Visualizing 3-Dimensional Manifolds

Year: 2013 Authors: Dugan J. Hammock

Core claim

Intersection slices of a 3-manifold in R^4 can be computed as level sets in parameter space and efficiently visualized in R^3 via projection.

Topics

3-dimensional manifolds, hyperplane slicing, computer visualization, projected isosurfaces

Domains

differential topology, manifold parametrization, level sets, higher-dimensional geometry, scientific visualization, geometric aesthetics, mathematical art

Methods

hyperplane intersection, parameter-space level sets, projection to R^3, computer plotting

Media

Mathematica, MATLAB, 3D plots, images

Paper text

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Proceedings of Bridges 2013: Mathematics, Music, Art, Architecture, Culture

Visualizing 3-Dimensional Manifolds

Dugan J. Hammock

Dept. of Mathematics, University of Massachusetts

Lederle Graduate Research Tower,

Amherst, MA 01003-9305, USA

hammock@math.umass.edu

Abstract

Given a parametrized 3-dimensional manifold sitting in 4-dimensional space, we wish to visualize it by looking at its intersections with 3-dimensional hyperplanes. The intersections are 2-dimensional surfaces in 4-space which can then be projected into 3-space for visualization. In this paper I present an algorithm for displaying these surfaces of intersection using computer plotting applications (e.g. Mathematica, MATLAB, etc.).

Methodology

Let $\phi :U\to \mathcal{M}\subset {\mathbb{R}}^4$ be a parametrization of a 3-manifold $\mathcal{M}$ where $U\subset {\mathbb{R}}^3$ is a region in the parameter space of $\phi$ . Let $f:{\mathbb{R}}^4\to \mathbb{R}$ be a smooth function whose differential $Df$ never vanishes, so that the level sets $f^{-1}(c)$ are 3-dimensional hypersurfaces in ${\mathbb{R}}^4$ . For this paper, $f$ is taken to be the dot product $f(\vec{v})=\vec{\eta }\cdot \vec{v}$ for some nonzero vector $\vec{\eta }\in {\mathbb{R}}^4$ ; the level sets $f^{-1}(c)$ are the hyperplanes in ${\mathbb{R}}^4$ perpendicular to $\vec{\eta }$ . Let $\pi :{\mathbb{R}}^4\to {\mathbb{R}}^3$ be some projection or mapping, for this paper $\pi$ is taken to be the projection onto the $xyz$ -hyperplane given by $\pi (x,y,z,w)=(x,y,z)$ .

Let $c\in \mathbb{R}$ be some value and consider the hypersurface $f^{-1}(c)\subset {\mathbb{R}}^4$ . We wish to compute the intersection ${\mathcal{M}}^{\prime }=(\mathcal{M}\cap f^{-1}(c))$ and then project this surface from the ambient space ${\mathbb{R}}^4$ to ${\mathbb{R}}^3$ via the map $\pi$ . The slice ${\mathcal{M}}^{\prime }$ is in general a 2-dimensional submanifold of $\mathcal{M}$ , and its projected image $\pi ({\mathcal{M}}^{\prime })=\pi (\mathcal{M}\cap f^{-1}(c))\subset {\mathbb{R}}^3$ is what we wish to observe as a 2-dimensional manifold in 3-space.

Note that if $\mathcal{M}$ is indeed parametrized by the patch $\phi :U\to \mathcal{M}$ , in particular if $\phi$ is onto, then every point $m\in \mathcal{M}$ has a preimage in the set $U$ , so $\phi (U)$ and $\mathcal{M}$ are equal as sets. We may take the slice ${\mathcal{M}}^{\prime }\subset \mathcal{M}$ and consider its preimage under $\phi$ as a subset $U^{\prime }\subset U$ in the parameter space of $\phi$ . Let $U^{\prime }$ be defined in this way; then $U^{\prime }:={\phi }^{-1}({\mathcal{M}}^{\prime })={\phi }^{-1}(f^{-1}(c)\cap \mathcal{M})=(f\circ \phi {)}^{-1}(c)\subset U\subset {\mathbb{R}}^3$ .

The original intersection ${\mathcal{M}}^{\prime }$ can be recovered by mapping the set $U^{\prime }\subset {\mathbb{R}}^3$ back into ${\mathbb{R}}^4$ via $\phi$ , since by definition we have $\phi (U^{\prime })=\phi ({\phi }^{-1}({\mathcal{M}}^{\prime }))={\mathcal{M}}^{\prime }$ . Mathematically, this fact is a trivial consequence of the stipulation that $\phi$ is onto. Computationally, however, this is important because $U^{\prime }=(f\circ \phi {)}^{-1}(c)$ can be computed directly as a level set (or “isosurface”) in the parameter space $U$ inside ${\mathbb{R}}^3$ . Once $U^{\prime }$ is computed, ${\mathcal{M}}^{\prime }=\phi (U^{\prime })$ is recovered by mapping $U^{\prime }$ back into ${\mathbb{R}}^4$ via $\phi$ ; from there we project ${\mathcal{M}}^{\prime }$ down to ${\mathbb{R}}^3$ via $\pi$ . The projected image of the slice ${\mathcal{M}}^{\prime }$ is the set $\pi ({\mathcal{M}}^{\prime })=\pi (\phi (U^{\prime }))=(\pi \circ \phi )(U^{\prime })=(\pi \circ \phi )\left((f\circ \phi {)}^{-1}(c)\right)$ . Thus, $\pi ({\mathcal{M}}^{\prime })$ is computed as the image of the isosurface $(f\circ \phi {)}^{-1}(c)\subset {\mathbb{R}}^3$ under the map $(\pi \circ \phi ):{\mathbb{R}}^3\to {\mathbb{R}}^3$ .

Hammock

Examples and Connections to Art

Figure 1: Slices of the manifolds $S^3$ , $S^2\times S^1$ , ${\mathbb{T}}^3$ , and two different ${\mathbb{T}}^2$ -bundles over $S^1$ . In the above parametrizations, $R_{ab}[\theta ]$ denotes the rotation in the ab-plane by angle $\theta$ , and ${\vec{\mathbf{e}}}_1=(1,0,0,0)$ denotes the $x$ -coordinate vector. Also shown are the $\{w=0\}$ slices of various 3-manifolds parametrized as fiber bundles over $S^1$ whose fibers are (oriented) genus-2 surfaces which perform any number of twists about two rotational planes as they trace around the base $S^1$ . These are all examples of aesthetically pleasing and geometrically interesting shapes that can be generated efficiently as slices of 3-dimensional manifolds.