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Topic: circle intersections...
Replies: 26   Last Post: Sep 1, 2013 10:49 PM

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Chris M. Thomasson

Posts: 48
Registered: 8/29/13
Re: circle intersections...
Posted: Aug 29, 2013 7:07 PM
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[cross-posted to {alt.math, sci.math}]

"Chris M. Thomasson" wrote in message
news:kvoe76$155$2@speranza.aioe.org...

I was wondering where I could find some programming
methods of clever circle intersection algorithms that use
a single square root operation. So far, all of the examples
I can find use more than one sqrt. Therefore, I have come
up with the following formula:


Solve[G=A-O,H=B-P,Q=-2*A^2+4*A*O-2*B^2+4*B*P-2*O^2-2*P^2,
J=sqrt[((-(C^2-F^2+G^2+H^2)^2)/(4*G^2+4*H^2)+C^2)/(G^2+H^2)],
K=(-G)*(F^2-C^2)/Q+A/2+O/2,L=(-H)*(F^2-C^2)/Q+B/2+P/2,M=K-H*J,
R=G*J+L, {G,H,Q,J,K,L,M,R}]


That discovers intersection(s) for circles c0(A, B, C) and c1(O, P, F)
where A, B, C, O, P and F are the respective x, y center and
radius components of circles c0 and c1. A point of intersection
is contained in M and R, which represent its x, y coordinates.


Here is a solve command for a concrete example:


Solve[A=200,B=500,C=200,O=300,P=500,F=200,G=A-O,H=B-P,
Q=-2*A^2+4*A*O-2*B^2+4*B*P-2*O^2-2*P^2,
J=sqrt[((-(C^2-F^2+G^2+H^2)^2)/(4*G^2+4*H^2)+C^2)/(G^2+H^2)],
K=(-G)*(F^2-C^2)/Q+A/2+O/2,L=(-H)*(F^2-C^2)/Q+B/2+P/2,
M=K-H*J,R=G*J+L, {A,B,C,O,P,F,G,H,Q,J,K,L,M,R}]


These examples work with Mathics Online (just copy and paste
them):

http://www.mathics.net


They should also work with Mathematica.



Anyway, I was wondering if somebody could help me find any
references to existing formulas that are equal to or better
than this one.


Also, I am looking for clever approximation techniques for
lengths of right triangles. Something like:

http://forums.parallax.com/showthread.php/147522-Dog-leg-hypotenuse-approximation



This is all for a fractal I created based on circle intersections; here
is a link:

https://plus.google.com/101799841244447089430/posts



I think I could run the approximation throughout the iteration of
the fractal, then episodically run an actual sqrt in order and/or
correct a set of error levels in a set of heuristics.


I have implemented the formula and put it online here:

http://webpages.charter.net/appcore/fractal/ttr/circle_isect



Thanks.




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