Geometric Illumination of Implicit Surfaces

TL;DR

This paper introduces a geometry-based framework for illuminating implicit surfaces by replacing polygonal meshes with algebraic representations. It constructs shadows from a point light using a terminator derived from the first polar $\mathcal{S}_L$ and a tangent cone $\mathcal{T}$, then partitions the scene via cylindrical algebraic decomposition into semi-algebraic subregions that are classified as illuminated or self-shaded. A nearest-subregion criterion based on light-ray intersections determines the visible components, and the method generalizes to multiple objects at the cost of higher CAD complexity. While offering precise, non-mesh-based visualizations and connections to occluding contours, the approach is computationally intensive due to the doubly exponential complexity of Gröbner basis calculations and CAD, particularly for high-degree surfaces.

Abstract

Illumination of scenes is usually generated in computer graphics using polygonal meshes. In this paper, we present a geometric method using projections. Starting from an implicit polynomial equation of a surface in 3-D or a curve in 2-D, we provide a semi-algebraic representation of each part of the construction. To solve polynomial condition systems and find constrained regions, we apply algebraic computational algorithms for computing the Gr{\" o}bner basis and cylindrical algebraic decomposition. The final selection of illuminated and self-shaded components for polynomial surfaces of a degree higher than three is discussed. The text is accompanied by visualizations of illumination of surfaces up to degree eight.

Figures (10)

• Figure 1: An initial setting of (left) a folium of Descartes $c: x^3 + y^3 - 6 xy=0$ and (right) a 5th degree surface $\mathcal{S}: x^2 + y^2 + z^4 (z - 1)=0$, and a point light source $L$.
• Figure 2: The 5th degree surface $\mathcal{S}$ with the first polar surface $\mathcal{S}_L$, terminator $t$, and tangent cone $\mathcal{T}$ with the point light source in $L=[1,0,2]$, and a detail in a different position on the shaded part separated by the first polar and terminator line. The equation of a polar is: $\mathcal{S}_L: 2 x + 3 x^2 + 3 y^2 + z^3 (-8 + 9 z)=0$. The tangent cone is given by a polynomial of degree 10 with 151 terms.
• Figure 3: The folium of Descartes and its first polar $c_L: 2 x^2 - x (6 + y) + y (-4 + 3 y)=0$ (green) with respect to the point light source $L=[6,4]$. Intersections of the curve $c$ and the first polar $c_L$ create the terminator.
• Figure 4: A polynomial curve $c: \gamma=((x - 1)^2 + (y - 3)^2 - 1) (x^3 + y^3 - 6 xy)$ consisting of a circle and folium of Descartes. The point light source $L$ is located at three different positions: (a) $[4,6]$, (b) $[1,\frac{1}{2}]$, and (c) $[0,3]$ corresponding to $\gamma(L)>0$, $\gamma(L)<0$, and $\gamma(L)=0$, respectively. The blue components are illuminated, the red components are shaded by the blue components, and the black component is separated by the first polar (green).
• Figure 5: A catalog of components: illuminated (blue), self-shaded (red), separated by the first polar (black); and their composition.