Meshes and Z-buffers
We're going back mesh objects, but we'll be using the PixApplet method for setting the color at each pixel that you have been using for Ray Tracing. And this time you'll be able to use a more general mesh structure. I'll be going over all of the material here in detail in class next Tuesday (Nov 9), and then you'll still have another week after that before you need to hand in this assignment.
As we discussed in class, a more 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 = { { v_{0},v_{1},v_{2},... }_{0}, { v_{0},v_{1},v_{2},... }_{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 two vertices are not the same. For example, here is a declaration for the vertices and the faces of a unit cube:
int[][] faces = { { 0, 1, 2, 3 }, { 4, 5, 6, 7 }, { 8, 9, 10, 11 }, { 12, 13, 14, 15 }, { 16, 17, 18, 19 }, { 20, 21, 22, 23 } }; double[][] vertices = { {-1,-1,-1,-1, 0, 0 }, {-1,-1, 1,-1, 0, 0 }, {-1, 1, 1,-1, 0, 0 }, {-1, 1,-1,-1, 0, 0 }, { 1,-1,-1, 1, 0, 0 }, { 1, 1,-1, 1, 0, 0 }, { 1, 1, 1, 1, 0, 0 }, { 1,-1, 1, 1, 0, 0 }, {-1,-1,-1, 0,-1, 0 }, { 1,-1,-1, 0,-1, 0 }, { 1,-1, 1, 0,-1, 0 }, {-1,-1, 1, 0,-1, 0 }, {-1, 1,-1, 0, 1, 0 }, {-1, 1, 1, 0, 1, 0 }, { 1, 1, 1, 0, 1, 0 }, { 1, 1,-1, 0, 1, 0 }, {-1,-1,-1, 0, 0,-1 }, {-1, 1,-1, 0, 0,-1 }, { 1, 1,-1, 0, 0,-1 }, { 1,-1,-1, 0, 0,-1 }, {-1,-1, 1, 0, 0, 1 }, { 1,-1, 1, 0, 0, 1 }, { 1, 1, 1, 0, 0, 1 }, {-1, 1, 1, 0, 0, 1 } };
compute normals, transforming normals:
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), or else you can use the following more general purpose method:
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. As before, you can use the MatrixInverter class to compute a 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 the math yet to do spline surfaces like teapots.
Z-BUFFERING:
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.
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 source objects, just as you did when you were doing ray tracing.
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:
(X_{BL}, Y_{BOTTOM}) (X_{BR}, Y_{BOTTOM})
You can see the process repesented here: First a polygon is split into triangles, and the each triangle is split into scan-line aligned trapezoids:
Each vertex of one of these scan-line aligned trapezoids will have both color and perspective z, or (r,g,b,pz).
If pz < zBuffer[X,Y] then replace the values at that pixel:
zBuffer[X,Y] ← pz
frameBuffer[X,Y] ← (r,g,b)
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:
value = a + t * (b - a)
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 - Y_{TOP}) / (Y_{BOTTOM} - Y_{TOP})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 - X_{LEFT}) / (X_{RIGHT} - X_{LEFT})