Calculation of Ellipse Arc Length
Calculation of Ellipse Arc Length

Parametric Equations of the Ellipse
Parametric Equations of Ellipse with respect to the Eccentric Angle
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)
Arc Length of Ellipse
Prismoidal Formula
Simpson's Rule
Simpson's Rule : Estimating the Arc Length of an Ellipse
Given any two stations x1 > x2
Semi-Axis lying on the x-axis = R
Semi-Axis lying on the y-axis = r
Parametric Angle at x1 : f1 = arccos (x1 ÷ R)
Parametric Angle at x2 : f2 = arccos (x2 ÷ R)
The interval between the angles is divided into twenty equal strips :
Δf = (f2f1) ÷ 2 , where f2 > f1
The values of y are determined :
y1 = Square Root [(R sin f1)2 + (r cos f1)2]
Angle f1 is incremented by Δf :
y2 = Square Root [(R sin (f1 + Δf))2 + (r cos (f1 + Δf))2]
For each successive value of y, angle f1 is incremented again by Δf :
y3 = Square Root [(R sin (f1 + 2 × Δf))2 + (r cos (f1 + 2 × Δf))2]
y4 = Square Root [(R sin (f1 + 3 × Δf))2 + (r cos (f1 + 3 × Δf))2]
... other terms ...
y21 = Square Root [(R sin (f1 + 20 × Δf))2 + (r cos (f1 + 20 × Δf))2]

The values of y are multiplied and summed as per Simpson's Rule :
Ellipse Arc Length =
f ÷ 3) × (y1 + 4y2 + 2y3 + 4y2 + 2y5 + ... + 2y19 + 4y20 + y21)
Arc Length of Ellipse
Scope : First Quadrant, 0 £ x £ R
Semi-Axis lying on the x-axis , R =
Semi-Axis lying on the y-axis ,  r =
( x1 > x2 )   ...   x1 =
x2 =

f1 =
( f2 > f1 )   ...   f2 =
Δf = ( f2f1 ) ÷ 20 =
Δf ÷ 3 =
f1 + Δf yn Multiplied yn
× 1 =
× 4 =
× 2 =
× 4 =
× 2 =
× 4 =
× 2 =
× 4 =
× 2 =
× 4 =
× 2 =
× 4 =
× 2 =
× 4 =
× 2 =
× 4 =
× 2 =
× 4 =
× 2 =
× 4 =
× 1 =
Sum of Multiplied yn =
f ÷ 3) × ( Sum of Multiplied yn ) = Arc Length

Intercept and General Forms of Ellipse Equations
Intercept and General Forms of Ellipse Equations
As the value of x approaches the value of the Semi-Axis lying on the x-axis, R, the divisor in the formula above approaches zero, returning an absurd result for the Ellipse Arc Length.

Arc Length of Ellipse

Note the Ellipse Arc Length as x approaches R

Sample Data: Ellipses revolved about Reference Axes
Joe Bartok
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