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~2 min readCollision Response

Linear + Angular: Putting Bodies in Their Right Form

Real rigid bodies have six degrees of freedom in 3D, three in 2D: two linear (x, y) and one rotational (theta about z). A 2D rigid body's state is:

position    p   (Vec2)
velocity    v   (Vec2)
orientation theta (scalar radians)
angular vel omega (scalar rad/s)
mass        m,  inv_mass = 1/m
inertia     I,  inv_I    = 1/I    (scalar in 2D)

Newton's Second Law - linear and angular

Linear	Angular
`F = m * a`	`tau = I * alpha`
`a = F / m`	`alpha = tau / I`
`v += a * dt`	`omega += alpha * dt`
`p += v * dt`	`theta += omega * dt`

Torque from a force applied at a point: tau = r x F (in 2D, the cross of two Vec2s is the scalar rx*Fy - ry*Fx). r is the offset from the center of mass to the application point.

Moment of inertia: common 2D shapes

I is the resistance to angular acceleration. For a uniform body in 2D:

Shape	I about center
solid disc (radius r)	`0.5 * m * r^2`
ring / hollow disc	`m * r^2`
solid rect (w x h)	`(1/12) * m * (w^2 + h^2)`
thin rod (length L)	`(1/12) * m * L^2` (about middle)
point mass at dist r	`m * r^2`

Static bodies use inv_I = 0 (infinite inertia — never rotate from applied torque).

Full impulse equation (with rotation)

The complete impulse magnitude at a contact, including angular terms, is the standard form you'll see in Box2D / Bullet:

j = -(1+e) * v_rel . n
    -----------------------------------
    1/m_a + 1/m_b + (r_a x n)^2 / I_a
                  + (r_b x n)^2 / I_b

The (r x n)^2 / I terms are the angular contribution: an off-center hit makes the body spin, which makes it harder to push linearly.

Exercise

Integrate one frame for a 2D rigid body subject to a force at an offset point. Output the new linear + angular state.

next: Practice

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