hyperbolism
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last updated: 2003-07-19 |
Given a curve C1, a point O, and a line l, then the hyperbolism
of C1 is the curve C2 formed by the points P as follows:
- let Q be a point on C1, then line m through OQ intersects l in R
- let n be a line parallel to line l through Q
- then P is the projection from R on m
Look at the witch
of Agnesi to illustrate the construction:
We can normalize the hyperbolism
to a curve C1 with respect to a point O and a line x = 1.
In this case holds that when C1 has equation f (x,y) = 0, the hyperbolism of C1
has as equation: f (x, xy) = 0.
The term xy makes the link to the hyperbola. The
inverse of the hyperbolism is the antihyperbolism:
when C2 is the hyperbolism of C1, then C1 is the antihyperbolism of C2.
Some examples are the following:
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