ray sphere intersection equation

Certainly, here is the equation for checking the intersection between a ray and a sphere in C++:

bool raySphereIntersect(Vector3 rayOrigin, Vector3 rayDirection, Vector3 sphereCenter, float sphereRadius) {
    Vector3 oc = rayOrigin - sphereCenter;
    float a = rayDirection.dot(rayDirection);
    float b = 2.0f * rayDirection.dot(oc);
    float c = oc.dot(oc) - sphereRadius * sphereRadius;
    float discriminant = b  b - 4  a * c;

    if (discriminant < 0) {
        return false; // No intersection
    } else {
        return true; // Intersection detected
    }
}

Explanation:

1. Initialization:
2. rayOrigin: Starting point of the ray.
3. rayDirection: Direction in which the ray extends.
4. sphereCenter: Center point of the sphere.

5. sphereRadius: Radius of the sphere.

6. Vector Calculation:

7. oc: Vector from the ray's origin to the sphere's center.

   • oc = rayOrigin - sphereCenter.

8. Coefficients Calculation:

9. a: Dot product of the ray direction with itself.
   • a = rayDirection.dot(rayDirection).
10. b: Dot product of the ray direction with oc, multiplied by 2.
    • b = 2.0f * rayDirection.dot(oc).

11. c: Dot product of oc with itself, subtracting the square of the sphere's radius.

    • c = oc.dot(oc) - sphereRadius * sphereRadius.

12. Discriminant:

13. discriminant: Calculation of the discriminant for the quadratic equation (b^2 - 4ac).

    • discriminant = b b - 4 a * c.

14. Intersection Check:

15. If discriminant is less than 0, there's no intersection (return false).
16. Otherwise, an intersection is detected (