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Orthotomic

Given a source $S$ and a curve $\gamma$, pick a point on $\gamma$ and find its tangent $T$. Then the Locus of reflections of $S$ about tangents $T$ is the orthotomic curve (also known as the secondary Caustic). The Involute of the orthotomic is the Caustic. For a parametric curve $(f(t),g(t))$ with respect to the point $(x_0, y_0)$, the orthotomic is

$\displaystyle x$ $\textstyle =$ $\displaystyle x_0-{2g'[f'(g-y_0)-g'(f-x_0)]\over f'^2+g'^2}$  
$\displaystyle y$ $\textstyle =$ $\displaystyle y_0+{2f'[f'(g-y_0)-g'(f-x_0)]\over f'^2+g'^2}.$  

See also Caustic, Involute


References

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, p. 60, 1972.




© 1996-9 Eric W. Weisstein
1999-05-26