Simpson's Rule: Estimating Ellipse Arc Lengths

Ellipse Eccentric Angle
Major Axis on x-axis : f = arccos (x/R)
Major Axis on y-axis : f = arccos (x/r)
Angular values expressed in radians
Major Axis on x-axis = 20     R = 10
Minor Axis on y-axis = 10     r = 5
x1 = 8     f1 = arccos (8/10) = .6435
x2 = 4     f2 = arccos (4/10) = 1.15928
Δf = (1.15928 – .6435)/4 = .12895
Ellipse Arc Length Formula: Major Axis on X-axis
1 × Ö ((10 sin (.6435 + 0 × .12895)) ² + (5 cos (.6435 + 0 × .12895)) ² = 7.21
4 × Ö ((10 sin (.6435 + 1 × .12895)) ² + (5 cos (.6435 + 1 × .12895)) ² = 31.38
2 × Ö ((10 sin (.6435 + 2 × .12895)) ² + (5 cos (.6435 + 2 × .12895)) ² = 16.87
4 × Ö ((10 sin (.6435 + 3 × .12895)) ² + (5 cos (.6435 + 3 × .12895)) ² = 35.81
1 × Ö ((10 sin (.6435 + 4 × .12895)) ² + (5 cos (.6435 + 4 × .12895)) ² = 9.38
(.12895/3) × (7.21 + 31.38 + 16.87 + 35.81 + 9.38) = 4.326

... Javascript Calculator : 4.32582


Ellipse Arc Length: Major Axis on X-axis
Major Axis on y-axis = 20     R = 10
Minor Axis on x-axis = 10     r = 5
x1 = 4.58258     f1 = arccos (4.58258/5) = .41151
x2 = 3     f2 = arccos (3/5) = .9273
Δf = (.9273 – .41151)/4 = .12895
Ellipse Arc Length Formula: Major Axis on Y-axis
1 × Ö ((5 sin (.41151 + 0 × .12895)) ² + (10 cos (.41151 + 0 × .12895)) ² = 9.38
4 × Ö ((5 sin (.41151 + 1 × .12895)) ² + (10 cos (.41151 + 1 × .12895)) ² = 35.81
2 × Ö ((5 sin (.41151 + 2 × .12895)) ² + (10 cos (.41151 + 2 × .12895)) ² = 16.87
4 × Ö ((5 sin (.41151 + 3 × .12895)) ² + (10 cos (.41151 + 3 × .12895)) ² = 31.38
1 × Ö ((5 sin (.41151 + 4 × .12895)) ² + (10 cos (.41151 + 4 × .12895)) ² = 7.21
(.12895/3) × (9.38 + 35.81 + 16.87 + 31.38 + 7.21) = 4.326

... Javascript Calculator : 4.32582


Ellipse Arc Length: Major Axis on Y-axis

Joe Bartok