The condition of hitting a point in a sector of a circle

The task is actually interesting because it is not so easy to determine whether a ray lies between two other rays. If you compare the value of the polar angle, you will have to correctly handle the case when the "cut" of the set of angle values ​​lies between your vectors. Example: if yours are in the gap (-180, 180], then beam 180 lies between rays 179 and -179.

If the rays are given by angles, we will need the operation of normalizing the angle: bringing the angle to the desired interval.

double mod(double n, double d) { if (d <= 0) throw new ArgumentException("denominator is negative"); double remainder = n % d; return (remainder >= 0) ? remainder : (remainder + d); } // return value is the same as =angle=, but in interval [basis, basis + 2*pi) double normalize(double angle, double basis) { var diff = mod(angle - basis, 2 * Math.PI); // diff is in [0, 2pi) return basis + diff; } 

Now, let these rays be given by the angles α₁ and α₂, and the desired ray - by angle β. What does it mean that the ray lies between these rays? After all, two beams set two angles. There are two possible answers:

1. β lies inside one of the angles that is less than π.

   Get the code:

    bool isWithinSmallerAngle(double a1, double a2, double b) { var spread = normalize(a2 - a1, 0); if (spread <= Math.PI) { var diffTo1 = normalize(b - a1, 0); return diffTo1 > 0 && diffTo1 < spread; } else { spread = 2 * Math.PI - spread; var diffTo2 = normalize(b - a2, 0); return diffTo2 > 0 && diffTo2 < spread; } } 

2. β lies inside the angle considered in the direction of increase from α₁ to α₂.

   Get the code:

    bool isWithinDirectedAngle(double a1, double a2, double b) { var spread = normalize(a2 - a1, 0); var diffTo1 = normalize(b - a1, 0); return diffTo1 > 0 && diffTo1 < spread; } 

The operation of converting to polar coordinates is rather heavy (it is necessary to use the atan2 function), so I would like to avoid it if possible. If the rays are given to us as codirectional vectors, then everything can be calculated as follows. We need the operation “to find out whether the vector data are in one half-plane with respect to the third vector” (I assume that there is an obvious vector structure):

 double dotProduct(vector v1, vector v2) { return v1.x * v2.x + v1.y * v2.y; } double crossProduct(vector v1, vector v2) { return v1.x * v2.y - v1.y * v2.x; } bool isInSameHalfPlane(vector v1, vector v2, vector axis) { var p1 = crossProduct(axis, v1); var p2 = crossProduct(axis, v2); return Math.Sign(p1) * Math.Sign(p2) == 1; } 

We again have two possible determinations of whether the ray lies between the other two.

1. The ray lies in the first quadrant of the oblique coordinate system defined by the vectors v ₁ and v ₂.

    // для параллельных векторов результат неопределён bool isWithinPositiveQuadrant(vector v1, vector v2, vector w) { return isInSameHalfPlane(v2, w, v1) && isInSameHalfPlane(v1, w, v2); } 

2. The ray lies in the corner swept up by a positive rotation from the vector v ₁ to the vector v ₂.

    bool isDirectedAngle(vector v1, vector v2, vector w) { var crossSign = Math.Sign(crossproduct(v1, v2)) if (crossSign > 0) { // угол от v1 к v2 меньше развёрнутого return isInSameHalfPlane(v2, w, v1) && isInSameHalfPlane(v1, w, v2); } if (crossSign < 0) { // угол от v1 к v2 больше развёрнутого return !isInSameHalfPlane(v2, w, v1) && !isInSameHalfPlane(v1, w, v2); } // здесь вектора параллельны. проверим, сонаправлены ли они var areCodirected = dotProduct(v1, v2) > 0; if (areCodirected) { // между сонаправленными векторами ничего не может лежать return false; } else { // угол между противопололжно направленными векторами - вся полуплоскость return crossProduct(v1, w) > 0; } } 

The part of the task that consists in checking the distances is trivial.