OSL - 2D Polar Coordinates

OSL
2D Polar Coordinates

Introduction

This tutorial looks at the basics of creating patterns defined within a 2D polar coordinate system. Using polar coordinates is somewhat unusual because the majority of effects applied by shaders generally depend on the direct use of 2D 'st' texture coordinates and/or the 3D coordinates of points, vectors and normals such as the global variables P, I and N. Before looking at the use of a 2D polar coordinate system this tutorial clarifies what are generically referred to as Cartesian coodinates.

2D Cartesian Coordinates

Locations within a 2D cartesian coordinate systems are measured relative to two axes that are perpendicular to each other. For example, figure 1 shows the coordinates (0.1, 0.45) of a point within a 'st' space. Although 'st' texture space uses two perpendicular axes it does not mean they must be planar. For example, 'st' texture space can be "drapped" around or over a curved surface.

Figure 1

3D Cartesian Coordinates

Figure 2 shows the coordinates (1.0, 2.0, 0.7) of a point within a 3D (xyz) cartesian coordinate space.

Figure 2

2D Polar Coordinates

Figure 3 illustrates how the location of a point, within a polar coordinate system, is specified by distance (R) and angle (theta). In the open shading language angles are measured in radians not degrees.

Figure 3

Relative to the center of the 1 x 1 polygon, shown in figure 3, the coordinates of the hilited box are,
R = 0.4
theta = 0.698 (ie. 40 degrees)
Since neither R or theta are global variables their values must be calculated by converting the 'st' (Cartesian) texture coordinates to their polar equivalents - listing 1.

Listing 1

void polar2D(float x, 
             float y, 
             float xcenter, 
             float ycenter,    
    output float R, 
    output float theta)
{
float   xdiff = x - xcenter,
        ydiff = y - ycenter;
R = sqrt(xdiff * xdiff + ydiff * ydiff);
theta = atan2(ydiff, xdiff);  // -PI to PI radians
theta = theta + M_2PI;        //  0 to 2PI radians
}

Drawing Curves

The School of Mathematics and Statistics at the University of St.Andrews, Scotland maintains a web site of "Famous Curves". Many of the curves are defined by formula that require the use of polar coordinates. For example, figure 4 uses the polar description of a curve called the "Trisectrix of Maclaurin"

Figure 4 - output of the Maclaurin shader

Listing 2 provides the code for the shader that produced the pattern shown above.

Listing 2 - maclaurin.osl

void polar2D(float x, 
             float y, 
             float xcenter, 
             float ycenter,    
    output float R, 
    output float theta)
{
float   xdiff = x - xcenter,
        ydiff = y - ycenter;
R = sqrt(xdiff * xdiff + ydiff * ydiff);
theta = atan2(ydiff, xdiff);  // -PI to PI radians
theta = theta + M_2PI;        //  0 to 2PI radians
}
  
shader
maclaurin(float    sc = 10         [[string help = "Scaling factor", float min = 0.01, float max = 20]],
          float    a = 2           [[string help = "Scaling coefficient", float min = 0, float max = 10]],
          float    b = 6           [[string help = "Lobes coefficient", float min = 0, float max = 10]],
          float    stroke = 0.03   [[string help = "Stroke thickness", float min = 0, float max = 1]],
          float    blur = 0.5      [[string help = "Stroke softness", float min = 0, float max = 1]],
          float    s_center = 0.5  [[string help = "Center in 's'", float min = -2, float max = 2]],
          float    t_center = 0.5  [[string help = "Center in 't'", float min = -2, float max = 2]],
          color    bakcolor = color(0,0,0.2)  [[string help = "Background color"]],
          color    patcolor = 1    [[string help = "Stroke color"]],
          float    s = 0           [[int lockgeom = 0]],
          float    t = 0           [[int lockgeom = 0]],
        output color resultRGB = 0)
{
float    R, theta;
float    ss = s * sc;
float    tt = t * sc;
  
// Convert the s,t location of the shading point 
// to polar coordinates.
polar2D(ss, tt, 0.5 * sc, 0.5 * sc, R, theta);
  
// Use the MacLaurin formula
float r = a * cos(b * theta)/cos(theta);
float blend = smoothstep(abs(R - r) - blur, abs(R - r), stroke);
resultRGB = mix(bakcolor, patcolor, blend);
}