CIRCLE_RENDERER(step_x) (SDL_Surface *pTarget, int iCenterX, int iCenterY, int iRadius, Uint32 iColour)
{
  int iRadiusSqr = iRadius * iRadius;

  for(int iX = 0; iX <= iRadius; ++iX)
  {
    int iY = (int)(sqrt((double)(iRadiusSqr - iX*iX)) + 0.5f); // circle formula; y^2 = r^2 - x^2
    PlotPixel(pTarget, iCenterX + iX, iCenterY + iY, iColour);
    PlotPixel(pTarget, iCenterX + iX, iCenterY - iY, iColour);
    PlotPixel(pTarget, iCenterX - iX, iCenterY + iY, iColour);
    PlotPixel(pTarget, iCenterX - iX, iCenterY - iY, iColour);
  }
}

CIRCLE_RENDERER(step_x_better) (SDL_Surface *pTarget, int iCenterX, int iCenterY, int iRadius, Uint32 iColour)
{
  int iRadiusSqr = iRadius * iRadius;
  int iY = 0;

  for(int iX = 0; iX <= iY; ++iX)
  {
    iY = (int)(sqrt((double)(iRadiusSqr - iX*iX)) + 0.5f); // circle formula; y^2 = r^2 - x^2

    PlotPixel(pTarget, iCenterX + iX, iCenterY + iY, iColour);
    PlotPixel(pTarget, iCenterX + iX, iCenterY - iY, iColour);
    PlotPixel(pTarget, iCenterX - iX, iCenterY + iY, iColour);
    PlotPixel(pTarget, iCenterX - iX, iCenterY - iY, iColour);
    PlotPixel(pTarget, iCenterX + iY, iCenterY + iX, iColour);
    PlotPixel(pTarget, iCenterX + iY, iCenterY - iX, iColour);
    PlotPixel(pTarget, iCenterX - iY, iCenterY + iX, iColour);
    PlotPixel(pTarget, iCenterX - iY, iCenterY - iX, iColour);
  }
}

CIRCLE_RENDERER(step_theta) (SDL_Surface *pTarget, int iCenterX, int iCenterY, int iRadius, Uint32 iColour)
{
  double fRadius = (double)iRadius;
  double fStep  = 1.0f / fRadius;
  double fLimit = PI / 2.0f;
  for(double fTheta = 0; fTheta < fLimit; fTheta += fStep)
  {
    int iX = (int)(cos(fTheta) * fRadius + 0.5f);
    int iY = (int)(sin(fTheta) * fRadius + 0.5f);
    PlotPixel(pTarget, iCenterX + iX, iCenterY + iY, iColour);
    PlotPixel(pTarget, iCenterX + iX, iCenterY - iY, iColour);
    PlotPixel(pTarget, iCenterX - iX, iCenterY + iY, iColour);
    PlotPixel(pTarget, iCenterX - iX, iCenterY - iY, iColour);
  }
}

CIRCLE_RENDERER(step_theta_smaller) (SDL_Surface *pTarget, int iCenterX, int iCenterY, int iRadius, Uint32 iColour)
{
  double fRadius = (double)iRadius;
  double fStep  = 0.5f / fRadius;
  double fLimit = PI / 2.0f;
  for(double fTheta = 0; fTheta < fLimit; fTheta += fStep)
  {
    int iX = (int)(cos(fTheta) * fRadius + 0.5);
    int iY = (int)(sin(fTheta) * fRadius + 0.5);
    PlotPixel(pTarget, iCenterX + iX, iCenterY + iY, iColour);
    PlotPixel(pTarget, iCenterX + iX, iCenterY - iY, iColour);
    PlotPixel(pTarget, iCenterX - iX, iCenterY + iY, iColour);
    PlotPixel(pTarget, iCenterX - iX, iCenterY - iY, iColour);
  }
}

CIRCLE_RENDERER(rotate_point) (SDL_Surface *pTarget, int iCenterX, int iCenterY, int iRadius, Uint32 iColour)
{
  // Start with a point at the top of the circle
  double fX = 0;
  double fY = (double)iRadius;
  // Calculate sin(0.5 / radius), sin(x) can be approximated to x for small x
  double fSin = 0.5f / fY;
  // Given that cos(x)^2 + sin(x)^2 = 1, estimate cos(0.5 / radius) using Newton's method
  double fTmp = fSin * fSin * 0.5 - 0.5;
  double fCos = 1 - fSin;
  fCos -= (fCos * fCos * 0.5 + fTmp) / (fSin + fCos);
  fCos -= (fCos * fCos * 0.5 + fTmp) / (fSin + fCos);

  // Repeat until rotated by 45 degrees (which is 1/8th of a circle, and 8 points are plotted per iteration)
  while(fX < fY)
  {
    // Convert X and Y to integers (rounding to nearest by adding 0.5 and rounding down)
    int iX = (int)(fX + 0.5f);
    int iY = (int)(fY + 0.5f);

    // Plot (X,Y) and the reflections of this in x and y
    PlotPixel(pTarget, iCenterX + iX, iCenterY + iY, iColour);
    PlotPixel(pTarget, iCenterX + iX, iCenterY - iY, iColour);
    PlotPixel(pTarget, iCenterX - iX, iCenterY + iY, iColour);
    PlotPixel(pTarget, iCenterX - iX, iCenterY - iY, iColour);
    // Rotate (X,Y) by 90 degrees and repeat (i.e. plot (Y,-X))
    PlotPixel(pTarget, iCenterX + iY, iCenterY + iX, iColour);
    PlotPixel(pTarget, iCenterX + iY, iCenterY - iX, iColour);
    PlotPixel(pTarget, iCenterX - iY, iCenterY + iX, iColour);
    PlotPixel(pTarget, iCenterX - iY, iCenterY - iX, iColour);

    // Rotate the point clockwise by (0.5 / radius) radians
    fTmp = fCos * fX + fSin * fY;
    fY = fCos * fY - fSin * fX;
    fX = fTmp;
  }

  // Note that not sin() nor cos() nor sqrt() are called (albiet sqrt() is estimated once)
}