Special Curves

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Greek_Math
Angle Trisection
Special Curves(Angle Trisection)

### Angle Trisection using Special Curves

#### The Cycloid of Ceva

Tomasso Ceva (1648 - 1737)
applied "Insertion method" by Archimedes to Angle trisection using a special curve called "Cycloidum anomalarum",
or "The Cycloid of Ceva".

The idea is shown in the figure shown below.

An equation of the curve in polar coordinate is:

**r = 1 + 2 cos(q)**,

and in rectangular coordinate:

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

See how the curve is drawn by **animation**.

********** cycloid_of_Ceva.dwg** ********

**To create this drawing and animation: **

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

Then from command line, type **draw_cycloid_Ceva_only **

#### Angle Trisection using the Cycloid of Ceva

The link CQPD are made up of 3 links CP, QD and DP. P slides along line connecting CQ, and D
slides along line CD(x-axis).When Q moves along a unit circle with its center at C, the locus of point P
makes a 4-leaves curve as shown. This is called "The Cycloid of Ceva".

See how the curve is drawn by **animation**.

Trisection Procedure:

Angle AOB is to be trisected.

Draw a line through D parallel to line AO.

It intersects the curve at point P.

Line PC trisects the angle AOB.

********** cycloid_of_Ceva_tri_desc.dwg** ********

This curve is of the sixth degree, and for a given value of y, it generally has 6 x-values as noticed
in the figure for trisection.

**To create this drawing and animation: **

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

Then from command line, type **Ceva_trisection **

#### Interpretation of 6 intersecting points

In order to explain the interpretation of 6 intersecting points (P, S, T and P',S', T')
the case of 60 degrees is shown in the figure. The 3 solutions for Trisecting angle
3q = 60 degrees are:

POB = q = 20 degrees

SOB = (1/3)(360 + 3q) = 120 + q = 140 degrees

BOU(clockwise) = (1/3)(720 + 3q) = 240 + q = 260 degrees,or,
100 degrees clockwise,where U is the point where the extension of line TO meets a unit circle.

Similarly P',S' and T' are for the case 3q = 180 - 60 = 120 degrees.

***** cycloid_Ceva_trisection_60_deg.dwg** ***

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Greek_Math
Angle Trisection
Special Curves(Angle Trisection)

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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