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: Generalizations: Any parabola can be transformed by a rigid motion (angles are not changed) into a parabola with equation

Replacing x gives the parametric representation of the parabola with the tangent slope as parameter:

with the still unknown n, which can be determined by inserting the coordinates of the parabola point.

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:

and respecting (II) leads to the slope depending parametric representation of the ellipse:

(For another proof: see Ellipse § Parametric representation.)

Eliminating the square root leads to

The constant term of a monic quadratic equation is always the product of its solutions.

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:

This is simultaneously a parametric representation of the orthoptic.

Elimination of the parameter t yields the implicit representation

Introducing the new parameter φ = t − ⁠5π/4⁠ one gets

(The proof uses the angle sum and difference identities.)

Hence: Below the isotopics for angles α ≠ 90° are listed.

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).

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

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

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

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

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:

To visualize the isoptics, see implicit curve.