Isoptic Curves of an Ellipse

​
animation
eccentricity of ellipse
0.8
view angle
65°
A
θ
-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
θ
. If this angle is
π/2
, the isoptic curve is called the orthoptic curve (the black dotted circle).
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.

Details

We use the equation of the ellipse with the semimajor axis equal to 1 and its eccentricity
ϵ
as the only parameter:
2
y
=(1-
2
ϵ
)(1-
2
x
)
.
If
θ
is the angle between two tangent lines (the view angle) to the ellipse starting from a point
{x,y}
, we have the equation
tan(θ)=-
2
-1+
2
y
+
2
ϵ
-
2
x
(-1+
2
ϵ
)
-2+
2
x
+
2
y
+
2
ϵ
. Converting to polar coordinates and solving for
r
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
π/2
, the other for angles >
π/2
.
Snapshot 1: the orthoptic curve of an ellipse with semimajor axis equal to 1 and eccentricity
ϵ
is a circle with radius
2-
2
ϵ
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.

External Links

Iso-Optic Curve of the Ellipse
Isoptic Curve (Wolfram MathWorld)
Orthoptic Curve (Wolfram MathWorld)
Spiric Section (Wolfram MathWorld)
Toric Section (Wolfram MathWorld)

Permanent Citation

Erik Mahieu
​
​"Isoptic Curves of an Ellipse"​
​http://demonstrations.wolfram.com/IsopticCurvesOfAnEllipse/​
​Wolfram Demonstrations Project​
​Published: February 25, 2012