Difference of plane/ray intersection with point/plane projection

Question

I found a solution for the ray plane intersection code in Wikipedia, which works and where I simply solve a linear equation system.

Later I found some code for a point onto plane projection, which is obviously implemented differently and also yields different solutions under certain conditions.

However, I do not really get what is the difference between a projection of a point along a vector and the intersection of a ray (build by the point and vector). In both cases I would expect just to find the point where the ray intersects the plane?!

Is there anywhere a figure to illustrate the difference?

struct Plane {
  glm::vec3 _normal;
  glm::vec3 _point;
};

glm::vec3 intersection(const Plane &p, const glm::vec3 &point, const glm::vec3 &direction) {
  // See: https://en.wikipedia.org/wiki/Line%E2%80%93plane_intersection
  const float t = glm::dot(p._normal, point - p._point) / glm::dot(p._normal, -direction);
  return point + t * direction;
}

glm::vec3 orthogonalProjection(const Plane &p, const glm::vec3 &point, const glm::vec3 &direction) {
  // from https://stackoverflow.com/questions/9605556/how-to-project-a-point-onto-a-plane-in-3d
  const float t = -(glm::dot(point, direction) - glm::dot(p.getOrigin(), p.getNormal()));
  return point+ t * direction;
}

score 1 · Accepted Answer · answered Nov 07 '18 at 15:44

A ray is an infinite line, so it has a direction. Intersecting a ray with a plane means finding where the line passes through the plane.

A point is a dimensionless dot suspended somewhere in space. Projecting a point onto a plane means shooting a ray that passes through the point and is perpendicular to the plane (called the "normal"), and seeing where it lands.

The ray already has a direction, the point doesn't. The direction chosen to project the point is the one perpendicular the plane simply because that's how the projection is defined.

So you can have a special case where the ray's direction and the plane's normal coincide, in which case intersecting the ray with the plane and projecting a point that happens to lie somewhere on the ray lead to the same result, but that's just a special case.

"The direction chosen to project the point is the one perpendicular the plane simply because that's how the projection is defined." Isn't this direction of the plane's normal? Then this would equal an intersection if inserting point and plane normal. — dgrat, Nov 07 '18 at 16:03
@dgrat yes that is equivalent to intersecting the plane with a ray that is perpendicular to the plane and passes through the point. That's the special case I described at the end. — Cyber, Nov 07 '18 at 16:05

score 0 · Answer 2 · answered Jan 21 '21 at 17:11

For those interested, here's a c# version of the intersection function above.

Thanks @dgrat you made my day.

    /// <summary>Calculates the intersection of a ray with a plane</summary>
    /// <param name="rayOrigin">Any point of the ray</param>
    /// <param name="rayDirection">The direction of the ray</param>
    /// <param name="planeOrigin">Any point of the plane</param>
    /// <param name="planeNormal">The normal of the plane</param>
    /// <returns>The intersection</returns>
    private Point3D Intersection(Point3D rayOrigin, Vector3D rayDirection, Point3D planeOrigin, Vector3D planeNormal)
    {
        var length = Vector3D.DotProduct(planeNormal, rayOrigin - planeOrigin) / Vector3D.DotProduct(planeNormal, -rayDirection);
        return rayOrigin + length * rayDirection;
    }

Difference of plane/ray intersection with point/plane projection

2 Answers2