Calculating and storing pixelated ellipse

Question

I was wondering if it is possible to create a function (arbitrary of language) that has as input a width and height.

This function would then calculate the biggest ellipse that would fit inside of the dimensions that it is given, and store this in a matrix such as these two examples;

In the left example, the width is 14 and height is 27, where the white part is the ellipse.

In the right example, the width is 38 and height is 21, where, once again, the white part is the ellipse.

Of course, the black and white parts can be seen as true/false values if they are part of the ellipse or not.

Answer 1

Yes it is possible. The process is called ellipse rasterization. Here few methods to do so:

let our image has xs,ys resolution so center (x0,y0) and semiaxises a,b are:

x0=xs/2
y0=y2/2
a =x0-1
b =y0-1 

1. using ellipse equation

   so 2 nested for loops + if condition deciding if you are inside or outside ellipse.

   for (y=0;y<ys;y++)
    for (x=0;x<xs;x++)
     if (((x-x0)*(x-x0)/(a*a))+((y-y0)*(y-y0)/(b*b))<=1.0) pixel[y][x]=color_inside;
      else pixel[y][x]=color_outside;
   

   You can optimize this quite a lot by pre-computing the parts of the equations only if thy change so some are computed just once others on each x iteration and the rest on each y iteration. Also is better to multiply instead of dividing.

2. using parametric ellipse equation

   x(t) = x0 + a*cos(t)
   y(t) = y0 + b*sin(t)
   t = <0,2.0*M_PI>  // for whole ellipse
   

   so one for loop creating quadrant coordinates and filling lines inside and outside for the 3 mirrors of the quadrant using only horizontal or only vertical lines. However this approach need a buffer to store the circumference points of one quadrant.

3. Using Bresenham ellipse algorithm

4. Using any Circle algorithm and stretch to ellipse

   so simply use square area of size of the lesser resolution from xs,ys render circle and than stretch back to xs,ys. If you do not stretch during rasterization than you might create artifacts. In such case is better to use the bigger resolution and stretch down but that is slower of coarse.