Isoptic

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Orthoptics of a circle, of some ellipses and hyperbolas

In the geometry of curves, an isoptic is the set of points for which two tangents of a given curve meet at a given angle. The orthoptic is the isoptic whose given angle is a right angle.

Without an invertible Gauss map, an explicit general form is impossible because of the difficulty knowing which points on the given curve pair up.

Example

Orthoptic of a parabola

Take as given the parabola (t,t²) and angle 90°. Find, first, τ such that the tangents at t and τ are orthogonal:

$(1,2t)\cdot (1,2\tau )=0\,$

$\tau =-1/4t\,$

Then find (x,y) such that

$(x-t)2t=(y-t^{2})\,$ and $(x-\tau )2\tau =(y-\tau ^{2})\,$

$2tx-y=t^{2}\,$ and $8tx+16t^{2}y=-1\,$

$x=(4t^{2}-1)/8t\,$ and $y=-1/4\,$

so the orthoptic of a parabola is its directrix.

The orthoptic of an ellipse is the director circle.

References

J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 58–59. ISBN 0-486-60288-5.

External links

Wikimedia Commons has media related to Isoptics.

v t e Differential transforms of plane curves

Unary operations	Evolute Involute Dual curve Inverse curve Parallel curve Isoptic

Unary operations defined by a point	Pedal & Contrapedal curves Negative pedal curve Pursuit curve Caustic

Unary operations defined by two points	Strophoid

Binary operations defined by a point	Roulette

Operations on a family of curves	Envelope

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