Source code in the processing programming language

float x, y;
float theta;
float rotation;

void setup() {
  size(300, 300);
  theta = 0;
  rotation = 0.1;
  background(255);
}

void draw() {
  translate(width/2.0, height/2.0);
  x = theta*cos(theta/PI); 
  y = theta*sin(theta/PI);
  point(x, y);   
  theta = theta + rotation;
  // check restart
  if (x>width/2.0) frameCount=-1;
}

float theta;
float rotation;

void setup() {
  size(300, 300);
  theta = 0;
  rotation = 0.1;
  background(255);
}

void draw() {
  translate(width/2.0, height/2.0);
  theta += rotation;
  rotate(theta/PI);
  point(theta, 0);
  // check restart
  if (theta>width/2.0) frameCount=-1;
}

PVector pv;
float rotation;

void setup() {
  size(300, 300);
  rotation = 0.1;
  pv = new PVector(rotation, 0);
  background(255);
}

void draw() {
  translate(width/2.0, height/2.0);
  pv.setMag(pv.mag()+rotation);
  println(pv.mag());
  pv.rotate(rotation/PI);
  point(pv.x, pv.y);
  // check restart
  if (pv.mag()>width/2.0) frameCount=-1;
}

float px, py, x, y;
float theta;
float rotation;

void setup() {
  size(300, 300);
  px = py = x = y = theta = 0;
  rotation = 0.1;
  background(255);
}

void draw() {
  translate(width/2.0, height/2.0);
  x = theta*cos(theta/PI); 
  y = (theta)*sin(theta/PI);
  line(x, y, px, py);
  theta = theta + rotation;
  px = x;
  py = y;
  // check restart
  if (px>width/2.0) frameCount=-1;
}

float x, y, px, py;
float theta;
float rotation;

void setup() {
  size(300, 300);
  x = y = px = py = theta = 0;
  rotation = 0.1;
  background(255);
}

void draw() {
  // find coordinates with rotating reference frame
  pushMatrix();  
  rotate(theta/PI);
  x = screenX(theta, 0);
  y = screenY(theta, 0);
  popMatrix();

  translate(width/2.0, height/2.0);
  theta += rotation;
  line(px, py, x, y);
  px = x;
  py = y;
  if (theta>width/2.0) frameCount=-1; // start over
}

theta = 0
rotation = 0.1

def setup():
    size(300, 300)
    background(255)

def draw():
    global theta
    translate(width / 2.0, height / 2.0)
    x = theta * cos(theta / PI)
    y = theta * sin(theta / PI)
    point(x, y)
    theta = theta + rotation
    # check restart
    if x > width / 2.0:
        background(255)
        theta = 0