Signature-inverse Theorem in Mesh Group-planes

Consider the Euclidean geometry of curves in the plane $\mathrm{E^{S\mathbb{E}}\simeq\mathbb{R}^{2,S\mathbb{E}}}$ where the underlying group of transformations is the special Euclidean group $\mathrm{S\mathbb{E}(2)=SO(2)\ltimes\mathbb{R}^{2}}$, containing all translations and rotations. One can also include reflections, leading to the Euclidean group $\mathrm{\mathbb{E}(2)=O(2)\ltimes\mathbb{R}^{2}}$.

Definition 2.1. The 2-point $\mathrm{\mathbb{E}}$-curvature of an ordinary mesh $\mathrm{\gamma^{\vartriangle}=\{p_{i}\}_{i\geq 3}\subset E^{\mathbb{E}}}$ is the function $\mathrm{\kappa_{\mathbb{E}}^{\vartriangle}:\gamma^{\vartriangle}\longrightarrow\mathbb{R}}$ given by $\mathrm{p_{i}\longmapsto\frac{4\bigtriangleup}{abc}(p_{i})}$, where $\mathrm{a(p_{i})\geqslant b(p_{i})\geqslant c(p_{i})}$ are the Euclidean distances in the one-neighborhood of $\mathrm{p_{i}}$ and $\mathrm{\bigtriangleup(p_{i})}$ is the area of the triangle whose vertices are the one-neighborhood of $\mathrm{p_{i}}$ given by

- Where $\mathrm{\gamma^{\vartriangle}}$ is equally spaced:

- Where $\mathrm{\gamma^{\vartriangle}}$ is unequally spaced:

Affine geometry is the study of geometric properties of the objects in the plane $\mathrm{E^{S\mathbb{A}}}$ which remain unchanged by signed area-preserving affine transformations $\mathrm{x\longmapsto Ax+b,A\in SL(2)}$ and $\mathrm{x,b\in\mathbb{R}^{2}}$ - called the equiaffine group $\mathrm{S\mathbb{A}(2)=SL(2)\ltimes\mathbb{R}^{2}}$. One can include reflections, leading to the extended equiaffine group $\mathrm{\bar{\mathbb{A}}(2)}$.

Moreover, the 3-point affine arc lengths of $\mathrm{\gamma^{\vartriangle}=\{p_{i}\}\subset E^{S\mathbb{A}}}$ are obtained as follows.

Theorem 2.3 agh2 . In the five-neighborhood of any point $\mathrm{p_{i}\in\gamma^{\vartriangle}}$, the 3-point $\mathbb{A}$-arc length $\mathrm{L_{k,l}=L(p_{k},p_{l})}$ from $\mathrm{p_{k}}$ to $\mathrm{p_{l}}$ is computed as follows.

-If $\mathrm{\frac{S}{F^{2/3}}(p_{i})\neq 0}$, then

-If $\mathrm{\frac{S}{F^{2/3}}(p_{i})=0}$, then

- Where $\mathrm{\gamma^{\vartriangle}}$ is equally spaced:

- Where $\mathrm{\gamma^{\vartriangle}}$ is unequally spaced:

According to agh ; agh1 , Curvature Theorem is correct in $\mathrm{M^{\mathbb{G}}}$, meaning congruent meshes have the same k-point $\mathrm{\mathbb{G}}$-curvature $\mathrm{\kappa_{\mathbb{G}}^{\vartriangle}}$. Now, we are going to investigate its inverse.

Example 1. Let $\mathrm{\gamma_{0-1}^{\vartriangle}=\{p_{1},p,p_{2}\}}$ and $\mathrm{\gamma_{0-2}^{\vartriangle}=\{q_{1},q,q_{2}\}}$ be two 3-point ordinary meshes of the unit circle $\mathrm{\gamma_{0}\subset\mathbb{R}^{2}}$ as shown in Figure 1.

Now consider two approximating meshes of $\mathrm{\gamma_{0}}$ with the same resolution (number of points) in which one is equally and the other one is unequally spaced. They have the same 2-point $\mathrm{\mathbb{E}}$-curvatures, while they are not congruent, meaning Curvature-inverse Theorem is not correct even for fine partitions.

Just like the Euclidean case, it is also easy to give some counterexamples in $\mathrm{M^{\mathbb{A}}}$, therefore, we have the following proposition.

proposition 3.1. Non-congruent meshes may have identical k-point $\mathrm{\mathbb{G}}$-curvatures.

According to agh1 , to validate Signature Theorem in terms of the current formulation, one just needs to consider the orientation-invariant version of it, meaning congruent meshes have the same orientation-invariant JINS. Now, we investigate the correctness of its inverse.

Let $\mathrm{R_{p}}$ be the radius of the circumcircle passing through the one-neighborhood of a point $\mathrm{p}$ on an ordinary mesh $\mathrm{\gamma^{\vartriangle}}$.

$\mathrm{R_{p_{1}}=R_{p_{2}}=}$ $\mathrm{R_{q_{1}}=R_{q_{2}}=\mathsf{R}}$       and       $\mathrm{R_{p_{3}}=R_{q_{3}}=\mathsf{r}}$.

Just like the Euclidean case, it is also easy to give some counterexamples in $\mathrm{M^{S\mathbb{A}}}$, therefore, we have the following corollary.

Proposition 3.2. Signature-inverse Theorem is not correct in $\mathrm{M^{\mathbb{G}}}$.

Remark. There are theorems with very simple conditions for Signature-inverse Theorem in $\mathrm{M^{\mathbb{G}}}$, but they are not in terms of the resulting JINSs. For example, the following theorem is the simplest.

From now on, this paper looks for conditions in terms of the resulting signatures to make Signature-inverse Theorem correct in $\mathrm{M^{\mathbb{G}}}$.

A mesh point $\mathrm{\gamma^{\vartriangle}\subset E^{\mathbb{E}}}$ is equally spaced, if all edges (two successive points) have the same Euclidean length. Also, in a 3-point equally spaced mesh, let $\mathrm{\mathsf{d}}$ and $\mathrm{\mathsf{d}\_}$ denote respectively the $\mathrm{\mathbb{E}}$-length of each edge and the $\mathrm{\mathbb{E}}$-distances between the first and end points. In addition, for a n-point equally spaced mesh, let $\mathrm{\mathsf{d}_{i}}$ and $\mathrm{\mathsf{d}_{\_i}}$ be respectively $\mathrm{\mathsf{d}}$ and $\mathrm{\mathsf{d}\_}$ in the one-neighborhood of its $\mathrm{i^{th}}$ point (in open meshes $\mathrm{1<i<n}$).

The following lemma plays a central role in our investigations.

Lemma 4.3 fel2 . The 2-point action $\mathrm{S\mathbb{E}(2)\times M^{2}\longrightarrow M^{2}}$ given by
$\mathrm{\hskip 156.49014ptg\cdot(p_{1},p_{2})\longmapsto(g\cdot p_{1},g\cdot p_{2})}$

$\square\succ$ Signature-inverse Theorem in terms of (1)

Lemma 4.4. For any $\mathrm{p,q\in E^{S\mathbb{E}}}$, there are at most 2 circles with a given radius $\mathrm{\mathsf{R}}$ passing through them, see FIG. 4.

However, it is not in terms of the $\mathrm{S\mathbb{E}}$-JINSs. It is also easy to check the following result.

Theorem 4.8. In $\mathrm{M^{S\mathbb{E}}}$, there exist two congruent classes of equally spaced 3-point meshes with curvature $\kappa$, angle $0<\theta_{0}<\pi$, and the same signature-direction denoted by $\mathrm{\gamma^{\vartriangle}_{\kappa,\theta_{0},SD}}$ and $\mathrm{\gamma^{\vartriangle}_{\kappa,\theta_{0},\sim SD}}$, see FIG. 8. Moreover, there is a unique equivalence class of them in $\mathrm{M^{\mathbb{E}}}$.

Theorem 4.9, Signature-inverse Theorem. Equally spaced ordinary meshes $\mathrm{\gamma^{\vartriangle}}$ and $\mathrm{\tilde{\gamma}^{\vartriangle}}$ in $\mathrm{M^{S\mathbb{E}}}$ with the same signature-direction and identical JINSs given by (1) are congruent.

Proof. Let $\mathrm{\gamma^{\vartriangle}=\{p_{i}\}_{1}^{n}}$ and $\mathrm{\tilde{\gamma}^{\vartriangle}=\{\tilde{p}_{i}\}_{1}^{n}}$. Also, without loss of generality, we assume that $\mathrm{\Xi_{S\mathbb{E}}^{\vartriangle}(\tilde{p}_{i})=\Xi_{S\mathbb{E}}^{\vartriangle}(p_{i})}$ for $\mathrm{1\leq i\leq n}$. By identity (1)

It is not also hard to check that Theorem 4.9 and Theorem 3.3 are the same.

$\square\succ$ Signature-inverse Theorem in terms of (2) - version 1

Definition 4.10. Two ordinary meshes $\mathrm{\gamma^{\vartriangle},\tilde{\gamma}^{\vartriangle}\subset{M}}$ have the same “angle-type” iff any pair of angles $\mathrm{\theta_{i}=\hskip 3.41432pt<(p_{i-1},p_{i},p_{i+1})}$ and $\mathrm{\tilde{\theta}_{i}=\hskip 3.41432pt<(\tilde{p}_{i-1},\tilde{p}_{i},\tilde{p}_{i+1})}$ are both acute, obtuse, or right.

Theorem 4.11. In $\mathrm{M^{S\mathbb{E}}}$, there exist at most 4 congruent classes of 3-point equally spaced ordinary meshes with curvature $\mathrm{\kappa}$ and distance $\mathrm{\mathsf{d\_}}$ as follows:

- If $\mathrm{\kappa\cdot\mathsf{d\_}=2}$; there are 2 congruent classes: one in $\mathrm{SD}$ denoted by $\mathrm{\gamma^{\vartriangle}_{\kappa,SD}}$ and another in $\mathrm{\sim SD}$ denoted by $\mathrm{\gamma^{\vartriangle}_{\kappa,\sim SD}}$, see FIG. 9.

- If $\mathrm{\kappa\cdot\mathsf{d\_}\neq 2}$; there are 4 congruent classes: two in $\mathrm{SD}$ denoted by $\mathrm{\{\gamma^{\vartriangle,1}_{\kappa,\mathsf{d\_},SD},\gamma^{\vartriangle,2}_{\kappa,\mathsf{d\_},SD}\}}$, and the other two in $\mathrm{\sim SD}$ denoted by $\mathrm{\{\gamma^{\vartriangle,1}_{\kappa,\mathsf{d\_},\sim SD},\gamma^{\vartriangle,2}_{\kappa,\mathsf{d\_}\sim SD}\}}$, see FIG. 10.

We, therefore, established the following results.

Theorem 4.12. In $\mathrm{M^{S\mathbb{E}}}$, there exist 2 congruent classes of 3-point equally spaced ordinary meshes with curvature $\mathrm{\kappa}$, distance $\mathrm{\mathsf{d\_}}$, and also the same angle-type $\mathrm{\theta}$. One in $\mathrm{SD}$ and another in $\mathrm{\sim SD}$ denoted respectively by $\mathrm{\gamma^{\vartriangle}_{\kappa,\theta,SD}}$ and $\mathrm{\gamma^{\vartriangle}_{\kappa,\theta,\sim SD}}$, see FIG. 11. Moreover, there is a unique congruent class of them in $\mathrm{M^{\mathbb{E}}}$.

Corollary 4.13. There exists a unique congruent class of 3-point equally spaced ordinary meshes with the same $\mathrm{\kappa}$, $\mathrm{\mathsf{d\_}}$, and angle-type $\mathrm{\theta}$ which are also in the same signature-direction.

Theorem 4.14, Signature-inverse Theorem. Equally spaced ordinary meshes $\mathrm{\gamma^{\vartriangle}}$ and $\mathrm{\tilde{\gamma}^{\vartriangle}}$ in $\mathrm{M^{S\mathbb{E}}}$ with the same signature-direction and angle-type which also have identical 2-point signatures in terms of (2) are congruent.

Proof. According to (2) and just like the Theorem 4.9 we have

$\mathrm{\hskip 76.82234ptg_{i}\cdot[p_{i},p_{i+1}]\longmapsto[\tilde{p}_{i},\tilde{p}_{i+1}]}\hskip 14.22636pt\mbox{and}\hskip 14.22636pt\mathrm{g_{i+1}\cdot[p_{i},p_{i+1}]\longmapsto[\tilde{p}_{i},\tilde{p}_{i+1}]}$.

We indeed established the following theorem for fine meshes as well.

$\square\succ$ Signature-inverse Theorem in terms of (2) - version 2

Definition 4.16. Let $\mathrm{\gamma^{\vartriangle}=\{p_{i}\}\subset M^{S\mathbb{E}}}$ denote an ordinary mesh. By “the signed angle” at a point $\mathrm{p_{i}}$ we mean $\mathrm{\vartheta_{i}=SS(p_{i})\cdot\theta_{i}}$ where $\mathrm{\theta_{i}=\hskip 3.41432pt<(p_{i-1},p_{i},p_{i+1})}$.

Definition 4.16, along with Corollary 4.13 and Theorem 4.14, gives the following results.

Theorem 4.17. There exists a unique congruent class of equally spaced 3-point ordinary meshes with $\mathrm{{\mathbb{E}}}$-curvature $\mathrm{\kappa}$, distance $\mathrm{\mathsf{d\_}}$, and the same signed angle-type $\mathrm{\vartheta}$, see FIG. 12.

Theorem 4.18, Signature-inverse Theorem. Equally spaced ordinary meshes $\mathrm{\gamma^{\vartriangle}}$ and $\mathrm{\tilde{\gamma}^{\vartriangle}}$ in $\mathrm{M^{S\mathbb{E}}}$ with the same signed angle-type and 2-point signature $\mathrm{\Xi_{S\mathbb{E}}^{\vartriangle}(\tilde{\gamma}^{\vartriangle})=\Xi_{S\mathbb{E}}^{\vartriangle}(\gamma^{\vartriangle})}$ obtained from (2) are congruent.

Now, we rewrite Theorem 4.8 in terms of the signed angles as follows.

Theorem 4.19. There is a unique congruent class of 3-point equally spaced meshes with Euclidean curvature $\kappa$ and signed-angle $\mathrm{0<\vartheta_{0}<\pi}$.

Theorem 4.19 establishes the following result.

Theorem 4.20. Equally spaced ordinary meshes $\mathrm{\gamma^{\vartriangle}}$ and $\mathrm{\tilde{\gamma}^{\vartriangle}}$ in $\mathrm{M^{S\mathbb{E}}}$ with identical signed angles $\mathrm{0<\tilde{\vartheta}_{0,i}=\vartheta_{0,i}<\pi}$ and 2-point Euclidean curvatures $\mathrm{\kappa_{\mathbb{E}}^{\vartriangle}({\tilde{\gamma}^{\vartriangle}})=\kappa_{\mathbb{E}}^{\vartriangle}({\gamma^{\vartriangle}})}$ are congruent.

Definition 4.21 agh . Let $\mathrm{\gamma^{\vartriangle}=\{p_{i}\}\subset M^{S\mathbb{E}}}$ be an ordinary mesh.

According to Aghayan agh1 ; agh3 , we have the following theorem.

Theorem 4.22, Signature Theorem in $\mathrm{M^{S\mathbb{E}}}$ agh ; agh1 ; agh2 . Congruent ordinary meshes have the same (orientation-invariant) 2-point $\mathrm{S\mathbb{E}_{m_{1}}^{m_{2}}}$-signature.

$\square\succ$ Signature-inverse Theorem in terms of (3) and (4)

In an unequally spaced 3-point mesh, let $\mathrm{\mathsf{d_{1}},\mathsf{d_{2}}}$ and $\mathrm{\mathsf{d\_}}$ denote, respectively, the $\mathrm{\mathbb{E}}$-distances between the first and middle points, the end and middle points, and the first and end points. Besides for an unequally spaced n-point mesh, let $\mathrm{(m_{1},m_{2})}$-$\mathrm{\mathsf{d_{1,i}}}$, $\mathrm{(m_{1},m_{2})}$-$\mathrm{\mathsf{d_{2,i}}}$ and $\mathrm{(m_{1},m_{2})}$-$\mathrm{\mathsf{d_{\_i}}}$ denote, respectively, the $\mathrm{\mathbb{E}}$-distances between the first and middle points, the end and middle points, and the first and end points of the $\mathrm{(m_{1},m_{2})}$-neighborhood of the $\mathrm{i^{th}}$ point (for open meshes $\mathrm{m_{1}<i<n-m_{2}+1}$ where $\mathrm{n>3}$). We also suppose that $\mathrm{(m_{1},m_{2})}$-$\mathrm{\mathsf{d_{2}}>(m_{1},m_{2})}$-$\mathrm{\mathsf{d_{1}}}$.

Lemma 4.23. There are at most 4 congruent classes of 3-point unequally spaced meshes with $\mathrm{\mathbb{E}}$-curvature $\kappa$ and the side lengths $\mathrm{\mathsf{d}_{1}}$ and $\mathrm{\mathsf{d}_{2}}$ classified as follows.

We, therefore, have the following theorem.

Theorem 4.24. In $\mathrm{M^{S\mathbb{E}}}$, there exists a unique congruent class of unequally spaced 3-point ordinary meshes with the following conditions.

Theorem 4.25, Signature-inverse Theorem. Unequally spaced ordinary meshes $\mathrm{\gamma^{\vartriangle}}$ and $\mathrm{\tilde{\gamma}^{\vartriangle}}$ in $\mathrm{M^{S\mathbb{E}}}$ with identical lengths $d_{i-1,i+1}$, the same signed (1,2)-angle type, and the same 2-point $\mathrm{S\mathbb{E}_{1}^{2}}$-signature in terms of (3) are congruent.

Proof. According to the definition of $\mathrm{S\mathbb{E}_{1}^{2}}$-signatures in terms of (3)