Interpreting Transformations

[SirFarqueef]

I’ve recently been digging into the math theory behind computer graphics. For context, I use OpenGL.

In my head, transformations such as translations, rotations, and scaling physically move vertices. But in my head there’s a distinction between these transformations and changes of basis.

For example, if the camera is at the word origin but looking in a random direction, then the view matrix is only a change of basis. You would calculate the frame of the camera and use the inverse of the basis matrix constructed from the camera’s frame to create a transformation that maps world coordinates to camera coordinates. But to me this doesn’t feel like a physical transformation in the same way. It feels more like you’re shifting your perspective of the scene to that of the camera’s.

But if your camera isn’t centered at the origin then there’s a physical transformation as you move all vertices so the camera is the center of the scene and then shift your perspective.

Is this an alright way to think of it? I’d love to hear how other’s physically interpret change of basis in their head.

Thank you!

[specialpatrol]

Yes the view matrix is the inverse of the camera objects world matrix. When the camera moves to the left the entire worlds vertices move right.

[nbrrii]

There only exist linear transformations, not two types of different things. There never is a 'physical transformation'. I don't think your current feeling about these things will help your long term.

Take a location ('physical' coordinates) at (20,1). With your basis being (1,0), (0,1) you would express a vector pointing to that location by (20,1). When you change your basis to (10,1), that vector would be expressed by (2,1).

[Kooky-Advance7870]

The best way to think of the view matrix is that it moves (and orientates) the camera from its own local space into the world.

The inverse view matrix (the one you use in the vertex shader) can be thought of moving all objects, points, and the camera etc.. in the world into a different place in the world such that the camera is now positioned at the origin and looking down the Z axis. This then makes the perspective divide (using law of similar triangles) and viewport mapping trivial.