Special Curves

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Trisecting an Angle
Special Curves

### Trisection using Special Curves

#### Parabola by Rene Descartes

Rene Descartes (1596 - 1650)
showed that Angle Trisection can be done by using Parabola. His idea is shown in the figure shown below.

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

##### *La Geometrie* by Rene Descartes

Quote from The Preface of the English translation of Rene Descartes *La Geometrie*:
"If a mathematician were asked to name the great epoch-making works in his science,he might
well hesitate in his decision concering the product of the nineteenth century; he might even
hesitate with respect to the eighteenth century; but as to the product of the sixteenth and
seventeenth centuries, and particularly as to the works of the Greeks in classical times,he would
have very definite views.
He would certainly include the works of Euclid, Archimedes, and Apollonius among the products of
the Greek civilization, which among those which contributed to the great renaissance of
mathematics in the seventeenth century, he would as certainly include *La Geometrie* of
Descartes and the *Principia* of Newton"

********** parabola_tri_desc.dwg** ********
He showed that both Delian and Trisection problems are equivalent to solving the cubic
equations, and its solution are obtained by using conics( parabola is one of them. )

In the case of trisection, the roots of the trisection equation:
**x**^{3} - 3x -2a = 0 ,
where a = cos(3q)
are represented by the x-coordinate values of intersection of (1) parabola
and (2) a circle with its center at point (a, 2).Thus,

(1) y = x^{2}

(2) x^{2} + y^{2} - 2ax - 4y = 0

By substituting (1) into (2) ,y is eliminated and the resulting equation only for x is

x ( x^{3} - 3x -2a ) = 0 , or ** x**^{3} - 3x -2a = 0

**To create this drawing and animation: **

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

Then from command line, type **parabola_trisection **

This will let users define angle to be trisected, draw parabola and circle. The rest of the
operaration will be done manually.

Example: AOB = 60 degrees case

1. Input 2<60. to specify point A

2. Drop a line (blue) perpendicular to y-axis. mark the mid point.

3. Find point C on the line y=1 drawing a line parallel line through the mid point.

4. With the center at C, draw a circle (green) with radius CO.

5. Intersection points between green circle and yellow parabola are P, Q & R.

6. Drop a line from point P perpendicular to x-axis.

7. This line intersects the unit circle at point T.

8. angle TOB trisects angle AOB.

Question: How about points Q & R ?

**** parabola_trisection_60_deg_case.dwg** **

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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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