First moments of a polyhedron clipped by a paraboloid

Abstract

We provide closed-form expressions for the first moments (i.e., the volume and volume-weighted centroid) of a polyhedron clipped by a paraboloid, that is, of a polyhedron intersected with the subset of the three-dimensional real space located on one side of a paraboloid. These closed-form expressions are derived following successive applications of the divergence theorem and the judicious parametrization of the intersection of the polyhedron's faces with the paraboloid. We provide means for identifying ambiguous discrete intersection topologies, and propose a corrective procedure for preventing their occurence. Finally, we put our proposed closed-form expressions and numerical approach to the test with millions of random and manually engineered polyhedron/paraboloid intersection configurations. The results of these tests show that we are able to provide robust machine-accurate estimates of the first moments at a computational cost that is within one order of magnitude of that of state-of-the-art half-space clipping algorithms.

Figures (8)

• Figure 1: A polyhedron ${\mathcal{P}\subset\mathbb{R}^3}$ intersected by the clipping region ${\mathcal{Q}\subset\mathbb{R}^3}$ located below a paraboloid surface ${\mathcal{S}}$.
• Figure 1: Illustration of the one-dimensional integration domains used for integrating the first, second, and third contributions to the moments.
• Figure 1: Unit cube centred at $1/21/21/2 - k^\intercal$ clipped by the elliptic paraboloid parametrically defined as $z = -x^2 - y^2$.
• Figure 2: Semi-major axis of the conic section generated by the intersection of the paraboloid $\mathcal{S}$, as defined in Eq. \ref{['def:paraboloid']}, when ${\alpha=|\beta|=1}$, with the plane implicitely defined by ${\mathbf{n}\cdot\mathbf{x} = -\mathbf{n}\cdot\mathbf{e}_z}$, with ${\mathbf{n} = \cos(\theta)0\sin(\theta)^\intercal}$, as a function of the angle $\theta$. The semi-major axis of the conic section tends to infinity when $\theta$ approaches a multiple of $\pi$, meaning that a parametrization of any arc of this conic section with trigonometric functions becomes singular if ${\mathbf{n}\cdot\mathbf{e}_z \to 0}$.
• Figure 2: Moments of the unit cube centred at $1/21/21/2 - k^\intercal$ clipped by the elliptic paraboloid parametrically defined as $z = -x^2 - y^2$, and the error in their estimation with $64$-bit floating point arithmetics. The volume moments and their estimation errors are scaled with respect to the moments at $k=3$. The surface area and its estimation error is scaled with respect to the surface area at $k=1$. The volume moments are computed from the analytical expressions derived in \ref{['sec:moments']}, whereas the surface area is computed from Eq. \ref{['eq:surface_area']} using an adaptive Gauss-Legendre quadrature rule. The $64$-bit machine-epsilon $\epsilon_{64} = 2^{-52}$ is shown as the dashed red line.