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I want to know the vertices of the arc where I have its StartPoint, Center and the EndPoint as well as the radius of the arc. I am aware that the arc which is drawn is by creating a small lines with the precision which is specified in the parameter. What I am trying to achieve is calculate the area of a polygon which may have an arc in it which can look like the image I have attached with this question.

  • SP = StartPoint of the Arc.
  • EP = EndPoint of the Arc.
  • CP = Center of the Arc.

enter image description here

Pankaj Nikam
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2 Answers2

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Knowing StartPoint, Center and the EndPoint of arc is not enough to define it uniquely. You have to knew some another parameter(s).

When arc is well defined, it is possible to calculate an area of circular segment geometrically

Edit: Because you also know radius R, we can calculate an area:

Theta = 2 * ArcSin(Distance_SPtoEP/(2*R))
Area = 1/2 * R * R * (Theta - Sin(Theta))

Quick check: 

R = 1, semicircle. 
Theta = 2 * ArcSin(2/2) = 2 * Pi/2 = Pi
Area = 1/2 * (Pi - 0) = Pi/2 - true

Edit2: It is simpler to connect SP and EP to get remaining polygon, than build polyline approximation of arc and calc area of hundred-vertice polygon.

Polyline approximation:
We want that arc-line distance doesn't excess some limit d. So we will calculate small arc angle A

d = R * (1-Cos(A/2))
A = 2 * ArcCos(1-d/R)

Now divide large arc to small pieces with angle A, and generate new vertices

MBo
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  • I do have its radius. Wouldn't that help me? – Pankaj Nikam Dec 23 '14 at 07:37
  • I guess that the above solution will work out just good. However I also have the remaining polygon to calculate the area. If I were to get the vertices of the arc, I could have used the algorithm specified here http://stackoverflow.com/questions/2034540/calculating-area-of-irregular-polygon-in-c-sharp – Pankaj Nikam Dec 23 '14 at 09:10
  • @PankajNikam I can't agree that there is a well defined radius. This is by no means a circular arc. –  Dec 23 '14 at 09:46
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Your drawing seems to indicate a half ellipse.

The long axis (A) would be SP-EP and the short axis (B) should be given. The area is Pi.A.B/8.