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Topic: Trisection
Replies: 19   Last Post: Feb 11, 2012 9:59 AM

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 mark Posts: 206 Registered: 12/6/04
RE: Trisection
Posted: Jul 15, 2002 2:05 PM

Thanks for the hard work Eric.

> I have algebraically checked this construction. It will effectively
converge
> to the trisection, with a cubic convergence rate. More precisely, if
we take A
> as the origin and C as the unit point on the X axis, 3*t the angle
to trisect,
> if h is the difference in abscissa between the first guess and the
result
> (i.e. the
> difference in cosine), and h' the same difference after one

iteration, we have:
>
> h' = h^3 / (48*sin(t)^2) + O(h^4)

construction line so I found my own.
In response to John's suggestion of relettering (below) I am reposting
my construction based on AOB.

>[It might also be
a good idea to change the lettering, since, to the extent that
>there's a
standard, it's usually an angle AOB that gets trisected.]

Near Exact Trisection:
1. Start with an unknown angle <90 deg., label the vertex O.
2. Draw an arc with origin at O crossing both lines of the angle at
points A and B.
3. Draw line AB making an isosceles triangle.
4. Using point A as the origin, draw an arc crossing line AB and the
earlier arc somewhere between ÃÂ¼ and ÃÂ½ way between points A and B.
Label where this new arc crosses line AB point D.
Label where this new arc crosses the first arc point E.
5. Draw line DE and extend it well past O . If line DE passes
exactly through point O (it wonÃÂt) stop, your first guess was an exact
trisection.
6. Extend line OA well past point O, step off 3 times length OA from
point O and label the new point F.
7. Swing an arc of length OF with O as the origin that crosses the
extended line DE near point F.
Label the intersection G.
8. Draw line GO and extend it to intersect the original arc from step
2.
Label the intersection EÃÂ.

Line OEÃÂ is a good trisection. However this is only the start.
Repeating the process from step 4 using AEÃÂ as the arc radius results
in a trisection to within 10E-11 degrees. Each subsequent iteration
improves the trisection by several orders of magnitude.

It takes an extra step, but an alternate construction using a line
from point B perpendicular to line OA (line AB is no longer necessary)
yields exactly the same results if your step 4 circle has its origin
at point B, is between 1/2 and 3/4 angle AOB, and point D is on the
new perpendicular line instead of line AC. This give a longer working
line for the construction.

Date Subject Author
6/20/02 Guest
7/3/02 Jacques
7/3/02 John Conway
7/3/02 mark
7/3/02 John Conway
7/3/02 Eric Bainville
7/3/02 John Conway
7/15/02 mark
7/16/02 mark
7/16/02 Rouben Rostamian
7/17/02 Eric Bainville
7/17/02 Steve Gray
7/18/02 mark
7/19/02 Eric Bainville
7/21/02 Steve Gray
7/22/02 Rouben Rostamian
7/23/02 mark
7/24/02 mark
7/19/02 Steve Gray
2/11/12 JO 753