Special Curves

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Trisecting an Angle
Special Curves

### Trisection using Special Curves

#### 1. Quadratrix-Hippias

Hippias's(about 460BC-about 400BC)
Trisection method is shown in the figure shown below.

The procedure for Trisection is:

Step 1:Draw the quadratrix , then select a point "A" on this curve to define angle AOB

Step 2:Draw a line from "A" parallel to AB and the intersection with line PO is point "C".

Step 3:Find a point "D" on line CO such that OD = CO/3

Step 4:Draw a line from "D" parallet to OB, and find a point "E" ,intersecting this curve.

Angle EOB trisects angle AOB.

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

************* quadratrix_tri_desc.dwg** ***********

**To create this drawing and animation: **

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

Then from command line, type **quadratrix_3 **

#### How to Draw a Quadratrix

OBQP is a square.
Divide both OP and BQ in equal parts N.

Divide a quarter circle BP into the same number N.
Point C,D & E arer such points.

Horizontal line CD(yellow) and polar line OE(red) intersect .

The locus of such points(cyan color) is the "Quadratrix".

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

************* quadratrix_curve_10_div.dwg** ***********

**To create this drawing and animation: **

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

Then from command line, type **quadratrix_2 **for drawing quadratrix for 1000 division.

test_1 & test_2 for drawing manually.

If OP & OB are x & y axis respectively, the curve is expressed as
**y = x tan(p*y/2)**

Using identity tan(a) = sin(a)/cos(a), and replacing (p/2)y = h

x = (2/p)*cos(h)*(h/sin(h))

As h approaches zero, both cos(h) and (h/sin(h)) approach 1.

So the x-coordinate of the point R,where the Quadratrix
intersect X-axis is (2/p).

This means that length OR is used to get p value, and we now have succeeded in "Squaring the Circle".
This is the reason why this curve is named "Quadratrix", i.e. curve for circle quadrature.

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Special Curves

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Last Updated Nov 22, 2006

Copyright 2006 Takaya Iwamoto All rights reserved.
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