Evolute of an Ellipse
Trace the centers of an ellipse’s osculating circles and discover the cusps of its evolute.
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Centers of Curvature of an Ellipse
The evolute of a curve is the locus of all its centers of curvature. For an ellipse, the radius of curvature changes as you move along the curve, being smallest at the major vertices and largest at the minor vertices. This variation traces out an astroid-like shape known as the evolute.
Parametrization and Curvature
Consider an ellipse parametrized by angle θ:
P(θ) = (a cos θ, b sin θ)
For a regular parametric curve, the curvature radius is R = ((x′)2 + (y′)2)3/2 / |x′y″ − y′x″|. Substituting the ellipse derivatives gives:
R = (a2 sin2 θ + b2 cos2 θ)3/2 / (ab)
The Center of Curvature
The center of curvature C is located along the inward normal vector at distance R from P. The normal vector points towards the concave side of the ellipse. The coordinates of the center of curvature are:
C(θ) = ( (a2 − b2) cos3 θ / a, (b2 − a2) sin3 θ / b )
As θ varies from 0 to 2π, C(θ) traces out the evolute. For unequal semiaxes the evolute has four cusps, corresponding to the four vertices. For equal semiaxes all centers coincide at the origin.
Worked example: Diagonal point of a 2×1 ellipse
Let Horizontal semiaxis a = 2, Vertical semiaxis b = 1, and Angle theta / pi = 0.25 (which corresponds to θ = π/4). We compute the coordinates, radius of curvature, and the center of curvature.
- Evaluate trigonometric functions at θ = π/4: cos(π/4) = 1/√2, sin(π/4) = 1/√2.
- Point on the ellipse: P = (2(1/√2), 1(1/√2)) = (√2, 1/√2) ≈ (1.414214, 0.707107).
- Numerator for R: 22 sin2(π/4) + 12 cos2(π/4) = 4(1/2) + 1(1/2) = 5/2 = 2.5.
- Radius of curvature: R = (2.5)3/2 / (2 · 1) ≈ 3.952847 / 2 ≈ 1.976424.
- Curvature: κ = 1 / R ≈ 0.505964.
- Center of curvature x-coordinate: (4 − 1)(1/√2)3 / 2 = 3(1/(2√2)) / 2 = 3 / (4√2) ≈ 0.530330.
- Center of curvature y-coordinate: (1 − 4)(1/√2)3 / 1 = -3(1/(2√2)) = -3 / (2√2) ≈ -1.060660.
Try the evolute experiment
Open the interactive Evolute of an Ellipse module to see the osculating circle and its center change dynamically as you trace the ellipse.
- Set Horizontal semiaxis a to 3 and Vertical semiaxis b to 2. Compare the ellipse and its four-cusp evolute. A center of curvature can lie outside the ellipse; at Angle theta / pi = 0.5 the center is (0, -2.5).
- Change Angle theta / pi from 0 to 0.5. At θ = 0 (major axis vertex), the osculating circle is small and fits snugly inside the narrow end. At Angle theta / pi = 0.5, or θ = π/2 (minor axis vertex), the circle is much larger, matching the flatter top edge of the ellipse.
- Observe how the center C traces the rose-colored evolute curve. As θ moves from 0 to 0.5, C travels from the cusp on the positive x-axis to the cusp on the negative y-axis.
- Set Horizontal semiaxis a to 2 and Vertical semiaxis b to 2. The ellipse becomes a circle. The radius of curvature is constantly 2, and the center of curvature C collapses to a single point at the origin (0, 0).
Limitations of the model
This ideal-ellipse model evaluates analytic formulas with floating-point arithmetic. The evolute drawing uses discrete straight segments, so the rendered curve is an approximation even when the analytic coordinates are known. For a = b, the evolute is the well-defined single point at the origin and every osculating circle coincides with the original circle. Positive semiaxes keep the curvature radius finite throughout the approved controls.
References
The formulas for the center of curvature and the evolute of an ellipse are standard results in differential geometry and calculus.
- Apollonius's Ellipse and Evolute Revisited - The Evolute — Mathematical Association of America (MAA) article explaining the geometric derivation of the evolute of an ellipse.
Experience it interactively
Adjust parameters, observe in real time, and build deep intuition with Riano’s interactive Evolute of an Ellipse module.
Try Evolute of an Ellipse on Riano →