Hyperplane families creating envelopes

Throughout this paper, let $n$ be a positive integer. Moreover, all manifolds, functions and mappings are of class $C^{\infty}$ unless otherwise stated.

A simple geometric mechanism: “the locus of intersections of perpendicular bisectors and normal lines”, often arises in many guises in Physical Sciences. For example, as Richard Feynman elegantly explained in [9], the orbit of a planet around the sun can be understood as a consequence of this mechanism under the assumption of the inverse-square law (see Figure 2 where the circle is the hodograph of the velocity vectors of a planet ​​, that is to say, the circle is a curve drawn by the end points of the vectors that are parallel to the velocity vectors and start at a fixed point $P$. The orbit of the planet is similar to the locus of intersections $B_{t}$ of the perpendicular bisectors of velocity vectors $\overrightarrow{PA_{t}}$ and the normal lines to the circle at $A_{t}$). This is an example in Celestial Mechanics. In the same book [9], one can find that even the historical discovery of atomic nucleus due to Ernest Rutherford can be explained as a consequence of this simple geometric mechanism (see Figure 2 where the center of circle $O$ is an atomic nucleus. The orbit of an $\alpha$ particle is the locus of intersections $B_{t}$ of the perpendicular bisectors of the segment $\overline{PA_{t}}$ and the normal lines to the circle at $A_{t}$ ). This is an example in Nuclear Physics.

In Crystallography, one can find such the mechanism in the so-called Wulff construction for the equilibrium shape of a crystal. A brief explanation of the Wulff construction is as follows. Given an equilibrium crystal, take an arbitrary point $P$ inside the crystal and fix it. Georg Wulff discovered in [20] the so-called Gibbs-Wulff theorem which asserts that the length from the fixed point $P$ to the foot of the perpendicular to the tangent space to the face of the crystal is proportional to its surface energy density of the face. Let $\gamma:S^{2}\to\mathbb{R}$ be the surface energy density function of the equilibrium crystal. The graph of $\gamma$ with respect to the polar coordinates about the point $P$ defines the mapping $g:S^{2}\to\mathbb{R}^{3}$. The mapping $g$ is often called the polar plot of $\gamma$ or the $\gamma$-plot or the Wulff plot. Set $f=2g$ and suppose that the image $f(S^{2})$ has the well-defined normal vectors at any point $f(x)$. Then, by the Gibbs-Wulff theorem, the accurate shape of the crystal surface is proportional to the shape obtained by our simple geometric mechanism: “the locus of the intersection of the perpendicular bisector of the vector $\overrightarrow{Pf(x)}$ and the normal line to $f(S^{2})$ at $f(x)$”. This is the Wulff construction and the constructed shape is called the Wulff shape. Notice that in general $f$ is a continuous mapping and thus from the viewpoint of rigorous mathematics, the Wulff construction is not a well-defined construction method in general. Nevertheless, Hoffman and Cahn showed in [11] that if $\gamma:S^{2}\to\mathbb{R}$ is differentiable, then the image $f(S^{2})$ has a well-defined normal vector at each point $f(x)$ and the set $\{\nabla\gamma(x)+\gamma(x)x\;|\;x\in S^{2}\}$ is exactly the shape obtained by our simple geometric mechanism for the point $P$ and the surface $f(S^{2})$. The Wulff construction and the Cahn-Hoffman formula in the plane is depicted in Figure 3. For details on the Wulff construction and Wulff shapes, see for instance [8, 10].

Moreover, it is a surprising fact that our simple geometric mechanism: “the locus of intersections of perpendicular bisectors and normal lines”can be applied even to Seismic Survey (see 7.14 (9) of [6]).

In Mathematics, our simple geometric mechanism: “the locus of intersections of perpendicular bisectors and normal lines”is called the anti-orthotomic of a mapping $f$ having a well-defined normal vector to its image at each point (for details on anti-orthotomics, see 7.14 of [6]. See also [15] where anti-orthotomics are generalized to frontals and [16] where more elementary explanation on anti-orthotomics can be found). In Mathematics as well, there are examples where anti-orthotomics are effectively applied (see [6]).

In order to understand better the powerfulness of the simple geometric mechanism, we would like to have more striking examples in Mathematics where anti-orthotomics are effectively applied. Namely, we want to seek mathematical problems which can be geometrically solved by our simple geometric mechanism though it seems difficult to solve them by other methods. This is the primitive motivation of this paper. In this paper, we show that the existence and uniqueness problem of envelopes for a given hyperplane family is one of such problems. Namely, we give a necessary and sufficient condition (see Definition 2) for a given hyperplane family to create an envelope. And then, we give a necessary and sufficient condition for the uniqueness of created envelopes if the given hyperplane family creates an envelope. It seems difficult to prove that the condition given in Definition 2 is actually a sufficient condition to create envelopes by other methods. In order to apply our simple geometric mechanism, we need some geometric objects to which the normal line can be reasonably well-defined at each point. Hyperplane families themselves are far from the reasonable geometric objects for our purpose. The reasonable geometric objects are frontals (the definition of frontal is given in the next paragraph). In order to obtain a frontal from a given hyperplane family, the mirror-image mapping will be locally introduced. Then, it turns out that if the given hyperplane family is creative (see Definition 2 below), then the mirror-image mapping is actually a frontal such that the normal line at each point intersects the corresponding hyperplane. Thus, we can apply the anti-orthotomic method developed in [15] to obtain Theorem 1 and Theorem 2. The existence and uniqueness problem of envelopes for a given hyperplane family can be easily interpreted as the existence and uniqueness problem of solutions for a certain type of system of first order differential equations with one constraint condition. In the author’s opinion, one of the most attractive features of our simple geometric mechanism is that it can make all solutions and their precise expressions clear in one shot by geometry without the need to solve the corresponding system of differential equations with one constraint condition.

Let $S^{n}$ be the $n$-dimensional unit sphere in the $(n+1)$-dimensional vector space $\mathbb{R}^{n+1}$. Given a point $P$ of $\mathbb{R}^{n+1}$ and an $(n+1)$-dimensional unit vector ${\bf n}\in S^{n}\subset\mathbb{R}^{n+1}$, the hyperplane $H_{\left(P,{\bf n}\right)}$ relative to $P$ and ${\bf n}$ is naturally defined as follows, where the dot in the center stands for the standard scalar product of two vectors $(X-P)$ and ${\bf n}$ in the vector space $\mathbb{R}^{n+1}$.

Let $N$ be an $n$-dimensional manifold without boundary. Given two mappings $\widetilde{\varphi}:N\to\mathbb{R}^{n+1}$ and $\widetilde{\nu}:N\to S^{n}$, the hyperplane family $\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$ relative to $\widetilde{\varphi}$ and $\widetilde{\nu}$ is naturally defined as follows.

A mapping $\widetilde{f}:N\to\mathbb{R}^{n+1}$ is called a frontal if there exists a mapping $\widetilde{\nu}:N\to S^{n}$ such that $d\widetilde{f}_{x}({\bf v})\cdot\widetilde{\nu}(x)=0$ for any $x\in N$ and any ${\bf v}\in T_{x}N$, where two vector spaces $T_{\widetilde{f}(x)}\mathbb{R}^{n+1}$ and $\mathbb{R}^{n+1}$ are identified. By definition, it is natural to call $\widetilde{\nu}:N\to S^{n}$ a Gauss mapping of the frontal $\widetilde{f}$. The notion of frontal has been recently investigated (for instance, see [13]). In this paper, as the definition of envelope created by a hyperplane family, the following is adopted.

By definition, any envelope $\widetilde{f}:N\to\mathbb{R}^{n+1}$ created by a hyperplane family $\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$ must be a frontal with Gauss mapping $\widetilde{\nu}:N\to S^{n}$. For details on envelopes created by families of plane regular curves, refer to [6]. In Chapter 5 of [6], several definitions for envelope are given. For a hyperplane family $\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$, Definition 1 is a generalization of their definition $E_{2}$ from a viewpoint of parametrization ($E_{2}$ envelope is a variety tangent to all lines of the given line family. Thus, in the case of plane, an envelope defined by Definition 1 is the same notion of $E_{2}$ envelope. For details on the definition $E_{2}$, see 5.12 of [6]). The following definition, which may be regarded as a higher dimensional generalization of $E_{1}$ from a viewpoint of parametrization ($E_{1}$ envelope is the set of the limits of intersections with nearby members of the given line family. For details on the definition $E_{1}$, see 5.8 of [6] and for the relation between Definition 2 in the plane case and $E_{1}$, see Subsection 2.3), is the key notion for this paper.

By definition, any frontal $\widetilde{f}:N\to\mathbb{R}^{n+1}$ with Gauss mapping $\widetilde{\nu}:N\to S^{n}$ is an envelope created by $\mathcal{H}_{(\widetilde{f},\widetilde{\nu})}$. Therefore, the notion of envelope created by a hyperplane family is the same as the notion of frontal. Moreover, it is clear that for any mapping $\widetilde{\nu}:N\to S^{n}$, a constant mapping $\widetilde{f}:N\to\mathbb{R}^{n+1}$ is an envelope created by $\mathcal{H}_{\left(\widetilde{f},\widetilde{\nu}\right)}$. On the other hand, for a constant mapping $\widetilde{\nu}:\mathbb{R}\to S^{1}$, if the line family $\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$ does not create an envelope then $\widetilde{\varphi}:\mathbb{R}\to\mathbb{R}^{2}$ must be not constant. From these elementary observations, it is natural to ask to obtain a necessary and sufficient condition for a given hyperplane family $\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$ to create an envelope $\widetilde{f}:N\to\mathbb{R}^{n+1}$ in terms of $\widetilde{\gamma}:N\to\mathbb{R}$ and $\widetilde{\nu}:N\to S^{n}$. Moreover, it is also desirable to solve the following two incidentally. “Suppose that a given hyperplane family $\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$ creates an envelope $\widetilde{f}:N\to\mathbb{R}^{n+1}$. Then, obtain a representation formula of $\widetilde{f}$.”  “Suppose that $n=1$. Then, find the precise relation between $E_{1}$ envelope and $E_{2}$ envelope.” In this paper, as an application of our simple geometric mechanism, all of these problems are solved as follows.

The key idea for the proof of Theorem 1 is to regard the given hyperplane family as a moving mirror parametrized by $x\in N$. Then, for any parameter $x_{0}\in N$, by taking a point $P\in\mathbb{R}^{n+1}$ outside the mirror $H_{\left(\widetilde{\varphi}(x_{0}),\widetilde{\nu}\left(x_{0}\right)\right)}$, the mirror-image

of $P$ by the mirror $H_{\left(\widetilde{\varphi}(x),\widetilde{\nu}(x)\right)}$ must have the same information as the mirror since the mirror is reconstructed as the perpendicular bisectors of the segment $\overline{Pf_{{}_{P}}(x)}$, where $x$ is a point in a sufficiently small neighborhood $U_{{}_{P}}$ of $x_{0}$.

Hence, investigation of the given hyperplane family $\mathcal{H}_{\left(\widetilde{\varphi}|_{{}U_{{}P}},\widetilde{\nu}|_{{}U_{{}P}}\right)}$ may be replaced with analyzing the associated mirror-image mapping $f_{{}_{P}}:U_{{}_{P}}\to\mathbb{R}^{n+1}$ (see Figure 4). This suggests applicability of results in [15] to the problem of this paper.

A sketch of the proof of Theorem 1 (1) may be given as follows. Suppose that the hyperplane family $\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$ is creative. Then, by definition, there exists a mapping $\widetilde{\Omega}:N\to T^{*}S^{n}$ having the form $\widetilde{\Omega}(x)=\left(\widetilde{\nu}(x),\widetilde{\omega}(x)\right)$ such that the equality $d\widetilde{\gamma}=\widetilde{\omega}$ holds as germs of $1$-form at $x_{0}$. Then, by investigating the Jacobian matrix of the mirror-image mapping $f_{{}_{P}}:U_{{}_{P}}\to\mathbb{R}^{n+1}$ at $x\in U_{{}_{P}}$ directly, it turns out that for any $x\in U_{{}_{P}}$ the non-zero vector

is perpendicular to the vector $d\left(f_{{}_{P}}\right)_{x}\left({\bf v}\right)$ for any ${\bf v}\in T_{x}N$, where $\mathbb{R}^{n+1}$, $T_{\widetilde{\nu}\left(x\right)}\mathbb{R}^{n+1}$ and $T^{*}_{\widetilde{\nu}\left(x\right)}\mathbb{R}^{n+1}$ are identified and $\frac{\partial}{\partial\Theta_{\left(i,\widetilde{\nu}(x)\right)}}=P_{\left(\widetilde{\nu}(x),\widetilde{\nu}\left(x_{0}\right)\right)}\left(\frac{\partial}{\partial\Theta_{i}}\right)$. Thus, $f_{{}_{P}}:U_{{}_{P}}\to\mathbb{R}^{n+1}$ is a frontal. From the construction, the mapping $\widetilde{f}_{{}_{P}}={\bf v}_{{}_{P}}+f_{{}_{P}}:U_{{}_{P}}\to\mathbb{R}^{n+1}$ must be exactly the same as the mapping $\widetilde{f}_{{}_{P}}$ given in Theorem 1 of [15]. Therefore, by Theorem 1 of [15] asserting that $\widetilde{f}_{{}_{P}}$ satisfies both conditions (a), (b) of Definition 1, $\widetilde{f}_{{}_{P}}$ is an envelope created by the hyperplane family $\mathcal{H}_{\left(\widetilde{\varphi}|_{U_{{}_{P}}},\widetilde{\nu}|_{U_{{}_{P}}}\right)}$. The mapping $\widetilde{f}_{{}_{P}}:U_{{}_{P}}\to\mathbb{R}^{n+1}$ is called the anti-orthotomic of $f_{{}_{P}}:U_{{}_{P}}\to\mathbb{R}^{n+1}$ relative to $P$. Calculation shows

Thus, unlike $f_{{}_{P}}(x_{0})$, the location $\widetilde{f}_{{}_{P}}(x_{0})$ does not depend on the particular choice of $P$. In other words, in order to discover the formula $(*)$, the role of $P$ is merely an auxiliary point just like an auxiliary line in elementary geometry (see Figure 5).

Since $x_{0}$ is an arbitrary point of $N$, the hyperplane family $\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$ creates an envelope $\widetilde{f}:N\to\mathbb{R}^{n+1}$.

Theorem 1 (2) is a direct by-product of the proof of Theorem 1 (1) (see Figure 5). Theorem 1 (3) seems to be not a direct by-product of the proof of Theorem 1 (1) although it can be proved relatively easily by using the above argument (see Subsection 2.3).

When $N=S^{n}$ and $\widetilde{\nu}:S^{n}\to S^{n}$ is the identity mapping, it is easily seen $\widetilde{\omega}(x)=\nabla\widetilde{\gamma}(x)$. Therefore, in the case that $N=S^{n}$ and $\widetilde{\nu}:S^{n}\to S^{n}$ is the identity mapping, Theorem 1 (2) has been known as the Cahn-Hoffman vector formula ([11]). Theorem 1 (2) is a comprehensive generalization of their formula. Any Wulff shape is clearly a convex body and conversely it is known that any convex body can be constructed by the Wulff construction (for instance, see [19]). There are many Wulff shapes such that the surface energy density functions $\gamma:S^{n}\to\mathbb{R}$ are not differentiable (convex polytopes are typical examples). Thus, for studing Wulff shapes having non-smooth surface energy functions, it is very significant to answer the following two problems: “(a) Generalize Cahn-Hoffman vector formula to the corresponding formula for any $\widetilde{\nu}:N\to S^{n}$”ȧnd “(b) Resolution of singularities of the boundary of a convex body having non-smooth boundary by a frontal $\widetilde{f}:S^{n}\to\mathbb{R}^{n+1}$”. By Theorem 1 (2), the problem (a) is completely solved. As for the problem (b), to the best of author’s knowledge, only the boundary of a square has been realized as a frontal $\widetilde{f}:S^{1}\to\mathbb{R}^{2}$ so far (see [15]). Although there are apparently no published proofs at present, it is a comparatively straightforward generalization of this result to show that the boundary of a convex polygon is realized as a frontal $\widetilde{f}:S^{1}\to\mathbb{R}^{2}$. However, even in the plane case, the problem (b) for the boundary of a convex body in general seems to be wrapped in mystery.

Moreover, Theorem 1 (2) might be useful even for the study of force problems in higher dimensional vector spaces. In [4], Petr Blaschke discovered that pedal coordinates are more suitable settings to study force problems in $\mathbb{R}^{2}$. Readers who want to confirm their usefulness are recommeded to refer to [4] (see also 7.24 (6) of [6] though this is not a force problem but a very suitable problem for understanding how useful pedal coordinates are). Theorem 1 (2) may be regarded as a higher dimensional generalization of pedal coordinates. Hence, it is expected that Theorem 1 (2) is a very suitable expression to study force problems etc. in all finite-dimensional vector spaces over $\mathbb{R}$. Example 4.2 (2) might be regarded as examples in which higher dimensional version of pedal coordinates are effectively used.

As an application of Theorem 1, a characterization for a hyperplane family to create a unique envelope is given as follows.

Under the assumption that $\Omega$ in Remark 1.1 (2) is immersive and some conditions are satisfied, a unique existence result of envelopes for hyperplane families has been obtained in [7]. Since their assumptions clearly imply that the creative condition defined in Definition 2 is satisfied and the set consisting of regular points of $\widetilde{\nu}$ is dense, their result follows from Theorem 1 and Theorem 2.

Notice that non-unique existence cases, too, are intriguing cases since Theorem 1 may be effectively applied even in such cases (see Example 4.2 (1), (2)).

This paper is organized as follows. Theorem 1 and Theorem 2 are proved in Section 2 and Section 3 respectively. In Section 4, examples are given. Section 5 is an appendix where an alternative proof of Theorem 1 except for Theorem 1 (3) is given. The alternative proof is a proof by a gauge theoretic approach. In order to avoid unnecessary complication, the alternative proof is given only in the case $n=1$. The author has no idea on how to prove Theorem 1 (3) by using the alternative proof.