Reliable Computation of the Singularities of the Projection in $\mathbb R^3$ of a Generic Surface of $\mathbb R^4$

Abstract

Computing efficiently the singularities of surfaces embedded in $\mathbb R^3$ is a difficult problem, and most state-of-the-art approaches only handle the case of surfaces defined by polynomial equations. Let F and G be $C^\infty$ functions from $\mathbb R^4$ to $\mathbb R$ and $\mathcal M=\{(x,y,z,t) \in \mathbb R^4 \, | \, F(x,y,z,t)=G(x,y,z,t)=0\}$ be the surface they define. Generically, the surface $\mathcal M$ is smooth and its projection $\Omega$ in $\mathbb R^3$ is singular. After describing the types of singularities that appear generically in $\Omega$, we design a numerically well-posed system that encodes them. This can be used to return a set of boxes that enclose the singularities of $\Omega$ as tightly as required. As opposed to state-of-the art approaches, our approach is not restricted to polynomial mapping, and can handle trigonometric or exponential functions for example.

7 Appendix: Proof of Lemma 1

Let $q\in \mathcal {M}$ be a cross-cap singularity of the projection $\mathfrak p : \mathcal {M} \mapsto \mathbb R^3$.

First, the condition $q \in \Sigma ^1(\mathfrak p )$ means that $d\mathfrak p (q)$ has corank 1. Since rank($\mathfrak p$) = 2 – corank($\mathfrak p$) = 1, the condition is also equivalent to $d\mathfrak p (q)$ has rank 1. In other words, the 2-dimensional tangent plane to $\mathcal {M}$ at q projects to a line, that is the direction of projection is in the tangent plane. Thus, the condition $q \in \Sigma ^1(\mathfrak p )$ of Definition 7 is equivalent to the first condition of Lemma 1: the direction of projection is in the tangent plane.

We now assume that the surface $\mathcal {M}$ is locally parameterized in a neighborhood of q by $(z,t) \mapsto (a(z,t),b(z,t),z,t)$, so that $\mathfrak p (z,t)= (a(z,t),b(z,t),z)$. The space $J^1(\mathcal {M},\mathbb R^3)$ is thus locally equal to $U\times \mathbb R^3 \times L(\mathbb R^2,\mathbb R^3)$ where U is a subset of $\mathbb R^2$ and L stands for the space of linear mappings. The 1-jet of a mapping $(f_1(z,t),f_2(z,t),f_3(z,t)): \mathcal {M} \mapsto \mathbb R^3$ is

$$\left( (z,t), (f_1(z,t),f_2(z,t),f_3(z,t)), \begin{pmatrix} f_{1z} &{} f_{1t} \\ f_{2z} &{} f_{2t} \\ f_{3z} &{} f_{3t}\end{pmatrix} \right) . $$

$\Sigma ^1$ is the subset of $J^1(\mathcal {M},\mathbb R^3)$ such that the matrix $\begin{pmatrix} f_{1z} &{} f_{1t} \\ f_{2z} &{} f_{2t} \\ f_{3z} &{} f_{3t}\end{pmatrix}$ has corank 1, that is has rank 1. Without loss of generality, if we assume $(f_{3z}, f_{3t})\ne (0,0)$, $\Sigma ^1$ is thus implicitly defined by the two equations: $\begin{vmatrix} f_{1z}&f_{1t} \\ f_{3z}&f_{3t} \end{vmatrix} = 0 \text { and } \begin{vmatrix} f_{2z}&f_{2t} \\ f_{3z}&f_{3t} \end{vmatrix}=0.$ One thus has $\Sigma ^1=\Phi ^{-1}(0)$ with

$$\begin{aligned} \Phi : J^1 (\mathcal { M}, \mathbb R^3)&\rightarrow \mathbb R^2&\\ \left( (z,t), (f_1(z,t),f_2(z,t),f_3(z,t)), \begin{pmatrix} f_{1z} &{} f_{1t} \\ f_{2z} &{} f_{2t} \\ f_{3z} &{} f_{3t} \end{pmatrix} \right)&\mapsto \begin{pmatrix} f_{1z} f_{3t} - f_{1t} f_{3z} \\ f_{2z} f_{3t} - f_{2t} f_{3z} \end{pmatrix}&\end{aligned}$$

According to [6, Lemma 4.3], $j^1\mathfrak p$ is transverse to $\Sigma ^1$ at q iff $\Phi \cdot j^1\mathfrak p$ is a submersion at q. On the other hand, $\Phi \cdot j^1\mathfrak p = \Phi \left( (z,t), (a(z,t),b(z,t),z), \begin{pmatrix} a_{z} &{} a_{t} \\ b_{z} &{} b_{t} \\ 1 &{} 0\end{pmatrix} \right) =-(a_t,b_t)$. This mapping is a submersion iff its Jacobian $\begin{pmatrix} a_{zt} &{} a_{tt} \\ b_{zt} &{} b_{tt} \end{pmatrix}$ is full rank, that is $a_{zt}b_{tt}-a_{tt}b_{zt}\ne 0$ which is exactly the second condition of Lemma 1.