Isoptic

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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.

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Example

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

<math>(1,2t)\cdot(1,2\tau)=0 \,</math>
<math>\tau=-1/4t \,</math>

Then find (x,y) such that

<math>(x-t)2t=(y-t^2) \,</math> and <math>(x-\tau)2\tau=(y-\tau^2) \,</math>
<math>2tx-y=t^2 \,</math> and <math>8t x+16t^2y=-1 \,</math>
<math>x=(4t^2-1)/8t \,</math> and <math>y=-1/4 \,</math>

so the orthoptic of a parabola is its directrix.

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External links

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