Raytracing: mirror reflection in object coordinates

Question

I'm having fun writing my own raytracer. I'm doing correctly a metal surface ray reflection on world coordinates:

• I have an incident ray in world coords, defined by origin_point and direction_vector;
• inverse_transform_matrix * origin_point gives me the ray origin in object space, and inverse_transform_matrix ^ direction_vector gives me the ray direction in object space (with ^ I mean matrix-vector multiplication with no translation).
• Now I do my calculations to get the hit point and hit normal vector, in object space;
• transform_matrix * hit_point gives me the hit point in world coords, and (inverse_transform_matrix_transposed ^ hit_normal).normalized() gives me the world_normal_vector with length 1 in world coordinates.

Now I can have the mirror-like reflected ray direction in world coords: reflected_dir_vector = direction_vector - 2*(direction_vector • world_normal_vector) * world_normal_vector (• is the dot product). The hit point in world coords is the origin point of my new ray.

Now I can check the angles of the incoming ray and reflected ray respect to the normal (still in world coordinates) using the cross product and they are equal; I can also export my vectors in a wavefront file, load it in Blender and see with my eyes that everything is fine - the angles between the vectors are ok and all the vectors are on the same plane. My eyes aren't a math proof but the rendered picture with these calculation is fine.

BUT:

I'd like to calculate the reflected ray direction in objected coordinates and after that go back to world coordinates, due to the way I want my code to work.

I can calculate the reflected vector direction in object space just like before but using object coords: reflected_dir_vector = object_coords_direction_vector - 2*(object_coords_direction_vector • object_coords_normal_vector) * object_coords_normal_vector. Now I don't get what matrix I should use to move this vector to world coordinates; I tried all my direct, inverse and trasponsed transformation matrices but all of them gives me a wrong reflected direction.

Where's my error?

I'm not showing code to not to make and excessively long post.

Thanks.

Answer 1

If your implementation of multiplication is correct you should use your local to global matrix to transform the reflection from local to world space. You should do: transformation_matrix ^ reflection_vector. Also make sure that your vectors are normalised.

transforming a normalised vector does not yield a normalised result.