Orthoptic (geometry)

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

The orthoptic of a parabola is its directrix.
Ellipse and its orthoptic (purple)
Hyperbola with its orthoptic (purple)

Examples:

  1. The orthoptic of a parabola is its directrix (proof: see parabola),
  2. The orthoptic of an ellipse x2/a2 + y2/b2 = 1 is the director circle x2 + y2 = a2 + b2 (see below),
  3. The orthoptic of a hyperbola x2/a2y2/b2 = 1, a > b, is the circle x2 + y2 = a2b2 (in case of ab there are no orthogonal tangents, see below),
  4. The orthoptic of an astroid x23 + y23 = 1 is a quadrifolium with the polar equation
(see below).

Generalizations:

  1. An isoptic is the set of points for which two tangents of a given curve meet at a fixed angle (see below).
  2. An isoptic of two plane curves is the set of points for which two tangents meet at a fixed angle.
  3. Thales' theorem on a chord PQ can be considered as the orthoptic of two circles which are degenerated to the two points P and Q.


Orthoptics (red circles) of a circle, ellipses and hyperbolas

Orthoptic of an ellipse and hyperbola[edit]

Ellipse[edit]

The ellipse with equation

can be represented by the unusual parametric representation[1]

where m is the slope of the tangent at a point of the ellipse. c+(m) describes the upper half and c(m) the lower half of the ellipse. The points a, 0)) with tangents parallel to the y-axis are excluded. But this is no problem, because these tangents meet orthogonal the tangents parallel to the x-axis in the ellipse points (0, ±b) Hence the points a, ±b) are points of the desired orthoptic (the circle x2 + y2 = a2 + b2).

The tangent at point c±(m) has the equation

If a tangent contains the point (x0, y0), off the ellipse, then the equation

holds. Eliminating the square root leads to

which has two solutions m1 and m2 corresponding to the two tangents passing (x0, y0). The free term of a reduced quadratic equation is always the product of its solutions. Hence, if the tangents meet at (x0, y0) orthogonally, the following equations hold:

The last equation is equivalent to

This means that:

  • The intersection points of orthogonal tangents are points of the circle x2 + y2 = a2 + b2.

Hyperbola[edit]

The ellipse case can be adopted nearly exactly to the hyperbola case. The only changes to be made are to replace b2 with b2 and to restrict m to |m| > b/a. Therefore:

  • The intersection points of orthogonal tangents are points of the circle x2 + y2 = a2b2, where a > b.

Orthoptic of an astroid[edit]

Orthoptic (purple) of an astroid

An astroid can be described by the parametric representation

.

From the condition

one recognizes the distance α in parameter space at which an orthogonal tangent to ċ(t) appears. It turns out that the distance is independent of parameter t, namely α = ± π/2. The equations of the (orthogonal) tangents at the points c(t) and c(t + π/2) are respectively:

Their common point has coordinates:

This is simultaneously a parametric representation of the orthoptic.

Elimination of the parameter t yields the implicit representation

Introducing the new parameter φ = t/4 one gets

(The proof uses the angle sum and difference identities.) Hence we get the polar representation

of the orthoptic. Hence:

Isoptic of a parabola, an ellipse and a hyperbola[edit]

Isoptics (purple) of a parabola for angles 80° and 100°
Isoptics (purple) of an ellipse for angles 80° and 100°
Isoptics (purple) of a hyperbola for angles 80° and 100°

Below the isotopics for angles α ≠ 90° are listed. They are called α-isoptics. For the proofs see below.

Equations of the isoptics[edit]

Parabola:

The α-isoptics of the parabola with equation y = ax2 are the branches of the hyperbola

The branches of the hyperbola provide the isoptics for the two angles α and 180° − α (see picture).

Ellipse:

The α-isoptics of the ellipse with equation x2/a2 + y2/b2 = 1 are the two parts of the degree-4 curve

(see picture).

Hyperbola:

The α-isoptics of the hyperbola with the equation x2/a2y2/b2 = 1 are the two parts of the degree-4 curve

Proofs[edit]

Parabola:

A parabola y = ax2 can be parametrized by the slope of its tangents m = 2ax:

The tangent with slope m has the equation

The point (x0, y0) is on the tangent if and only if

This means the slopes m1, m2 of the two tangents containing (x0, y0) fulfil the quadratic equation

If the tangents meet at angle α or 180° − α, the equation

must be fulfilled. Solving the quadratic equation for m, and inserting m1, m2 into the last equation, one gets

This is the equation of the hyperbola above. Its branches bear the two isoptics of the parabola for the two angles α and 180° − α.

Ellipse:

In the case of an ellipse x2/a2 + y2/b2 = 1 one can adopt the idea for the orthoptic for the quadratic equation

Now, as in the case of a parabola, the quadratic equation has to be solved and the two solutions m1, m2 must be inserted into the equation

Rearranging shows that the isoptics are parts of the degree-4 curve:

Hyperbola:

The solution for the case of a hyperbola can be adopted from the ellipse case by replacing b2 with b2 (as in the case of the orthoptics, see above).

To visualize the isoptics, see implicit curve.

External links[edit]

Notes[edit]

References[edit]

  • Lawrence, J. Dennis (1972). A catalog of special plane curves. Dover Publications. pp. 58–59. ISBN 0-486-60288-5. 
  • Odehnal, Boris (2010). "Equioptic Curves of Conic Sections". Journal for Geometry and Graphics. 14 (1): 29–43. 
  • Schaal, Hermann (1977). "Lineare Algebra und Analytische Geometrie". III. Vieweg: 220. ISBN 3-528-03058-5. 
  • Steiner, Jacob (1867). Vorlesungen über synthetische Geometrie. Leipzig: B. G. Teubner. Part 2, p. 186. 
  • Ternullo, Maurizio (2009). "Two new sets of ellipse related concyclic points". Journal of Geometry. 94: 159–173.