Given a circle
x^2+y^2=r^2
Evaluate the number of points with integer coordinates ( both x and y ) inside the circle ( i.e. points within the area enclosed by the curve)
(In reply to
for integral r only (spoiler) by Charlie)
Actually, the algorithm, and the formula from OEIS, seem to work well even for non-integers, and that seems intuitive as the formula accounts for the integral nature of the grid by using the floor function.
Running this though give me an example of the importance of noting the possibility of rounding errors not taking consideration of points exactly on the circumference of the circle. Initially I was counting by .01 for the below, but integral r was not showing the points at the end of each radius as being exactly on the circle. I changed to use a variable that incremented by 1 each time but went to 1000, and divided by 100. That fixed the rounding problem for thee, but there may be other points exactly on the circumference that this does not fix.
r interior circumference total OEIS formula
0.30 1 0 1 1
0.31 1 0 1 1
0.32 1 0 1 1
0.33 1 0 1 1
0.34 1 0 1 1
0.35 1 0 1 1
0.36 1 0 1 1
0.37 1 0 1 1
0.38 1 0 1 1
0.39 1 0 1 1
0.40 1 0 1 1
0.41 1 0 1 1
0.42 1 0 1 1
0.43 1 0 1 1
0.44 1 0 1 1
0.45 1 0 1 1
0.46 1 0 1 1
0.47 1 0 1 1
0.48 1 0 1 1
0.49 1 0 1 1
0.50 1 0 1 1
0.51 1 0 1 1
0.52 1 0 1 1
0.53 1 0 1 1
0.54 1 0 1 1
0.55 1 0 1 1
0.56 1 0 1 1
0.57 1 0 1 1
0.58 1 0 1 1
0.59 1 0 1 1
0.60 1 0 1 1
0.61 1 0 1 1
0.62 1 0 1 1
0.63 1 0 1 1
0.64 1 0 1 1
0.65 1 0 1 1
0.66 1 0 1 1
0.67 1 0 1 1
0.68 1 0 1 1
0.69 1 0 1 1
0.70 1 0 1 1
0.71 1 0 1 1
0.72 1 0 1 1
0.73 1 0 1 1
0.74 1 0 1 1
0.75 1 0 1 1
0.76 1 0 1 1
0.77 1 0 1 1
0.78 1 0 1 1
0.79 1 0 1 1
0.80 1 0 1 1
0.81 1 0 1 1
0.82 1 0 1 1
0.83 1 0 1 1
0.84 1 0 1 1
0.85 1 0 1 1
0.86 1 0 1 1
0.87 1 0 1 1
0.88 1 0 1 1
0.89 1 0 1 1
0.90 1 0 1 1
0.91 1 0 1 1
0.92 1 0 1 1
0.93 1 0 1 1
0.94 1 0 1 1
0.95 1 0 1 1
0.96 1 0 1 1
0.97 1 0 1 1
0.98 1 0 1 1
0.99 1 0 1 1
1.00 1 4 5 5
1.01 5 0 5 5
1.02 5 0 5 5
1.03 5 0 5 5
1.04 5 0 5 5
1.05 5 0 5 5
1.06 5 0 5 5
1.07 5 0 5 5
1.08 5 0 5 5
1.09 5 0 5 5
1.10 5 0 5 5
1.11 5 0 5 5
1.12 5 0 5 5
1.13 5 0 5 5
1.14 5 0 5 5
1.15 5 0 5 5
1.16 5 0 5 5
1.17 5 0 5 5
1.18 5 0 5 5
1.19 5 0 5 5
1.20 5 0 5 5
1.21 5 0 5 5
1.22 5 0 5 5
1.23 5 0 5 5
1.24 5 0 5 5
1.25 5 0 5 5
1.26 5 0 5 5
1.27 5 0 5 5
1.28 5 0 5 5
1.29 5 0 5 5
1.30 5 0 5 5
1.31 5 0 5 5
1.32 5 0 5 5
1.33 5 0 5 5
1.34 5 0 5 5
1.35 5 0 5 5
1.36 5 0 5 5
1.37 5 0 5 5
1.38 5 0 5 5
1.39 5 0 5 5
1.40 5 0 5 5
1.41 5 0 5 5
1.42 9 0 9 9
1.43 9 0 9 9
1.44 9 0 9 9
1.45 9 0 9 9
1.46 9 0 9 9
1.47 9 0 9 9
1.48 9 0 9 9
1.49 9 0 9 9
1.50 9 0 9 9
1.51 9 0 9 9
1.52 9 0 9 9
1.53 9 0 9 9
1.54 9 0 9 9
1.55 9 0 9 9
1.56 9 0 9 9
1.57 9 0 9 9
1.58 9 0 9 9
1.59 9 0 9 9
1.60 9 0 9 9
1.61 9 0 9 9
1.62 9 0 9 9
1.63 9 0 9 9
1.64 9 0 9 9
1.65 9 0 9 9
1.66 9 0 9 9
1.67 9 0 9 9
1.68 9 0 9 9
1.69 9 0 9 9
1.70 9 0 9 9
1.71 9 0 9 9
1.72 9 0 9 9
1.73 9 0 9 9
1.74 9 0 9 9
1.75 9 0 9 9
1.76 9 0 9 9
1.77 9 0 9 9
1.78 9 0 9 9
1.79 9 0 9 9
1.80 9 0 9 9
1.81 9 0 9 9
1.82 9 0 9 9
1.83 9 0 9 9
1.84 9 0 9 9
1.85 9 0 9 9
1.86 9 0 9 9
1.87 9 0 9 9
1.88 9 0 9 9
1.89 9 0 9 9
1.90 9 0 9 9
1.91 9 0 9 9
1.92 9 0 9 9
1.93 9 0 9 9
1.94 9 0 9 9
1.95 9 0 9 9
1.96 9 0 9 9
1.97 9 0 9 9
1.98 9 0 9 9
1.99 9 0 9 9
2.00 9 4 13 13
2.01 13 0 13 13
2.02 13 0 13 13
2.03 13 0 13 13
2.04 13 0 13 13
2.05 13 0 13 13
2.06 13 0 13 13
2.07 13 0 13 13
2.08 13 0 13 13
2.09 13 0 13 13
2.10 13 0 13 13
2.11 13 0 13 13
2.12 13 0 13 13
2.13 13 0 13 13
2.14 13 0 13 13
2.15 13 0 13 13
2.16 13 0 13 13
2.17 13 0 13 13
2.18 13 0 13 13
2.19 13 0 13 13
2.20 13 0 13 13
2.21 13 0 13 13
2.22 13 0 13 13
2.23 13 0 13 13
2.24 21 0 21 21
2.25 21 0 21 21
2.26 21 0 21 21
2.27 21 0 21 21
2.28 21 0 21 21
2.29 21 0 21 21
2.30 21 0 21 21
2.31 21 0 21 21
2.32 21 0 21 21
2.33 21 0 21 21
2.34 21 0 21 21
2.35 21 0 21 21
2.36 21 0 21 21
2.37 21 0 21 21
2.38 21 0 21 21
2.39 21 0 21 21
2.40 21 0 21 21
2.41 21 0 21 21
2.42 21 0 21 21
2.43 21 0 21 21
2.44 21 0 21 21
2.45 21 0 21 21
2.46 21 0 21 21
2.47 21 0 21 21
2.48 21 0 21 21
2.49 21 0 21 21
2.50 21 0 21 21
2.51 21 0 21 21
2.52 21 0 21 21
2.53 21 0 21 21
2.54 21 0 21 21
2.55 21 0 21 21
2.56 21 0 21 21
2.57 21 0 21 21
2.58 21 0 21 21
2.59 21 0 21 21
2.60 21 0 21 21
2.61 21 0 21 21
2.62 21 0 21 21
2.63 21 0 21 21
2.64 21 0 21 21
2.65 21 0 21 21
2.66 21 0 21 21
2.67 21 0 21 21
2.68 21 0 21 21
2.69 21 0 21 21
2.70 21 0 21 21
2.71 21 0 21 21
2.72 21 0 21 21
2.73 21 0 21 21
2.74 21 0 21 21
2.75 21 0 21 21
2.76 21 0 21 21
2.77 21 0 21 21
2.78 21 0 21 21
2.79 21 0 21 21
2.80 21 0 21 21
2.81 21 0 21 21
2.82 21 0 21 21
2.83 25 0 25 25
2.84 25 0 25 25
2.85 25 0 25 25
2.86 25 0 25 25
2.87 25 0 25 25
2.88 25 0 25 25
2.89 25 0 25 25
2.90 25 0 25 25
2.91 25 0 25 25
2.92 25 0 25 25
2.93 25 0 25 25
2.94 25 0 25 25
2.95 25 0 25 25
2.96 25 0 25 25
2.97 25 0 25 25
2.98 25 0 25 25
2.99 25 0 25 25
3.00 25 4 29 29
3.01 29 0 29 29
3.02 29 0 29 29
3.03 29 0 29 29
3.04 29 0 29 29
3.05 29 0 29 29
3.06 29 0 29 29
3.07 29 0 29 29
3.08 29 0 29 29
3.09 29 0 29 29
3.10 29 0 29 29
3.11 29 0 29 29
3.12 29 0 29 29
3.13 29 0 29 29
3.14 29 0 29 29
3.15 29 0 29 29
3.16 29 0 29 29
3.17 37 0 37 37
3.18 37 0 37 37
3.19 37 0 37 37
3.20 37 0 37 37
3.21 37 0 37 37
3.22 37 0 37 37
3.23 37 0 37 37
3.24 37 0 37 37
3.25 37 0 37 37
3.26 37 0 37 37
3.27 37 0 37 37
3.28 37 0 37 37
3.29 37 0 37 37
3.30 37 0 37 37
3.31 37 0 37 37
3.32 37 0 37 37
3.33 37 0 37 37
3.34 37 0 37 37
3.35 37 0 37 37
3.36 37 0 37 37
3.37 37 0 37 37
3.38 37 0 37 37
3.39 37 0 37 37
3.40 37 0 37 37
3.41 37 0 37 37
3.42 37 0 37 37
3.43 37 0 37 37
3.44 37 0 37 37
3.45 37 0 37 37
3.46 37 0 37 37
3.47 37 0 37 37
3.48 37 0 37 37
3.49 37 0 37 37
3.50 37 0 37 37
3.51 37 0 37 37
3.52 37 0 37 37
3.53 37 0 37 37
3.54 37 0 37 37
3.55 37 0 37 37
3.56 37 0 37 37
3.57 37 0 37 37
3.58 37 0 37 37
3.59 37 0 37 37
3.60 37 0 37 37
3.61 45 0 45 45
3.62 45 0 45 45
3.63 45 0 45 45
3.64 45 0 45 45
3.65 45 0 45 45
3.66 45 0 45 45
3.67 45 0 45 45
3.68 45 0 45 45
3.69 45 0 45 45
3.70 45 0 45 45
3.71 45 0 45 45
3.72 45 0 45 45
3.73 45 0 45 45
3.74 45 0 45 45
3.75 45 0 45 45
3.76 45 0 45 45
3.77 45 0 45 45
3.78 45 0 45 45
3.79 45 0 45 45
3.80 45 0 45 45
3.81 45 0 45 45
3.82 45 0 45 45
3.83 45 0 45 45
3.84 45 0 45 45
3.85 45 0 45 45
3.86 45 0 45 45
3.87 45 0 45 45
3.88 45 0 45 45
3.89 45 0 45 45
3.90 45 0 45 45
3.91 45 0 45 45
3.92 45 0 45 45
3.93 45 0 45 45
3.94 45 0 45 45
3.95 45 0 45 45
3.96 45 0 45 45
3.97 45 0 45 45
3.98 45 0 45 45
3.99 45 0 45 45
4.00 45 4 49 49
4.01 49 0 49 49
4.02 49 0 49 49
4.03 49 0 49 49
4.04 49 0 49 49
4.05 49 0 49 49
4.06 49 0 49 49
4.07 49 0 49 49
4.08 49 0 49 49
4.09 49 0 49 49
4.10 49 0 49 49
4.11 49 0 49 49
4.12 49 0 49 49
4.13 57 0 57 57
4.14 57 0 57 57
4.15 57 0 57 57
4.16 57 0 57 57
4.17 57 0 57 57
4.18 57 0 57 57
4.19 57 0 57 57
4.20 57 0 57 57
4.21 57 0 &n
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Posted by Charlie
on 2023-06-12 13:35:10 |
re: for integral r only (spoiler)
