Colin Maclaurin (1698-1746) used a curve called "Trisectrix" to accomplish Angle Trisection.
His idea is shown in the figure shown below.
=(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(x2 + y2) and cosq by y/sqrt(x2 + y2)
y2 = x2(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 :y2 = (x + 1)2(2x - 1) /(3 + 2x)
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 ********
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
All questions/suggestions should be sent to Takaya Iwamoto
Last Updated Nov 22, 2006
Copyright 2006 Takaya Iwamoto All rights reserved.