved to Clipping space and projected to NDC space with the lookAt matrix.

var eye = lookAt([0,0,-1],[0,0,0],[0,1,0]);
var proj =  this.Frustum(-0.5,0.5,-0.5,0.5,1,20);
proj = mult(proj,eye);
this.gl.uniformMatrix4fv(this.projLoc, false,flatten(proj));