Define Circle Knowing Two Points and ARC LENGTH Only.

[Ryank]

I am having trouble doing exactly what the title says. I have two points and the arc length between them (this is a bending beam type of a situation). Essentially I know where the ends of the beam are and how long the bent beam is and I need to get the equation of the circle. And yes I realize there would be two distinct solutions depending on positive of negative curvature. Any help would be amazing, thanks!

 

[berkeman]

I am having trouble doing exactly what the title says. I have two points and the arc length between them (this is a bending beam type of a situation). Essentially I know where the ends of the beam are and how long the bent beam is and I need to get the equation of the circle. And yes I realize there would be two distinct solutions depending on positive of negative curvature. Any help would be amazing, thanks!

What is the context of the question? Could you post a sketch? Thanks.

 

[Ryank]

There doesn't really need to be a context. You know the length of a chord on the circle and the length of the arc segment between them. I did some digging and did finally manage to find two solutions. One uses Newton's approximation method and the other uses and taylor expansion:

http://norman.rasmussen.co.za/24/radius-from-arc-and-chord-length/
http://www.mathforum.com/dr.math/faq/faq.circle.segment.html#1

 

[hotvette]

Looks like case 1 from Dr. Math that requires Newton's Method is the most straightforward.

 

[Studiot]

Perhaps this sketch will help, it's basic trigonometry.

Since you have fixed two points on the circumference you not only know the arc length you also know the chord length.