arXiv subjects
Publications and source records attributed to Takashi Nishimura.
In this paper, on envelopes created by sphere families in Euclidean 3-space, all four basic problems (existence problem, representation problem, problem on the number of envelopes, problem on relationships of definitions) are solved.
We show that, under an additional mild assumption, on the class of generic frontals, any involution whose fixed point set is exactly the same as the fixed point set of the Legendre involution must be the Legendre involution (Theorem 2 in §1). Moreover, its natural complexification (Corollary 1 in §3) is simultaneously shown.
There is a widespread method to represent the envelope when a given hyperplane family creates an envelope. However, one sometimes encounters cases when the widespread method fails to represent the desired envelope precisely, and is confused. At the same time, one wants to find a correct method to draw the envelope precisely. In this article, focused on straight line families in the plane, an easy to understand explanation is given on the recently discovered correct method to represent the envelope precisely. Moreover, it is explained when and why the widespread method fails to represent the precise shape of envelope as well.
In this paper we investigate the relationships between envelopes of circle families and some special curves in the plane, such as evolutes, pedals, evolutoids and pedaloids.
In this paper, on envelopes created by circle families in the plane, all four basic problems (existence problem, representation problem, problem on the number of envelopes, problem on relationships of definitions) are solved.
A simple geometric mechanism: "the locus of intersections of perpendicular bisectors and normal lines", often arises in many guises in Nonlinear Sciences. In this paper, a new application of this simple geometric mechanism is given. Namely, we show that this mechanism gives answers to all four basic problems on envelopes created by hyperplane families (existence problem, representation problem, equivalence problem of definitions, uniqueness problem) at once.
In this paper, it is shown that for any two non-empty closed (resp., open) and spherical convex subsets $\mathcal{W}_1, \mathcal{W}_2$ of $S^n$, the intersection $\mathcal{W}_1\cap \mathcal{W}_2$ is empty if and only if the subset $\{P\in S^n\; |\; P\cdot Q>0 \mbox{ for any } Q\in \mathcal{W}_1 \mbox{ and } P\cdot R<0 \mbox{ for any } R\in \mathcal{W}_2\}$ is non-empty, open (resp., closed) and spherical convex.
Let $f: N^n\to \mathbb{R}^{n+1}$ be a frontal with its Gauss mapping $ν: N\to S^n$ and let $P\in \mathbb{R}^{n+1}$ be a point such that $(f(x)-P)\cdot ν(x) \ne 0$ for any $x\in N$. In this paper, for the mapping $\widetilde{f}: N\to \mathbb{R}^{n+1}$ defined by $$ \widetilde{f}(x)=f(x)-\frac{||f(x)-P||^2}{2(f(x)-P) \cdot ν(x)}ν(x), $$ the following four are shown. (1) $\widetilde{f}$ is a frontal with its Gauss mapping $\widetildeν(x)=\frac{f(x)-P}{||f(x)-P||}$ at $\widetilde{f}(x)$. (2) $\widetilde{f}$ is the unique anti-orthotomic of $f$ relative to $P$. (3) The property $(\widetilde{f}(x)-P)\cdot \widetildeν(x)\ne 0 $ holds for any $x\in N$. (4) The equality $||\widetilde{f}(x)-P||=||\widetilde{f}(x)-f(x)||$ holds for any $x\in N$. Moreover, three applications of the main result are given. As the first application, a generalization of Cahn-Hoffman vector formula is given. The second application is to clarify an optical meaning of anti-orthotomics. The third application gives a criterion to be a front for a given frontal.
In this paper, it is shown that for an $n$-dimensional spherical unit speed curve $γ: I\to S^n$, a given point $P \in S^n$ and a point $s_0$ of the open interval $I$, the spherical orthotomic curve-germ $ort_{γ, P}: (I, s_0)\to S^n$ of $γ$ relative to $P$ is $\mathcal{L}$-equivalent to the spherical pedal curve-germ $ped_{γ, P}: (I, s_0)\to S^n$ of $γ$ relative to $P$ (resp., the spherical dual curve-germ ${\bf u}_n: (I, s_0)\to S^n$ of $γ$) if and only if $P\ne \pm{\bf u}_n(s_0)$ (resp., if $P= \pm{\bf u}_n(s_0)$).
For the n-dimensional spherical pedal curve $ped_{γ,P}$ with respect to an n-dimensional spherical unit speed curve $γ$ and a given point $P \in S^n$, we define the spherical orthotomic curve of $γ$ relative to the point $P$, and classify singularities of spherical orthotomic curves.
In this paper, it is shown that, for any equidimensional $C^\infty$ map-germ $f: (\mathbb{R}^n,0)\to (\mathbb{R}^n,0)$, the map-germ $F: (\mathbb{R}^n, 0) \to \mathbb{R}^n\times\mathbb{R}^{\ell}$ defined by $F(x)=\left(f(x), μ_1(x){|Jf|^2(x)}, \cdots, μ_\ell(x){|Jf|^2(x)}\right)$ is always a frontal; where $μ_i$ is a $C^\infty$ function-germ and $|Jf|$ is the Jacobian-determinant of $f$. Moreover, it is also shown that when the multiplicity of $f$ is less than or equal to $3$, any frontal constructed from $f$ must be $\mathcal{A}$-equivalent to a frontal $F$ of the above form.
Singular quadratic mappings creating Kato's chaos are given.
For any given Wulff shape $\mathcal{W}$, we can define the unique continuous function $S^{n}\to \mathbb{R}_{+}$ called convex integrand, denoted by $γ_{{}_{\mathcal{W}}}$. In this paper, we show that, for any Wulff shapes $\mathcal{W}_{1}$ and $\mathcal{W}_{2}$, the equality $d(γ_{{}_{\mathcal{W}_{1}}}, γ_{{}_{\mathcal{W}_{2}}})= h(\mathcal{W}_{1}, \mathcal{W}_{2})$ holds, where $d$ is the maximum distance of the function space consisting of convex integrands and $h$ is the Pompeiu-Hausdorff distance of the space consisting of Wulff shapes. Moreover, applications of this result are given.
In this paper, we investigate simultaneous properties of a convex integrand $γ$ and its dual $δ$. The main results are the following three. (1) For a $C^\infty$ convex integrand $γ: S^n\to \mathbb{R}_+$, its dual convex integrand $δ: S^n\to \mathbb{R}_+$ is of class $C^\infty$ if and only if $γ$ is a strictly convex integrand. (2) Let $γ: S^n\to \mathbb{R}_+$ be a $C^\infty$ strictly convex integrand. Then, $γ$ is stable if and only if its dual convex integrand $δ: S^n\to \mathbb{R}_+$ is stable. (3) Let $γ: S^n\to \mathbb{R}_+$ be a $C^\infty$ strictly convex integrand. Suppose that $γ$ is stable. Then, for any $i$ $(0\le i\le n)$, a point $θ_0\in S^n$ is a non-degenerate critical point of $γ$ with Morse index $i$ if and only if its antipodal point $-θ_0\in S^n$ is a non-degenerate critical point of the dual convex integrand $δ$ with Morse index $(n-i)$.
In this paper, the following three are shown. (1) For a $C^\infty$ convex integrand $γ: S^n\to \mathbb{R}_+$, its dual convex integrand $δ: S^n\to \mathbb{R}_+$ is of class $C^\infty$. (2) For a stable convex integrand $γ: S^n\to \mathbb{R}_+$, its dual convex integrand $δ: S^n\to \mathbb{R}_+$ is stable. (3) Let $γ: S^n\to \mathbb{R}_+$ be a stable convex integrand. Then, for any $i$ $(0\le i\le n)$, $θ_0\in S^n$ is a non-degenerate critical point of $γ$ with Morse index $i$ if and only if $-θ_0\in S^n$ is a non-degenerate critical point of the dual convex integrand $δ$ with Morse index $(n-i)$.
Let $γ: S^n\to \mathbb{R}_+$ be a continuous function and let $\mathcal{W}_γ$ be the Wulff shape associated with $γ$. In this paper, we show that if the boundary of $\mathcal{W}_γ$ is a $C^1$ submanifold, then $γ$ must be the convex integrand of $\mathcal{W}_γ$.
Any generalized distance-squared mapping of equidimensional case has singularities, and their singularity types are wrapped into mystery in higher dimensional cases. Any generalized distance-squared mapping of equidimensional case is not injective. Nevertheless, in this paper, it is shown that the non-singular property or the injective property of a mapping is preserved by composing a generic generalized distance-squared mapping of equidimensional case.
A spherical Wulff shape is the spherical counterpart of a Wulff shape which is the well-known geometric model of a crystal at equilibrium introduced by G. Wulff in 1901. As same as a Wulff shape, each spherical Wulff shape has its unique dual. The spherical dual transform for spherical Wulff shapes is the mapping which maps a spherical Wulff shape to its spherical dual Wulff shape. In this paper, it is shown that the spherical dual transform for spherical Wulff shapes is an isometry with respect to the Pompeiu-Hausdorff metric.