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.

  Orthoptic of the parabola (its directrix)
  Orthoptic of the ellipse (its director circle)
  Minimum bounding box of the ellipse (circumscribed by the orthoptic circle)
  Major and minor axes of the ellipse
  Orthoptic of the hyperbola (its director circle)
  xy-axes and hyperbolic asymptotes

Examples:

  1. The orthoptic of a parabola is its directrix (proof: see below),
  2. The orthoptic of an ellipse is the director circle (see below),
  3. The orthoptic of a hyperbola is the director circle (in case of ab there are no orthogonal tangents, see below),
  4. The orthoptic of an astroid 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.

Orthoptic of a parabola

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Any parabola can be transformed by a rigid motion (angles are not changed) into a parabola with equation  . The slope at a point of the parabola is  . Replacing x gives the parametric representation of the parabola with the tangent slope as parameter:   The tangent has the equation   with the still unknown n, which can be determined by inserting the coordinates of the parabola point. One gets  

If a tangent contains the point (x0, y0), off the parabola, then the equation   holds, 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   which is the equation of the directrix.

Orthoptic of an ellipse and hyperbola

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Ellipse

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Let   be the ellipse of consideration.

  1. The tangents to the ellipse   at the vertices and co-vertices intersect at the 4 points  , which lie on the desired orthoptic curve (the circle  ).
  2. The tangent at a point   of the ellipse   has the equation   (see tangent to an ellipse). If the point is not a vertex this equation can be solved for y:  

Using the abbreviations

  (I)

and the equation   one gets:   Hence

  (II)

and the equation of a non vertical tangent is   Solving relations (I) for   and respecting (II) leads to the slope depending parametric representation of the ellipse:   (For another proof: see Ellipse § Parametric representation.)

If a tangent contains the point  , off the ellipse, then the equation   holds. Eliminating the square root leads to   which has two solutions   corresponding to the two tangents passing through  . The constant term of a monic quadratic equation is always the product of its solutions. Hence, if the tangents meet at   orthogonally, the following equations hold:

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

  The last equation is equivalent to   From (1) and (2) one gets:

The intersection points of orthogonal tangents are points of the circle  .

Hyperbola

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

The intersection points of orthogonal tangents are points of the circle  , where a > b.

Orthoptic of an astroid

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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:

The orthoptic of an astroid is a quadrifolium.

Isoptic of a parabola, an ellipse and a hyperbola

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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

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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

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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.

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Notes

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References

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  • 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" (PDF). Journal for Geometry and Graphics. 14 (1): 29–43.
  • Schaal, Hermann (1977). Lineare Algebra und Analytische Geometrie. Vol. III. Vieweg. p. 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 (1–2): 159–173. doi:10.1007/s00022-009-0005-7. S2CID 120011519.