How to determine whether a point (X,Y) is contained within an arc section of a circle (i.e. a Pie slice)?

Question

Imagine a circle. Imagine a pie. Imagine trying to return a bool that determines whether the provided parameters of X, Y are contained within one of those pie pieces.

What I know about the arc:

I have the CenterX, CenterY, Radius, StartingAngle, EndingAngle, StartingPoint (point on circumference), EndingPoint (point on circumference).

Given a coordinate of X,Y, I'd like to determine if this coordinate is contained anywhere within the pie slide.

Answer 1

Check:

1. The angle from the centerX,centerY through X,Y should be between start&endangle.
2. The distance from centerX,centerY to X,Y should be less then the Radius

And you'll have your answer.

Answer 2

I know this question is old but none of the answers consider the placement of the arc on the circle.

This algorithm considers that all angles are between 0 and 360, and the arcs are drawn in positive mathematical direction (counter-clockwise)

First you can transform to polar coordinates: radius (R) and angle (A). Note: use Atan2 function if available. wiki

R = sqrt ((X - CenterX)^2 + (Y - CenterY)^2)

A = atan2 (Y - CenterY, X - CenterX)

Now if R < Radius the point is inside the circle.

To check if the angle is between StartingAngle (S) and EndingAngle (E) you need to consider two possibilities:

1) if S < E then if S < A < E the point lies inside the slice

2) if S > E then there are 2 possible scenarios

• if A > S

then the point lies inside the slice

• if A < E

then the point lies inside the slice

In all other cases the point lies outside the slice.