Figure 8-15: A cubical formation of points created by looping in three directions in .NET

Printing QR Code in .NET Figure 8-15: A cubical formation of points created by looping in three directions
Figure 8-15: A cubical formation of points created by looping in three directions
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8.6.2 Spherical Formations
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As 2 discussed, the parametric equations of a circle are:
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x=r*cos(theta), y=r*sin(theta) theta is the parameter that changes, whereas r is the radius of the circle. The parametric equations of a circle in a three-dimensional world should be exactly the same, including a z coordinate that will be 0 if the circle is placed on the x-y plane. Consider the following:
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x = v cos(u) y = v sin(u) z = 0 u = [0, 2*Pi) ,
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This describes a circle of radius v in the xy plane. Suppose that we vary v from 0 to some constant radius r. We would obtain a series of concentric circles in the xy plane. These circles correspond to what would happen if we sliced a hypothetical sphere with radius r perpendicular to the z-axis. For some v, we see that such a sphere has a z coordinate equal to sqrt(r2 v2). It follows that these equations
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x = v cos(u) y = v sin(u) z = sqrt(r2-v2) u = [0, 2*Pi) v = [0, r]
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are the parametric equations of a hemisphere above and including the xy plane. To get the whole sphere, we could mirror it in the xy plane and let the square root take on both positive and negative values, but this is not very elegant. Rather, consider replacing v with r cos(v). As this new v goes from 0 to p/2, r*sin(v) goes from 0 to r. So, let v go from p/2 to p/2, and z = sqrt(r 2 v 2)cos2(v)) = r*cos(v). Hence
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x = r sin(v) cos(u) y = r sin(v) sin(u) z = r cos(v) u = [0, 2*Pi) v = [0, Pi]
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are the parametric equations of the whole sphere of radius r in 3D. The algorithm for a parametric sphere in 3D is shown in the following source code and in Figure 8-16.
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1 2 3 4 5 6 size(500,500, P3D); //setup the screen camera(-15,15,-15,0,0,0,0,0,1); //get a viewpoint for(int i=0; i<360; i+=10) for(int j=0; j<360; j+=10){ float x = 10 * sin(radians(i)) * cos(radians(j)); float y = 10 * sin(radians(i)) * sin(radians(j));
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float z = 10 * cos(radians(i)); point(x,y,z);
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Figure 8-16: A spherical formation using sampled points
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While a point is the simplest way of visualizing the position of three coordinate numbers in 3D space, in some cases, this information is not enough. Instead, we can connect consecutive points to form segment that will appear overall as geodesic lines. This technique is shown in the following code:
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 size(500,500, P3D); //setup the screen camera(-15,15,-15,0,0,0,0,0,1); //get a viewpoint background(255); for(int i=0; i<360; i+=10) for(int j=0; j<360; j+=10){ float x = 10 * sin(radians(i)) * cos(radians(j)); float y = 10 * sin(radians(i)) * sin(radians(j)); float z = 10 * cos(radians(i)); float xn = 10 * sin(radians(i+10)) * cos(radians(j)); float yn = 10 * sin(radians(i+10)) * sin(radians(j)); float zn = 10 * cos(radians(i+10)); float xu = 10 * sin(radians(i)) * cos(radians(j+10)); float yu = 10 * sin(radians(i)) * sin(radians(j+10)); float zu = 10 * cos(radians(i)); line(x,y,z, xn,yn,zn); line(x,y,z, xu,yu,zu); }
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The result is shown in Figure 8-17.
Figure 8-17: A spherical formation using line segments connecting sampled points
8.6.3 Superquadrics
As we saw in the last section, the parametric equation of a sphere is:
x = r cos(v) cos(u) y = r cos(v) sin(u) z = r sin(v) u = [-Pi, Pi) v = [-Pi/2, Pi/2]
This representation can be seen as part of a more generalized set of representations, where the sphere is just one instance. These representations are described through the following parametric equations:
x = rx cosn(v) cose(u) y = ry cosn(v) sine(u) z = rz sinn(v) u = [-Pi, Pi) v = [-Pi/2, Pi/2]
The set of objects that are produced through such representations are called superquadrics. The interesting part of these objects is their ability to transform