Rough sketch of sphere vs arc sector intersection test

sphere_vs_arc_sector_intersection.cpp

struct Sphere
{
  float2 pos;
  float radius;
};

struct ArcSector
{
  float2 pos;
  float radius;
  float2 dir; // center line direction from pos towards dir vector, normalized.
  float angle; // radians, assume symmetricly spreading out angle +/- away from direction vector.
}

bool sphere_sphere_intersect(Sphere s1, Sphere s2); // exercise for reader

// Intersects a sphere with a half-plane that passes through the origin.
bool sphere_intersects_halfplane(Sphere s, float planeNormal)
{
  return dot2(s.pos, planeNormal.xy) - s.radius <= 0;
}

void intersect_sphere_arc(Sphere s, ArcSector a)
{
  if (!sphere_sphere_intersect(s, Sphere(a.pos, a.radius))) return false;

  Sphere sphereLocal(s.pos - a.pos, s.radius);

  float dirAngle = atan2(a.dir.y, a.dir.x);
  float2 edge1 = float2(cos(dirAngle+angle), sin(dirAngle+a.angle));
  float2 edge2 = float2(cos(dirAngle-angle), sin(dirAngle-a.angle));

  float2 normal1 = float2(-edge1.y, edge1.x); // rotate 90 deg CCW
  float2 normal2 = float2(edge2.y, -edge1.x); // rotate 90 deg CW

  if (a.angle <= pi/2) // convex arc sector
  {
    if (!sphere_intersects_halfplane(sphereLocal, normal1)) return false;
    if (!sphere_intersects_halfplane(sphereLocal, normal2)) return false;
  }
  else if (!sphere_intersects_halfplane(sphereLocal, normal1) // concave arc sector
        && !sphere_intersects_halfplane(sphereLocal, normal2)) return false;

  return true;
}