Isoptic Curves of an Ellipse

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A -isoptic curve of an ellipse (the red curve) is the geometrical locus of the points from which the ellipse can be viewed at a fixed angle . If this angle is , the isoptic curve is called the orthoptic curve (the black dotted circle).

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All isoptic curves of a circle are also circles, but the isoptic curves of other ellipses are not ellipses nor are they conic sections; they are spiric curves [1], a special case of toric sections.

The animation rotates a viewpoint along the isoptic curve together with its two tangent lines to the ellipse.

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Contributed by: Erik Mahieu (February 2012)
Open content licensed under CC BY-NC-SA


Snapshots


Details

We use the equation of the ellipse with the semimajor axis equal to 1 and its eccentricity as the only parameter: .

If is the angle between two tangent lines (the view angle) to the ellipse starting from a point , we have the equation . Converting to polar coordinates and solving for gives us the equations of the isoptics. Because of the square root, there are two solutions and two isoptic curves. One is for viewing angles smaller than , the other for angles > .

Snapshot 1: the orthoptic curve of an ellipse with semimajor axis equal to 1 and eccentricity is a circle with radius

Snapshots 2 and 3: isoptics with acute viewing angles are outside this circle; those with obtuse viewing angles are inside

Snapshot 4: the -isoptic curve of an ellipse is the ellipse itself; the two tangents coincide

References

[1] T. Dana-Picard, N. Zehavi, and G. Mann, "From Conic Intersections to Toric Intersections: The Case of the Isoptic Curves of an Ellipse," The Mathematics Enthusiast, 9(1-2), 2012 pp. 59–76.

[2] B. Odehnal. "Equioptic Curves of Conic Sections." Institute of Discrete Mathematics and Geometry. (Feb 19, 2010) www.geometrie.tuwien.ac.at/odehnal/tr200.pdf.

[3] A. Miernowski and W. Mozgawa, "On Some Geometric Conditions for Convexity of Isoptics", Rendiconti del Seminario Matematico, 55(2), 1997 pp. 93–98.



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