6. Boxes can rotate#
6.1. Another degree of freedom#
The circles in Two rigid balls share an impact only translate: their centers move. Now replace them with rigid square cross-sections. A square also has an orientation, and an impact away from its center can change its rotation. This is a two-dimensional side view, not a simulation of three-dimensional cubes. Nothing deforms, and there are no scripted spins or sideways kicks.
Click or tap inside the scene to create one square above that horizontal position. Its starting angle is random, but its linear and angular velocities are zero. The square falls under gravity, then contacts can make it turn. A dot near one corner makes rotation easier to see. New squares start at least 2.3 m above the floor, or higher if needed to avoid an occupied spawn space. Near a wall, their horizontal centers move inward just enough to fit. If no clear position exists above that spot, the request is visibly refused. There is no box-count cap and no silent deletion; available space and device performance are the practical limits. Clear all boxes starts over. Focus the scene and press Space or Enter, or use Drop in the middle, for keyboard access.
If the embed cannot start, open the boxes on their own. JavaScript and Canvas 2D are required, not WebGL. Both classic scripts are local; opening the standalone HTML from a file does not require fetching a module or a remote library.
6.2. Position, angle and inertia#
Let \(\mathbf p=(p_x,p_y)\) be a square’s center in metres, with positive \(y\) upward. Let \(\mathbf v=(v_x,v_y)\) be its velocity in m/s. The orientation \(\theta\) is an angle in radians, positive counterclockwise; angular velocity \(\omega\) measures its change in rad/s. The mass \(m\) is 1 kg. The width \(a\) and height \(b\) are both 0.35 m. The moment of inertia \(I\) measures resistance to angular acceleration about the center, perpendicular to this plane. For a uniform rectangle, applying the mass-integration method in OpenStax’s moment-of-inertia examples separately to the two coordinate directions gives
The rectangle’s four local corners are \((\pm a/2,\pm b/2)\). For a local corner \((u_x,u_y)\), rotate and then translate to find its world position \((q_x,q_y)\):
The renderer and collision geometry use the same orientation. There is no circle proxy or axis-aligned bounding box pretending to be the square. Axis-aligned bounds only help search for candidate pairs; contact uses the oriented polygon faces.
Let \(\mathbf F\) be total force in newtons and \(\tau\) total torque about the center in N m. A force acting at a displacement \(\mathbf r=(r_x,r_y)\) from the center has torque \(\tau=r_xF_y-r_yF_x\). This signed scalar is the two-dimensional version of a cross product. Our continuous equations are
A dot means time derivative, not multiplication. Gravity acts through the center: \(\mathbf F=(0,-m g)\) with \(g=9.81\ \mathrm{m/s^2}\). It therefore causes no torque here. Between contacts a newly spawned square falls without turning, even though its initial orientation may be tilted.
6.3. Contact velocity is not center velocity#
Consider a touching point \(\mathbf q\) on bodies A and B. Define the lever arms \(\mathbf r_A=\mathbf q-\mathbf p_A\) and \(\mathbf r_B=\mathbf q-\mathbf p_B\), in metres. The velocity of a point combines translation and rotation. For either body,
Let \(\mathbf n\) be the unit normal from A toward B. Define their relative normal contact velocity \(u_n\) by taking the dot product of the point-velocity difference with that normal:
Negative means approaching. A dot product multiplies matching components and adds them. Define \(c_A=r_{Ax}n_y-r_{Ay}n_x\) and similarly \(c_B\); these are signed lever arms in metres. An impulse of magnitude \(J_n\) in N s acts along the normal on B and oppositely on A. For one isolated approaching contact, the effective inverse mass \(k_n\) (in 1/kg) and restitution rule are
The dimensionless coefficient \(e\) is 0.25 here; impacts below 1 m/s use zero restitution to avoid tiny repeated rebounds. A fixed wall or floor has zero inverse mass and inverse inertia. The impulse changes the states by
This is how a collision creates rotation. If the normal passes through the center, its signed lever arm is zero and its normal impulse causes no spin. See Baraff’s rigid-body notes for the underlying impulse derivation.
6.4. A worked off-center impact#
Take a 1 kg square with the inertia above, downward center velocity 2 m/s, and no spin, hitting a fixed floor at a point 0.10 m to the right of its center. For this calculation only, omit tangential friction. Choose A as the floor and B as the square, so \(\mathbf n=(0,1)\) and \(c_B=0.10\ \mathrm m\). Then \(u_n=-2\ \mathrm{m/s}\) and
The outgoing center velocity is about -0.32192 m/s vertically, still downward, while the angular velocity becomes +8.21918 rad/s. The contact point’s vertical velocity is \(-0.32192+0.10(8.21918)=0.50\) m/s upward, exactly the expected restitution speed \(e(2)\). Do not apply the restitution rule to the center alone. Translational plus rotational kinetic energy falls from 2 J to about 0.74144 J. The floor supplies an external impulse, so this square’s momentum alone is not conserved. This isolated example is not a promise about a multi-contact pile.
6.5. Friction, a pile, and the actual solver#
The demo uses Planck 1.5.1, a locally vendored MIT-licensed Box2D-derived engine. Contact generation uses separating face axes and clipped polygon edges to create a manifold: up to two touching points for each pair. The velocity constraints include the rotational terms above. Two face contacts can balance torque rather than pretending all load acts at a single corner. Several contacting squares must be solved together through repeated sweeps, not independent one-shot collisions.
Let \(\mu=0.6\) be the dimensionless friction coefficient and \(J_t\) the tangential impulse in N s. Coulomb friction bounds its magnitude by \(|J_t|\leq\mu J_n\). Within this bound, the solver tries to stop sliding contact; otherwise sliding remains. Tangential impulses can also cause torque. This is contact friction, not airborne damping, and the same material is used on boxes and container faces.
The model wrapper uses fixed 1/240 s steps with 12 velocity iterations and 8 position iterations per step. Gravity updates velocity before position, a semi-implicit Euler scheme, not Runge–Kutta. Planck warm-starts persistent contacts with previously computed impulses and uses its two-point normal block solver where appropriate. Position correction reduces remaining penetration separately; it is not an extra restitution impulse. All boxes enable bullet continuous collision detection, including fast box-box pairs. See Planck’s simulation documentation and the pinned solver source.
Numerical contact is not exact geometry. The library uses a 0.005 m linear slop and 0.010 m polygon skin. Drawn faces can therefore show small gaps; the engine tolerates small contact errors rather than chasing roundoff. Tests allow up to 0.015 m negative manifold separation, not arbitrary overlap. The outer container has a floor, two walls and a ceiling, all fixed. Spawn checks use oriented polygon overlap with an extra 0.025 m local margin. Each added body starts with new zero velocities, and existing bodies are never relocated to make room.
A supported island can sleep after motion remains below the library’s thresholds (0.01 m/s and 2 degrees/s) for 0.5 s. Sleeping bodies retain mass and inertia. An awake body hitting a supported pile wakes the contacting island; sleep is a numerical rest approximation, not gluing boxes in place. Gravity, friction, impact loss and contact impulses supply the behavior, with no artificial global velocity damping or energy rescaling. Spawning adds potential energy externally; energy need not decrease when you add a box.
At most 0.1 wall seconds and 24 physics steps are processed per animation frame. Hidden tabs discard time budgets. A slow device can slow playback instead of taking one large unstable step. Clear rebuilds the world. The energy readout sums center kinetic energy, spin energy and gravitational potential \(m g p_y\) relative to the floor. Numerical integration and position correction can introduce small energy errors; this is a stable teaching approximation, not exact energy conservation or a full 3D model.
Focused tests exercise rectangular inertia, rotated vertices, actual polygon contacts, centered versus off-center rotation, free-pair momentum and energy, restitution, fast confinement, sleeping stacks and waking, overlap-free spawning beyond 20 boxes, occupied-space refusal and bounded frame work. The preceding one-ball and two-ball simulations remain separate examples.