Parametric Equations of the Ellipse
Differentiating with respect to the Eccentric Angle
R = Semi-Axis lying on the x-axis
r = Semi-Axis lying on the y-axis Differentiating with respect to f dy/df = d(r · sinf)/df = r · cosf dx/df = d(R · cosf)/df = R · (– sinf) |
Simpson's Rule : Estimating the Arc Length of an Ellipse
Given any two stations x_{1} > x_{2}
Semi-Axis lying on the x-axis = R Semi-Axis lying on the y-axis = r Parametric Angle at x_{1} : f_{1} = arccos (x_{1} ÷ R) Parametric Angle at x_{2} : f_{2} = arccos (x_{2} ÷ R) The interval between the angles is divided into twenty equal strips : Δf = (f_{2} – f_{1}) ÷ 2 , where f_{2} > f_{1} The values of y are determined : y_{1} = Square Root [(R sin f_{1})^{2} + (r cos f_{1})^{2}] Angle f_{1} is incremented by Δf : y_{2} = Square Root [(R sin (f_{1} + Δf))^{2} + (r cos (f_{1} + Δf))^{2}] For each successive value of y, angle f_{1} is incremented again by Δf : y_{3} = Square Root [(R sin (f_{1} + 2 × Δf))^{2} + (r cos (f_{1} + 2 × Δf))^{2}] y_{4} = Square Root [(R sin (f_{1} + 3 × Δf))^{2} + (r cos (f_{1} + 3 × Δf))^{2}] ... other terms ... y_{21} = Square Root [(R sin (f_{1} + 20 × Δf))^{2} + (r cos (f_{1} + 20 × Δf))^{2}] The values of y are multiplied and summed as per Simpson's Rule : Ellipse Arc Length = (Δf ÷ 3) × (y_{1} + 4y_{2} + 2y_{3} + 4y_{2} + 2y_{5} + ... + 2y_{19} + 4y_{20} + y_{21}) |