Arc-Polar Coordinates

original example by Karsten Schmidt
toxiclibs.js by Kyle Phillips

Quick demo & test for working with 2D polar coordinates, interpolation and circle segments. Might be useful for some…

/*
 * Copyright (c) 2006-2009 Karsten Schmidt
 *
 * This library is free software; you can redistribute it and/or
 * modify it under the terms of the GNU Lesser General Public
 * License as published by the Free Software Foundation; either
 * version 2.1 of the License, or (at your option) any later version.
 *
 * http://creativecommons.org/licenses/LGPL/2.1/
 *
 * This library is distributed in the hope that it will be useful,
 * but WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
 * Lesser General Public License for more details.
 *
 * You should have received a copy of the GNU Lesser General Public
 * License along with this library; if not, write to the Free Software
 * Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA  02110-1301  USA
 */

scope the Vec2D object locally so that you can refer to it the way you naturally would in java/processing

var Vec2D = toxi.geom.Vec2D;

float theta=0;
float innerRadius=150;
float outerRadius=180;
int numSteps=10;

void setup() {
  size(640,380);
}

void draw() {
  background(200);
  translate(width/2,height/2);

¶

translate mouse position into polar coordinates in the polar space the vector components are interpreted as: x = radius y = angle
```
  Vec2D mousePos=new Vec2D(mouseX,mouseY).sub(width/2,height/2).toPolar();
```

ensure we always interpolate the angle over the smaller difference

  if (abs(theta-mousePos.y)>PI) {
    if (theta>mousePos.y) {
      theta-=TWO_PI;
    }
    else {
      mousePos.y-=TWO_PI;
    }
  }

¶

interpolate to the new angle, adaptive speed based on current velocity
```
  theta+=(mousePos.y-theta)*abs(mousePos.y-theta)*0.2;
```
¶

avoid “over-spinning”
```
  theta%=TWO_PI;
```

create the arc as tri strip

  noStroke();
  beginShape(TRIANGLE_STRIP);
  Vec2D p=new Vec2D();
  for(float i=0,t=theta-PI*0.25; i<numSteps; i++) {

¶

convert theta back into cartesian coordinate space the radius of 1 means the resulting vector will be normalized
```
    p.set(1,t).toCartesian();
```

scale point to inner/outer radius

    vertex(p.x*innerRadius,p.y*innerRadius);
    vertex(p.x*outerRadius,p.y*outerRadius);
    t+=HALF_PI/numSteps;
  }
  endShape();

draw comparison vector to mouse vs. arc positions

  mousePos.toCartesian();
  stroke(255,0,0);
  line(0,0,mousePos.x,mousePos.y);
  Vec2D arcPos=new Vec2D(innerRadius,theta).toCartesian();
  stroke(0,0,255);
  line(0,0,arcPos.x,arcPos.y);
}