Isoptic

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:

The ellipse with equation

can be represented by the unusual parametric representation

where $m$ is the slope of the tangent at a point of the ellipse. $c \to + (m)$ describes the upper half and $c \to - (m)$ the lower half of the ellipse. The points $(\pm a, 0)$ ) 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 $(0, \pm b)$ Hence the points $(\pm a, \pm b)$ are points of the desired orthoptic (the circle $x 2 + y 2 = a 2 + b 2$ ).

The tangent at point $c \to \pm (m)$ has the equation

If a tangent contains the point $(x 0, y 0)$ , off the ellipse, then the equation

holds. Eliminating the square root leads to

which has two solutions $m 1$ and $m 2$ corresponding to the two tangents passing $(x 0, y 0)$ . The free term of a reduced quadratic equation is always the product of its solutions. Hence, if the tangents meet at $(x 0, y 0)$ orthogonally, the following equations hold:

The last equation is equivalent to

This means that:

The ellipse case can be adopted nearly exactly to the hyperbola case. The only changes to be made are to replace $b 2$ with $- b 2$ and to restrict $m$ to $| m | > b / a$ . Therefore:

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