Isoptic

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

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 $8t x+16t^2y=-1 \,$

$x=(4t^2-1)/8t \,$ 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

• 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)

This geometry-related article is a stub. You can help Wikipedia by expanding it.v · Categories: Curves Geometry stubs Wikimedia Foundation. 2010. Игры ⚽ Поможем решить контрольную работу Hyperpolarization Hyperpolarization (physics) Look at other dictionaries: List of curves — This is a list of curves, by Wikipedia page. See also list of curve topics, list of surfaces, Riemann surface. Algebraic curves*Cubic plane curve *Quartic plane curve *Quintic plane curve *Sextic plane curveRational curves*Ampersand curve… …   Wikipedia Hypocycloid — In geometry, a hypocycloid is a special plane curve generated by the trace of a fixed point on a small circle that rolls within a larger circle. It is comparable to the cycloid but instead of the circle rolling along a line, it rolls within a… …   Wikipedia List of mathematics articles (I) — NOTOC Ia IA automorphism ICER Icosagon Icosahedral 120 cell Icosahedral prism Icosahedral symmetry Icosahedron Icosian Calculus Icosian game Icosidodecadodecahedron Icosidodecahedron Icositetrachoric honeycomb Icositruncated dodecadodecahedron… …   Wikipedia Sinusoidal spiral — In geometry, the sinusoidal spirals are a family of curves defined by the equation in polar coordinates:r^n = a^n cos(n heta),where a is a nonzero constant and n is a rational number other than 0. With a rotation about the origin, this can also… …   Wikipedia Orthoptic — Orthopic may refer to: Orthoptic, the set of points for which two tangents of a given curve meet at a right angle, a type of isoptic. A form of eye exercise designed to correct vision Orthoptics, the diagnosis and treatment of defective eye… …   Wikipedia 18+ © Academic, 2000-2024 Contact us: Technical Support, Advertising Dictionaries export, created on PHP, Joomla, Drupal, WordPress, MODx. Mark and share Search through all dictionaries Translate… Search Internet Share the article and excerpts Direct link … Do a right-click on the link above and select “Copy Link”