Rectangle physics in 2D. Am I doing this right?

Question

I'm writing a 2D game, in which I would like to have crate-like objects. These objects would move around, like real crates do. I have a hypothetical idea of how I would like to achieve that:

Basically I'd store the boxes' corners' coordinates with their force and velocity unit vectors, and in every update I'd basically do the following steps:
1. Apply the forces(gravity, from collisions, etc..) accordingly.
2. Modify velocity vector based on the force.
3. Move every corner of the box, like so:
4. I repeat nr 3. for every corner, so I get the real movement of the cube.

My questions are: Is this approach heading in the right direction? Is this theory even correct? If not, what would be the correct way to move a box around based on vectors in a 2D environment?

Just to clarify: I'm only dragging corner "A" in the picture, but I want to repeat the dragging for every other corner, with their own vectors. By "dragging" I mean the algorithm I just stated.

Answer 1

Keeping each corner's coordinate and speed makes no sense as you would be storing lots of redundant information. Boxes are rigid objects, which means that there are constraints that must be satisfied at any time instant, namely the distance between any two given corners is fixed. This also translates to a constraint that links the velocities of all four corners and so they are not independent values. With rigid bodies the movement of any point is the sum of two independent movements - the linear movement of the centre of mass (CM) and the rotation around a fixed axis - often, but not always, chosen to be the one that goes through the CM. Hence you only need to store the position and the velocity of the crate's CM (which coincides with the geometric centre of the crate) as well as the angle of rotation and the rate of rotation around the CM.

As to the motion, the gravity field is a constant vector field and hence cannot induce rotation in symmetric objects like those rectangular crates. Instead it only produces accelerated vertical motion of the CM. This is also what happens due to all external forces - one has to take their vector sum and apply it to the CM. Only external forces whose direction does not go through the CM give torque and so cause rotation. Such forces are any external pushes/pulls or reaction forces that arise when crates collide with each other or hit the ground / a wall. Computing torque due to external forces is easy but computing reaction forces could be quite involving process because of the constrained dynamics that has to be employed. Once the torque has been computed, one has to divide it by the moment of inertia of the create in order to get the angular acceleration. Often it is more convenient to use another axis and not the one that goes through the CM - Steiner's theorem can be employed in this case in order to compute the moment of inertia around that other axis.

To summarise:

• all forces, acting on the create, are first added together (as vectors) and the resultant force (divided by the mass of the create) determines the linear acceleration of the CM;
• the torque of all forces is computed and then used to determine the angular acceleration around a given axis.

See here for some sample problems of rigid body motion and how the physics is actually worked out.

Given your algorithm, if by "velocity vector" you actually mean "the velocity of CM", then 1 would be correct - all corners move in the same direction (the linear motion of the CM). But 2 would not be always correct - the proper angle of rotation would depend on the time the torque was applied (e.g. the simulation timestep), and one has to take into account that the lever arm length changes in between as the crate rotates.