Orthoptic (geometry)


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.
Examples:
  1. The orthoptic of a parabola is its directrix,
  2. The orthoptic of an ellipse is the director circle ,
  3. The orthoptic of a hyperbola,, is the circle ,
  4. The orthoptic of an astroid is a quadrifolium with the polar equation
Generalizations:
  1. An isoptic is the set of points for which two tangents of a given curve meet at a fixed angle .
  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 can be considered as the orthoptic of two circles which are degenerated to the two points and.

    Orthoptic of a parabola

Any parabola can be transformed by a rigid motion into a parabola with equation. The slope at a point of the parabola is. Replacing gives the parametric representation of the parabola with the tangent slope as parameter: The tangent has the equation with the still unknown, which can be determined by inserting the coordinates of the parabola point. One gets
If a tangent contains the point, off the parabola, then the equation
holds, which has two solutions and corresponding to the two tangents passing. The free term of a reduced quadratic equation is always the product of its solutions. Hence, if the tangents meet at orthogonally, the following equations hold:
The last equation is equivalent to
which is the equation of the directrix.

Orthoptic of an ellipse and hyperbola

Ellipse

Let be the ellipse of consideration.
' The tangents of ellipse at neighbored vertices intersect at one of the 4 points, which lie on the desired orthoptic curve.
' The tangent at a point of the ellipse has the equation . If the point is not a vertex this equation can be solved:
Using the abbreviations and the equation one gets:
Hence and the equation of a non vertical tangent is
Solving relations for and respecting leads to the slope depending parametric representation of the ellipse:
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. The free term of a reduced quadratic equation is always the product of its solutions. Hence, if the tangents meet at orthogonally, the following equations hold:
The last equation is equivalent to
From ' and ' one gets:
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 to. Therefore:
An astroid can be described by the parametric representation
From the condition
one recognizes the distance in parameter space at which an orthogonal tangent to appears. It turns out that the distance is independent of parameter, namely. The equations of the tangents at the points and are respectively:
Their common point has coordinates:
This is simultaneously a parametric representation of the orthoptic.
Elimination of the parameter yields the implicit representation
Introducing the new parameter one gets
Hence we get the polar representation
of the orthoptic. Hence:
Below the isotopics for angles are listed. They are called -isoptics. For the proofs see [|below].

Equations of the isoptics

; Parabola:
The -isoptics of the parabola with equation are the branches of the hyperbola
The branches of the hyperbola provide the isoptics for the two angles and .
; Ellipse:
The -isoptics of the ellipse with equation are the two parts of the degree-4 curve
.
; Hyperbola:
The -isoptics of the hyperbola with the equation are the two parts of the degree-4 curve

Proofs

; Parabola:
A parabola can be parametrized by the slope of its tangents :
The tangent with slope has the equation
The point is on the tangent if and only if
This means the slopes, of the two tangents containing fulfil the quadratic equation
If the tangents meet at angle or, the equation
must be fulfilled. Solving the quadratic equation for, and inserting, 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 and.
; Ellipse:
In the case of an ellipse 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, 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 with .
To visualize the isoptics, see implicit curve.