Orthoptic
Orthopic may refer to:
See also
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.
The orthoptic of
Remark: In medicine there exists the term orthoptic, too.
Orthoptic of an ellipse and hyperbola
Ellipse
The ellipse with equation 
can be represented by the unusual parametric representation
where 
is the slope of the tangent at a point of the ellipse. 
describes the upper half and 
the lower half of the ellipse. The points 
) 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 
Hence the points 
are points of the desired orthoptic (circle 
).
The tangent at point 
has the equation
If a tangent contains the point 
, off the ellipse, then the equation
holds. Eliminatig the square root leads to
which has two solutions 
corresponding to the two tangents passing point . The free term of a reduced quadratic equation is always the product of its solutions. Hence, if the tangents meet at point orthogonal, the following equations hold:
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