
Introduction
This tutorial presents a relatiely simple way of drawing a regular polgon with
any number of sides (greater than two). Figure 1 illustrates the approach which
is based on the observation that the distance (dist) of a point (P0) to the red edge of
the polygon is the same when P0 and the edge are rotated so that the edge is
aligned to the vertical axis. If the radius of the inscribed circle of
the polygon is known, it is trivial to calculate the distance of P1
to the rotated edge  shown below in blue.
Evaluating the distance based on the xdifference is faster than calculating
the distance using a "pointtoline" function  refer to the tutorial
"Vectors: Shortest Distance from a
Point to a Line". Also, note the length of the polygon side is not taken
into account.
Figure 1
To calculate angle required to rotate the blue line to the red line we must know in
which "segment" the point P0 is located. First, the 'st' texture coordinates
of P0 are converted to polar coordinates. In figure 2 assume
theta is 125 degrees. Dividing theta by the "interior" angle of 72 degrees
gives a ratio of 1.74. Using the floor() function yields a value of 1.0.
Since P is in "segment" 1 an angle of,
1 x 72 + 36 degrees (segment * interior_angle + half_interior_angle )
is required to rotate P0 to P1. Assuming the inscribed radius is 0.35 and P1.x of, say
0.37, the distance is,
dist = 0.37  0.35 (abs(inscribed_radius  P1.x) )
Figure 2
If the distance is within an acceptable tolerance the original s,t location
(ie. point P0) should receive the foreground color, otherwise, it should ba
assigned the background color.
Listing 1  polygon.sl

surface
polygon( float numSides = 4;
float innerRad = 0.25;
float line = 0.02;
color bakcolor = 1;
color pencolor = color(0, 0, 0.2) )
{
//_____ Convert to polar coordinates______
// Note: positive theta is clockwise.
float x = s  0.5, y = t  0.5;
float R = sqrt(x * x + y * y);
float theta = atan(y, x); // PI to PI radians
if(theta < 0.0)
theta += 2 * PI; // 0 to 2PI radians
//________________________________________
// Find the quadrant and the angle to counterrotate the
// st location so that it's 's' distance from a "vertical"
// polygon edge.
// Note: positive rotation is anticlockwise.
float interior = (2 * PI)/numSides;
float quadrant = floor(theta/interior);
float rotation = (quadrant * interior + interior/2);
// Because the rotate() function uses 3D points the s,t are
// used to define a 3D point (shadingPnt). The axis of rotation
// is defined by the center of the polygon and z_axis.
point stPnt = point(s  0.5, 0.5  t, 0);
point origin = point(0, 0, 0);
point z_axis = point(0, 0, 1);
point rotPnt = rotate(stPnt, rotation, origin, z_axis);
// Antialias the line
float blend = smoothstep(innerRad  line/2, innerRad, rotPnt[0]) *
(1  smoothstep(innerRad, innerRad + line/2, rotPnt[0]));
// Finally, apply the surface color
color surfcolor = mix(bakcolor, pencolor, blend);
Oi = Os;
Ci = Oi * Cs * surfcolor;
}

