Now that we've visited ray tracing, we're going back mesh objects. We'll continue to use the PixApplet method for setting the color at each pixel that you have been using for ray tracing.


To review what we discussed in class earlier in the semester, a general way to store polyhedral meshes is to use a vertex array and a face array. Each element of the vertex array contains point/normal values for a single vertex:

vertices = { { x,y,z, nx,ny,nz }0, { x,y,z, nx,ny,nz }1, ... }

Each element of the face array contains an ordered list of the vertices in that face:

faces = { { v0,v1,v2,... }0, { v0,v1,v2,... }1, ... }

A vertex is specified by both its location and its normal vector direction. As I said in class, if vertex normals are different, then we'll adopt the convention that two vertices are not the same - even if x,y,z are the same.


If you want to create the illusion of smooth interpolated normals when you're approximating rounded shapes (like spheres and cylinders) you can do so either by using special purpose methods for that particular shape (like using the direction from the center of a sphere to each vertex - as we discussed in class), or else you can use the following more general purpose method:

  1. For each face, approximate the gradient for that face by taking the cross product between v1-v0 and v2-v1, where v0, v1 and v2 are the first three vertices in that face.

  2. For each vertex, sum up the gradients for all faces that contain that vertex. Normalizing this sum gives a good approximation to the normal vector at this vertex.

To transform a normal vector, you need to transform it by the transpose of M-1, where M is the matrix that you are using to transform the associate vertex location. Here is a MatrixInverter class that you can use to compute the matrix inverse.

As usual, try to make fun, cool and interesting content. See if you can make various shapes that are built up from things like tubes and cylinders or tori, as in the applet I showed in class. We haven't done all the math yet that we would need to do spline surfaces like teapots.


When rendering a phong-shaded scene you will want to have a list of Materials, where each Material object gives the data that you need to simulate a particular kind of surface using the Phong algorithm (eg: DiffuseColor, SpecularColor, etc.).

You will also want to have a list of Light sources, where each light source Li has a direction (x,y,z) and a illuminance color (r,g,b):

Li = { x , y , z , r , g , b }


The Z-buffer algorithm is a way to get from triangles to shaded pixels.

You use the Z buffer algorithm to figure out which thing is in front at every pixel when you are creating fully shaded versions of your mesh objects, such as when you use the Phong surface shading algorithm.

The algorithm starts with an empty zBuffer, indexed by pixels [X,Y], and initially set to zero for each pixel. You also need an image FrameBuffer filled with background color. This frameBuffer can be the pix[] array you currently use for ray tracing.

The general flow of things is:

A note about linear interpolation:

In order to interpolate values from the triangle to the trapezoid, then from the trapezoid to the horizontal span for each scan-line, then from the span down individual pixels, you'll need to use linear interpolation.

Generally speaking, linear interpolation involves the following two steps:

In order to compute t, you just need your extreme values and the intermediate value where you want the results. For example, to compute the value of t to interpolate from scan-line Y_TOP and Y_BOTTOM to a single scan-line Y:

t = (double)(Y - YTOP) / (YBOTTOM - YTOP)
Similarly, to compute the value of t to interpolate from pixels X_lEFT and X_RIGHT to values at a single pixel X:

t = (double)(X - XLEFT) / (XRIGHT - XLEFT)