Konami SCC

File:Orthoptic locus of a circle, ellipses and hyperbolas.gif

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

File:Isoptic.png

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

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External links

• Mathworld
• Jan Wassenaar's Curves
• "Courbe Isoptique" at Encyclopédie des Formes Mathématiques Remarquables (in French)
• "Courbe Orthoptique" at Encyclopédie des Formes Mathématiques Remarquables (in French)