Special Curves

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Trisecting an Angle
Special Curves

### Trisection using Special Curves -Archimedian-Spiral Curves

### Archimedian Spiral

Archimedes (287 BC - 212 BC)
used Spiral (called Archimedian Spiral) for Trisection. His idea is shown in the figure shown below.

#### How to draw Archimedian Spiral manually

Formula for Archimedean Spiral in polar coordinate is

** r = a q**

where "a" is an constant. Here it is set to be (2/p) . Then the radii of this curve
where it cuts X-Y axis is 1, 2, 3 ,4 etc as it goes around anti-clockwise.

The curve is drawn by connecting the intersecting points between concentric
circles and lines passing through the center of those circles.

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

******* draw_Archimedian_spiral_base.dwg** *****

There is a similarity
between this curve and Quadratrix. In the case of Quadratrix ,Y-axis length
changes linearly, whereas in Spiral case, length of radius changes linearly
with angle value.

**To create this drawing and animation : **

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

Then from command line, type ** (concentric_circles) & (multiple_rays) **

#### How to draw Archimedian Spiral semi_automatically

***************** spiral_a_1.dwg** ***************
***************** spiral_a_2.dwg** ***************
**To create these drawings : **

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

Then from command line, type ** test_1 & test_2 **

#### Angle Trisection by Archimedian Spiral

Similarity of Spiral to Quadratrix suggests that this curve can also be used for angle trisection.

The reason it can be used for angle trisection is that since the change in angle is proportional to the change
in its radius, division of angle is equivalent to the division of length, which will be an easy process.

You can see the process in animation.

*************** spiral_tri_desc.dwg** *************

**To create this drawing and animation : **

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

Then from command line, type ** spiral_4 **

#### Samples of Archimedes' Spiral

3 Archimedian Spirals after 1 rotation , 6 rotations, and 128 rotations are shown here.
As the distance between curves gets closer and closer, Moire fringe-like pattern developes.

*************** spiral_2_pi.dwg** ****************
*************** spiral_12_pi.dwg** ***************
*************** spiral_128_pi.dwg** *************

**To create these drawings : **

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

Then from command line, type ** spiral_3 **

In response to the prompt message "How many rounds (def = 2 ) ?" Type in 1, 6 & 64 to get these pictures.

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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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