Special Curves

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Trisecting an Angle
Special Curves

### Trisection using Special Curves

#### The Hyperbola-Pappus

Pappus (280 - 350)
showed that "Hyperbola" can be used for Angle Trisection. His idea is shown in the figure shown below.

The general formula for this hyperbola is given by

**y**^{2} + (1-e^{2})x^{2}-2kx + k^{2} = 0

where k = distance of F1 to directrix (X-coordinate value of point F1)
=half the distance between B & C.

and e = eccentricity = 2 in this case.

In general, e> 1 for hyperbola

You can see the process in **animation**.

For detail, go to the section on **Conics**.

******** Pappus_hyperbola_tri_desc.dwg** ******

**To create this drawing and animation: **

** Load Pappus_hyperbola.lsp (load "Pappus_hyperbola")**

Then from command line, type **pappus_1 **

Proof of the trisection is straight forward :

Since the hyperbola is drawn such that half of BC is equal to BF_{1}(= CF_{2}),
these 3 arcs have the same length. Therefore angle F_{2}OC = COB = BOF_{1} = (1/3)angle F_{1}OF_{2}.

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Trisecting an Angle
Special Curves

All questions/suggestions should be sent to Takaya Iwamoto

Last Updated Nov 22, 2006

Copyright 2006 Takaya Iwamoto All rights reserved.
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