hindsight.md

️🧐 Hindsight and Follow-Ups

Floating Point-Rounding Errors in ACCD (2025. Feb 26.)

Additive Continuous Collision Detection (ACCD) works only if we can precisely compute the gap distance at any sub-step time. With our single-precision solver, the computed distance may be inaccurate when the gap is extremely tight, potentially leading to pass-through, as illustrated in the figure below.

Our paper does not attempt to fundamentally fix this issue, but our cubic energy helps to drive the distance curve away from such a numerically dangerous zone.

Note that this issue alone is not easy to confirm in simulation because, for such tight gaps (if any), ill-conditioned systems appear before this error emerges.

Quadratic Barrier Function (Section 5.4)

In the paper we presented the following barrier as a quadratic energy counterpart:

$$\begin{equation} \psi_{\mathrm{quad}}(g,\hat{g},\kappa) = \frac{\kappa}{2 \hat{g}} \left(g - \hat{g}\right)^2, \nonumber \end{equation}$$

However, soon after the publication, we realized that this was not the best counterpart since its curvature is a constant $\kappa / \hat{g}$. Our elasticity-inclusive stiffness assumes that the curvature should be on the order of $\kappa$, implying that its magnitude does not align. Since our cubic energy yields a curvature of $2 \kappa$ at $g = \hat{g}/2$, a more suitable counterpart would be:

$$\color{red} \begin{equation} \psi^{\mathrm{new}}_{\mathrm{quad}}(g,\hat{g},\kappa) = \kappa \left(g - \hat{g}\right)^2, \nonumber \end{equation} \color{black}$$

In hindsight, we discovered that with this change, the majority of artifacts arising from the use of quadratic barriers have improved, but objectionable issues persist. We show an example and discuss why.

📉 Artifacts

When the above new quadratic barrier is used, visual artifacts may emerge when contacts are lightly touched as shown in Figure A.

Figure A: Domino scene. Noticeable snags occur when one domino pushes the next ones.

🔄 The Sources of Artifacts

One of the most important differences between quadratic and cubic barriers is how the curvature varies from $g = \hat{g}$ to $g = 0$. For our cubic barrier, the curvature starts from zero at $g = \hat{g}$ and gradually increases to $4\kappa$ at $g = 0$.

In contrast, the quadratic barrier produces its maximum curvature everywhere in $g < \hat{g}$. In other words, our cubic barrier gradually stiffens the system as the gap shrinks to zero, while the quadratic barrier maxes out the stiffness immediately when the barrier is turned on. This can be seen in Figure B.

Note

You can think of this sort of like an ill-configured CPU fan controller, where the fan always runs at full throttle despite low CPU usage. Ideally, the fan should spin in accordance with the CPU temperature, much like our cubic energy where stiffness gradually increases. The quadratic energy, on the other hand, acts like this ill-configured controller; it gets the job done but is mostly overwhelming.

As a result, when $g \approx \hat{g}$ (that is, contacts are lightly touched), the conditioning of the system unnecessarily stiffens, leading to possible artifacts.

Figure B: Visualizing the transition of the magnitude of both our cubic barrier and a quadratic counterpart.

Tilted Slope (Section 5.8)

In Section 5.8, we stated that the slope was tilted at an angle of $30^\circ$.

However, this was incorrect. The actual tilt was set up such that gravity is split into two orthogonal components, with their ratio being 2:1, as shown in Figure C.

Figure C: Actual tilt configuration used in the paper example.

That is, the actual tilt angle is approximately $26.57^\circ$. We chose this setup because, in this way, the sliding object eventually stops its motion when $\mu > 0.5$ and keeps sliding if $\mu < 0.5$.