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Endpoint of an Arc

Date: 06/25/2001 at 15:26:01
From: Mike Bryson
Subject: Endpoint of an arc

I'm sure this is basic, but I've been out of school for many years and 
I have searched extensively through your archives for this answer. 
I've found many similar answers but not the exact one I need. So, 
given the center of the circle, the angle of the arc, the radius 
of the circle, and the starting point of the arc, how do I determine 
the end point of the arc using cartesian coordinates?


Date: 06/26/2001 at 11:17:07
From: Doctor Rob
Subject: Re: Endpoint of an arc

Thanks for writing to Ask Dr. Math, Mike.

Let the center of the circle be (h,k), the angle of the arc be A, the
radius of the circle be r, and the starting point of the arc be (a,b).  
Let angle B be defined by

   cos(B) = (a-h)/r,
   sin(B) = (b-k)/r.

Then there are two ending points (c,d) for the arc, corresponding to
clockwise and counterclockwise directions around the circle. They
are given by

   c = h + r*cos(B-A),
   d = k + r*sin(B-A),

   c = h + r*cos(B+A),
   d = k + r*sin(B+A),


Another way to find them is to solve following two simultaneous
quadratic equations for (x,y):

   (x-h)^2 + (y-k)^2 = r^2,
   (x-a)^2 + (y-b)^2 = 4*r^2*sin^2(A/2).

The first equation is the original circle, and the second equation is 
also a circle, the set of points whose distance from the starting 
point (a,b) is equal to the length of the chord subtending the arc.  
The points of intersection of these two circles will be the two points 
mentioned above. There will be just two real solution pairs.

- Doctor Rob, The Math Forum   
Associated Topics:
High School Conic Sections/Circles
High School Coordinate Plane Geometry
High School Geometry

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