8

between different primitive forms, that is, from a cube to a cylinder and from a sphere to a rhomboid. For example, in the preceding equations, if rx = ry = rz = 10 and n=1 and e=1, then we have a sphere of radius 10 (see the first set of equations above). In the following code, we show the algorithm for creating superquadrics:

int samples = 20; float a1 = 10., a2 = 10., a3 = 10.; float u1 = 0., u2 = 20., v1 = 0., v2 = 20.; float dU = (u2 - u1) / samples; float dV = (v2 - v1) / samples; float n = 1., e = 1.; void setup(){ size(500,500, P3D); //setup the screen camera(-20,20,-20,0,0,0,0,0,1); //get a viewpoint float u = u1; for(int i=0; i<samples; i++){ float v = v1; for(int j=0; j<samples; j++){ float x = a1 * sqCos (u, n) * sqCos (v, e); float y = a2 * sqCos (u, n) * sqSin (v, e); float z = a3 * sqSin (u, n); point(x,y,z); v += dV; } u += dU; } } float sign ( float x ) { if ( x < 0 )return -1; if ( x > 0 )return 1; return 0; } float sqSin( float v, float n ) { return sign(sin(v)) * pow(abs(sin(v)),n); } float sqCos( float v, float n ) { return sign(cos(v)) * pow(abs(cos(v)),n); }

which is a cube because we start with n = e = 0. The chart shown in Figure 8-18 illustrates how a superquadric transforms for different values of n and e. The behavior of a superquadric for negative or large values of n and e can be quite interesting, as shown in Figure 8-19.

8

Summary

In this chapter, you have been introduced to the concept of 3D space. At this point, you should be able to define a point and project it using an orthographic or perspective method. Also, you saw how to use Processing graphic commands

8

to place points, lines, and boxes in 3D space and how to select them. We also introduced the concept of parameterization, and we showed how to create a sphere and a superquadric. The purpose was to show the use of parametric equations in 3D space.

1. Using the analytical equation of a sphere, that is, x2 + y2 + z2 = r2, construct a sphere that has a radius of 10. 2. A superhyperboloid is defined by the following parametric equations:

x = rx sec^n(v) cos^e(u) y = ry sec^n(v) sin^e(u) z = rz tan^n(v) u = [0, 2*Pi) v = [-Pi/2, Pi/2]

Modify the existing superquadric code to handle superhyperboloids. Explore the different values of e and n. 3. Write the code that will produce the following pattern of 36 1 1 1 cubes in a circular arrangement of a radius of 10 units:

8

size(500,500, P3D); camera(-5,5,-20,0,0,0,0,0,1); for(int phi=0; phi<360; phi+=10){ float x = float y = }

4. Write the code that will produce a spiral staircase as shown here:

5. Create a spiral curve, using the curveVertex() command (try to use as few lines of code as possible):

8

6. In a perspective projection, we use the following equations to extract the x and y coordinates on the screen: xp = x t, y p = y t where t= 1 1+ z/d

Modify and implement the following equations to project on the screen: x p = x sin(t) + y cos(t) y p = y sin(t) + x cos(t)

So far, you have dealt only with points or sets of points in a 3D space. You have seen how to project points on a 2D plane of sight and also how to arrange discrete points in the formation of cubes, spheres, and superquadrics. This chapter shows you how to construct more complex objects such as faces, solids, and groups of solids. You will also look into the identity and memory of these objects, that is, selecting and transforming their position. The next sections define objects as collections of faces and faces as collections of points. This structural arrangement enables you to store information in a hierarchical way and to get access to subelements of objects by, for example, selecting a face or moving a vertex or a face within an object.