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A perspective projection is defined by a center of projection and a plane of projection . The projector rays connect the points in the scene with the center of projection, thus highlighting thecorresponding points in the plane of projection. The [link] shows a section where the plane of projection produces a straight line whose abscissa is $-d$ , and the center of projection is in the origin.
By similarity of two triangles it is easy to realize that the point having ordinate $y$ gets projected onto the plane in the point having ordinate ${y}_{\mathrm{p}}=-\left(\frac{yd}{z}\right)$ .In general, the projection of a point having homogeneous coordinates $\left(\begin{array}{c}x\\ y\\ z\\ 1\end{array}\right)$ onto a plane orthogonal to the $z$ axis and intersecting such axis in position $-d$ is obtained, in homogeneous coordinates, by multiplication with the matrix $\begin{pmatrix}1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & -\left(\frac{1}{d}\right) & 0\\ \end{pmatrix}$ . The projected point becomes $\left(\begin{array}{c}x\\ y\\ z\\ -\left(\frac{z}{d}\right)\end{array}\right)$ , which can be normalized by multiplication of all its element by $-\left(\frac{d}{z}\right)$ . As a result, we obtain $\left(\begin{array}{c}-\left(\frac{xd}{z}\right)\\ -\left(\frac{yd}{z}\right)\\ -d\\ 1\end{array}\right)$
Parallel views are obtained by taking the center of projection back to infinity ( $$∞ ). In this way, the projector rays are all parallel.
The orthographic projection produces a class of parallel views by casting projection rays orthogonal to the planeof projection. If such plane is positioned orthogonally to the $z$ axis and passing by the origin, the projection matrix turns out to beparticolarly simple: $\begin{pmatrix}1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ \end{pmatrix}$ . Among orthographic projections, the axonometric projections are based on the possibility to measure the object along three orthogonalaxes, and on the orientation of the plane of projection with respect to these axes. In particular, in the isometric projection the projections of the axes form angles of $120\xb0$ . The isometric projection has the property that equal segments on the three axes remain equal when theyare projected onto the plane. In order to obtain the isometric projection of an object whose main axes areparallel to the coordinate axes, we can first rotate the object by $45\xb0$ about the $y$ axis, and then rotate by $\arctan \left(\frac{1}{\sqrt{2}}\right)=35.264\xb0$ about the $x$ axis.
We can talk about oblique projection every time the projector rays are oblique (non-orthogonal) to the projection plane. In order to deviate the projector rays from the normal direction by the angles $\theta $ and $\phi $ we must use a projection matrix $\begin{pmatrix}1 & 0 & -\tan \theta & 0\\ 0 & 1 & -\tan \phi & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ \end{pmatrix}$
As we have seen, Processing has a local illumination model, thus being impossible to cast shadows directly. However, bymanipulating the affine transformation matrices we can cast shadows onto planes. The method is called flashing in the eye , thus meaning that the optical center of the scene is moved to the point where the light source ispositioned, and then a perspective transformation is made, with a plane of projection that coincides with the planewhere we want to cast the shadow on.
The following program projects on the floor the shadow produced by a light source positioned on the $y$ axis. The result is shown in [link]
size(200, 200, P3D);
float centro = 100;float yp = 70; //floor (plane of projection) distance from center
float yl = 40; //height of light (center of projection) from centertranslate(centro, centro, 0); //center the world on the cube
noFill();box(yp*2); //draw of the room
pushMatrix();fill(250); noStroke();
translate(0, -yl, 0); // move the virtual light bulb highersphere(4); //draw of the light bulb
stroke(10);popMatrix();
pushMatrix(); //draw of the wireframe cubenoFill();
rotateY(PI/4); rotateX(PI/3);box(20);
popMatrix();// SHADOW PROJECTION BY COMPOSITION
// OF THREE TRANSFORMATIONS (the first one in// the code is the last one to be applied)
translate(0, -yl, 0); // shift of the light source and the floor back// to their place (see the translation below)
applyMatrix(1, 0, 0, 0,0, 1, 0, 0,
0, 0, 1, 0,0, 1/(yp+yl), 0, 0); // projection on the floor
// moved down by yltranslate(0, yl, 0); // shift of the light source to center
// and of the floor down by ylpushMatrix(); // draw of the cube that generate the shadow
fill(120, 50); // by means of the above transformationsnoStroke();
rotateY(PI/4); rotateX(PI/3);box(20);
popMatrix();
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