Sorry, no pictures. Explanation here.
This construction, due to Free Jamison (see the reference at the top of this page) is a more accurate variant of the construction described in a simpler construction.
Consider the circular arc AB centered at O, shown in the diagram above. Assume the angle AOB is between 0 and 180 degrees. To trisect AOB, do:
The line OE is an approximate trisector of the angle AOB.
Let α and β be the sizes of the angles AOB and EOB, respectively. The angle FOD equals α/8 by the construction, therefore the angle FCD, which is half the central angle FOD, is equal to α/16. The triangle DOC is isosceles, therefore the angle ODC also equals α/16.
In the triangle OED, let x and y be the sizes of the angles OED and EOD, respectively. Since the sum x+y of the triangle's internal angles equals the triangle's external angle ODC, we have x+y=α/16. Let us note, however, that the angle y equals DOB minus EOB. Thus y=3α/8−β, whence x=β−5α/16.
In the triangle OED, the side DE is twice the side OD by the construction, therefore the law of sines gives siny=2sinx. Consequently, sin(3α/8−β)=2sin(β−5α/16). Solving this for β we arrive at: β=516α+arctansin(a/16)2+cos(a/16)=13α−1212⋅34α3+O(α5)=13α−1331776α3+O(α5).
We see that the trisection error e(α)=α/3−β is given by: e(α)=148α−arctansin(a/16)2+cos(a/16). (This formula is also given in Jamison's article.) The function e(a) is monotonically increasing in α. The worst error on the interval 0≤α≤π/2 is e(π/2) = 0.0000117 radians = 0.00067 degrees. The worst error on the interval 0≤α≤π is e(π) = 0.000093756 radians = 0.00537 degrees. Quite impressive!
This applet was created by
Rouben Rostamian
using
David Joyce's
Geometry
Applet on
July 22, 2002.
Cosmetic revisions on June 7, 2010.
Go to list of trisections |
![]() ![]() |