I wrote a program to find all the circles with at least three points located at the integers grid. I started with smallest boxes (width >= height) calculating the radius of a circle with points located at:
- (0, 0)
- (x, width) where 0 < x <= height
- (height, y) where 0 < y <= width
Each box gives (w+1)*(h+1) - 1 real circles, several with repeated radius (circles with 4 or more points in the lattice) and avoid the one with radius infinite. I checked all circles with width <= W, avoid repetitions and sort them. Then I find the smallest radius for boxes of width W+1 and reject from the sorted list all the findings greater than W+1 smallest. This way I assure my reduced list has no missing circles.
First circle's radii (squared) I got are:
1/2, 1/1, 5/4, 25/18, 25/16, 2/1, 5/2, 25/9...
Each circle and radius is identified by four integers (width, height, x, y), width >= height, x >= width, y >= height.
My javascript program and HTML/SVG results can be seen here: jsfiddle
- Has this sequence already a name or another use?
- Is my procedure correct (no missing circles)?