3-D Shadows of 4-D Algebraic Hypersurfaces in a 4-D Perspective

TL;DR

A general method to find shadow boundaries in an arbitrary dimension and apply it in a three- and four-dimensional space is described and a system of polynomial equations to construct occluding contours of algebraic surfaces in a 4-D perspective is designed.

Abstract

The paper is focused on the four-dimensional visualization of hypersurfaces represented by implicit equations without their parametrization. We describe a general method to find shadow boundaries in an arbitrary dimension and apply it in a three- and four-dimensional space. Furthermore, we design a system of polynomial equations to construct occluding contours of algebraic surfaces in a 4-D perspective. The method is presented on a composed 3-D scene and three 4-D cases with gradual complexity. In general, our goal is to improve the understanding of spatial properties in a four-dimensional space.

Figures (16)

• Figure 1: Initial setting of hypersurfaces $\mathcal{S}, \mathcal{P}$, and a point light source $L$.
• Figure 2: The polar hypersurface $\mathcal{S}_L$ of a hypersurface $\mathcal{S}$ with respect to a point light source $L$ and its terminator $c$.
• Figure 3: The tangent hypercone $\mathcal{T}$ to a hypersurface $\mathcal{S}$ through a light source $L$ and the shadow cast on $\mathcal{P}$.
• Figure 4: (left) Roman surface: $-2 (x-2) (y-1) z+(x-2)^2 (y-1)^2+(x-2)^2 z^2+(y-1)^2 z^2=0$ and (right) Cross-cap: $\left((x+1)^2+y^2\right) \left((x+1)^2+z^2\right)+(x+1)^2+\frac{y^4}{4}+y^2 z=0$, their parts separated by the first polars, and polar boundaries projected to a plane $z+2=0$. Without omission of the self-shaded parts.
• Figure 5: A 2-D situation of a Cassini oval (degree 4) and its self-shading from the point light source. The green curve is the first polar, dividing the plane into two areas. The area that does not contain the light source is excluded. The subcones in 2-D case are plane angles bounded by the rays from the light source. The blue arcs represent illuminated parts, and the red arcs are in the self-shade.