3D Engine - Projection Matrix

Question

Hi
Im basically building a 3d engine (very simple one) using software rendering (no opengl or directx) for learning purposes.
The 3d engine has one simple rendering method thats suppose to render a point from world space to the 2d space (monitor)
i believe i got the rendering working properly, but it has no perspective
i believe i need a perspective matrix
can you pros help me build one
  public static Pixel2D render(Camera c, Pixel3D p) {
    //matrix transformations (with euler style rotation)
    //fm = pm * cm * wm * point
    double fm[][];
    double om[][] = createMatrix(4, 1);
    double cm[][];
    double ctm[][] = createIdentityMatrix(4); //camera translation
    double crxm[][] = createIdentityMatrix(4); //camera x-axis rotation
    double crym[][] = createIdentityMatrix(4); //camera y-axis rotation
    double crzm[][] = createIdentityMatrix(4); //camera z-axis rotation
    double pm[][] = createIdentityMatrix(4); //perspective matrix
    double rx = Math.toRadians(c.getRoll());
    double ry = Math.toRadians(c.getPitch());
    double rz = Math.toRadians(c.getYaw());
    //camera matrix
    //translation
    ctm[0][3] = c.getX();
    ctm[1][3] = c.getY();
    ctm[2][3] = c.getZ();
    //roll
    crxm[1][1] = Math.cos(rx);
    crxm[1][2] = Math.sin(rx);
    crxm[2][1] = -Math.sin(rx);
    crxm[2][2] = Math.cos(rx);
    //pitch
    crym[0][0] = Math.cos(ry);
    crym[0][2] = -Math.sin(ry);
    crym[2][0] = Math.sin(ry);
    crym[2][2] = Math.cos(ry);
    //yaw
    crzm[0][0] = Math.cos(rz);
    crzm[0][1] = Math.sin(rz);
    crzm[1][0] = -Math.sin(rz);
    crzm[1][1] = Math.cos(rz);
    cm = multiplyMatrix(crxm, crym);
    cm = multiplyMatrix(cm, crzm);
    cm = multiplyMatrix(cm, ctm);
    //projection matrix
    pm[3][2] = 1;
    pm[3][3] = 0;
    //object matrix (point)
    om[0][0] = p.getX();
    om[1][0] = p.getY();
    om[2][0] = p.getZ();
    om[3][0] = 1;
    //we have no world matrix
    //fm = multiplyMatrix(pm, cm);
    //fm = multiplyMatrix(cm, om);
    fm = multiplyMatrix(cm, om);
    return new Pixel2D( (int) fm[0][0], (int) fm[1][0], p.getColour());
  }for complete source code so far: http://williamchan.brixonsolutions.com/myworks/java/graphics3d/source.php

Answer 1

I can't remember the details of doing a perspective transformation, but what I do remember is that you add an extra column to your data to put it into a 4d space, called homogenous coordinates. Then you apply a linear transformation that maps the viewpoint into a plane in 4d. This causes all the lines that project out from the viewer to be parallel. You can then lose the 4th dimension and the resulting x and y are in perspective and the z is the distance from the viewer. The phrase to google for is "homogenous transformation".