Special Curves

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Trisecting an Angle
Special Curves

### Trisection using Special Curves

#### Trisectrix by Maclaurin

Colin Maclaurin
(1698-1746)
used a curve called "Trisectrix" to accomplish Angle Trisection.

His idea is shown in the figure shown below.

##### Equation of Trisectrix

In polar coordinate: ** r = CP - CQ = 2cos(q) - 1/2cos(q) **
=(2cos2q)/(2cosq) + 1/(2cosq) = 2QS + CQ = CR.

Therefore QS = SR, and OQ = OR.

********** Maclaurin_tri_desc.dwg** ********

In rectangular coordinate:

Assuming temporalily that point C is the origin of X-Y axis

by replacing r by sqrt(x^{2} + y^{2}) and cosq by y/sqrt(x^{2} + y^{2})

**y**^{2} = x^{2}(3 - 2x) /(1 + 2x)

Now shifting the X-coordinate orign from C to O (distance of 1) (replacing x by x+1)

Trisectrix equation is :**y**^{2} = (x + 1)^{2}(2x - 1) /(3 + 2x)

##### Characteristics of of Trisectrix

This curve intersects Y -axis at x = -1 and 1/2, and as angle QCO approaches 90 degree, point R
approaches line x = -3/2,both up and down.
Why this curve can be used for trisecting angle becomes clear when this figure is compared with the following drawing
used in deriving Trisection equation.

********** trisection_equation_desc.dwg** ********

#### Example of Trisection-60 degree case

Angle **AOB** is to be trisected.

There are 3 trisection solutions.

They are one third of

given angle q (= 60) --->**RCO** = 20 deg

360 + q --->**180 - OCS** = 140 deg

2x360 + q --->**RCT** = 260 deg (shown as -100 deg).

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

********** Maclaurin_60_deg.dwg** ********

**To create this drawing and animation: **

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

Then from command line, type **DRAW_MACLAURIN_ONLY ** for drawing Trisectrix curve

#### Trisection:

After drawing Trisectrix,define an anlge AOB to be trisected.
#### References

1. Dudley,Underwood :"The Trisectors" The Mathematical Association of America,1994.p 12-13

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Last Updated Nov 22, 2006

Copyright 2006 Takaya Iwamoto All rights reserved.
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