3D Viewing Pipeline

3D Matrix Transformations

The first change that was made to the graphics system to accomodate 3D operations was the addition of 3D transformation functions. The following transforms are now supported:

Translate3D -- translates in all 3 dimensions
Scale3D -- scales in all 3 dimensions
Rotate3DX -- Rotates about the x-axis
Rotate3DY -- Rotates about the y-axis
Rotate3DZ -- Rotates about the z-axis
RotateZ -- Rotates about the z-axis
Rotate3DXYZ -- Rotates about the x, y, and z-axes
ShearZ -- Shears x and y
PerspProj -- Performs the matrix transform necessary for perspective projection

All of these transforms have been adapted to work in the hierarchical modeling environment created in the last lab.

3D Perspective Viewing Pipeline

Two main changed were made to accomodate a 3D perspective viewing pipeline. First, a new structure was created to hold all of the parameters that describe a 3D view. These parameters are:

VRP -- View Reference Point
VPN -- View Plane Normal
VUP -- View Up Vector
d -- Projection Distance
du, dv -- View Window Extent in x and y
F, B -- Front and Back Clip Planes
screenX, screenY -- desired number of pixels in image

The second change involved creating a function to perform the appropriate transformations for a perspective projection. The steps are as follows (taken from Bruce Maxwell's lab 6 homepage):

Translate the VRP to the origin: VTM = T(- vrp.x, - vrp.y, - vrp.z).
Align the coordinate axes:
- Normalize the VPN and VUP.
- Create UVEC = VUP x VPN.
- Redo VUP' = VPN x UVEC.
- VTM = RotateXYZ(UVEC, VUP', VPN) * VTM.
Translate the COP (represented by the projection distance) to the origin: VTM = T(0, 0, d) * VTM.
Scale to the canonical view volume [CVV]:
- Let B = B + d.
- VTM = Scale( 2 * d / (du * B), 2 * d / (dv * B), 1 / B) * VTM.
Project onto the image plane:
- Let d = d / B.
- VTM = PerspProj(d) * VTM.
Scale to the image size: VTM = Scale( - screenX / (2 * d), - screenY / (2 * d), 1.0) * VTM.
Translate the lower left corner to the origin: VTM = T(screenX/2, screenY/2) * VTM.

Required Images

There were two required images for this lab. Both contain a cube with opposite corners at (0, 0, 0) and (1, 1, 1). The first image is 100x100. The COP is located at (0.5, 0.5, -4), the VRP is located at (0.5, 0.5, -2), and the view window size is (du, dv) = (1, 1). This image can be seen below.

The second required image is of the same cube, but this time it is viewed with 3-point perspective.

Extensions

Several extensions were implemented. First, three animated gifs were created to demonstrate the new 3D perspective viewing pipeline. The first of these images performs a "drive-by" by moving in the x direction across the world.

The next image is an animated gif as the view moves around the cube in an elliptical orbit. The VPN is always pointed at the center of the cube.

Finally, the third image alters the value of d linearly and then sets du = dv = 2d/(d+2). This is similar to changing from a wide angle to a telephoto lens on a camera. Casey insists that I use the word "non-trivial" to describe this transformation.

The second extension that we performed was to make all of our code operational within our hierarchical modeling system. Now, all of the 3D transforms and 3D perspective viewing can be using within modules. All of our extensions worked properly.

What are the (x,y) values for the eight corners of the cube in the first required image? The (x,y) values for the eight corners in the first image are:
- (25, 25)
- (75, 25)
- (75, 75)
- (25, 75)
- (30, 30)
- (70, 30)
- (70, 70)
- (30, 70)
How does modifying the distance between the COP and the VRP affect the appearance of the cube in the first required image?
Moving the COP closer to the VRP makes the cube look smaller, while moving the COP farther from the VRP makes the cube look bigger. Moving the COP out to infinity is theequivalent of parallel projection.

How does modifying the direction of VUP modify the appearance of the cube in the first required image?
Modifying the direction of the VUP rotates the cube about an axis going from the viewer to the center of the cube (an axis into the image).

How does modifying the size of the view window modify the appearance of the cube in the first required image?
Making the view window larger makes the cube appear smaller, and vice versa. This is because increasing the view window makes the cube take up less space in relation to the image.

What extensions did you do for this assignment, how did you do them, and how well did they work?
Scroll up for a description of the extensions.