Local Turn-Boundedness: A Curvature Control for a Good Digitization

Abstract

This paper focuses on the classical problem of the control of information loss during the digitization step. The properties proposed in the literature rely on smoothness hypotheses that are not verified by the curves including angular points. The notion of turn introduced by Milnor in the article On the Total Curvature of Knots generalizes the notion of integral curvature to continuous curves. Thanks to the turn, we are able to define the local turn-boundedness. This promising property of curves do not require smoothness hypotheses and shares several properties with the par(r)-regularity, in particular well-composed digitizations. Besides, the local turn-boundedness enables to constraint spatially the continuous curve in function of its digitization.

1 Introduction

The loss of information caused by a digitization process is inevitable. Therefore a fundamental point concerns the control of this information loss. Indeed, the border of a compact connected shape S of the Euclidean plane can be arbitrarily far from the digitization of S or can oscillate around this latter border. Therefore, hypotheses on the border of the shape S, which is a Jordan curve denoted by $\mathcal C$, are needed. One of the most used hypothesis, called par(r)-regularity, was introduced by Pavlidis in [7]. It demands that any point $c\in \mathcal C$ has an interior osculating disk entirely included in the interior of $\mathcal C$ except for the point c and an exterior osculating disk entirely included in the exterior of $\mathcal C$ except for the point c. It has been used to prove some preserving topology properties [2, 5, 7] or to control the behavior of the projection from the digitized curve to the continuous curve $\mathcal C$ [3, 4]. However, the notion can be hard to manipulate in geometry and most of the authors using this notion add the assumption that the curve $\mathcal C$ is of class $C^2$. The par(r)-regularity encompasses two ideas: the border of the shape has a curvature bounded from above and the shape has a positive minimal thickness. In this article, relaxing the assumption of a bounded curvature, we develop a new notion called local turn-boundedness that is defined on continuous curves, including polygons. The local turn-boundedness relies on the notion of turn adapted to regular curves and polygons, firstly introduced by Milnor [6] to study the geometry of knots. The main properties and definitions around the notion of turn are recalled in Sect. 2. The local turn-boundedness involves the Euclidean distance between any two points of a curve and the turn of this curve between these two points. The properties of locally turn-bounded curves are given and illustrated in Sect. 3.

2 Turn of a Simple Curve

Although the notion was introduced by Milnor in [6], the definitions and properties given in this section come from the book of Alexandrov and Reshetnyak [1]. As presented in Proposition 2, the turn extends to continuous curves the notion of integral curvature already defined for regular curves.

Terminology and Notations

• Let $c \in \mathbb R^2$ and $r\ge 0$, B(c, r) is the open disk of center c and radius r.

• A parametrized curve is a continuous application from an interval [a, b] of $\mathbb R$ ($a<b$) to $\mathbb R^2$. It is simple if it is injective on [a, b) and closed if $\gamma (b)=\gamma (a)$. A (geometric) curve is the image of a parametrized curve. A Jordan curve is a simple closed curve.

• For a simple parametrized curve $\gamma$, an order is defined on the points of the associated geometric curve $\mathcal C$ by:

  $$\gamma (\alpha )\le _\gamma \gamma (\beta )\Leftrightarrow \alpha \le \beta $$

  and $\le _\gamma$ is noted $\le$ if there is no ambiguity. A simple parametrized curve $\gamma$ with such an order is called oriented curve and its image is called oriented geometric curve. A chain is a finite increasing sequence of points on an oriented geometric curve $\mathcal C$.

• A polygonal line with vertices $x_0$, ..., $x_{N}$ is denoted by $[x_0x_1 ...x_{N}]$ (if $x_N=x_0$, the polygonal line is a polygon). A polygonal line L is inscribed into an oriented geometric curve $\mathcal C$ if the vertices of the polygonal line L form a chain. For a Jordan curve, a polygonal line L is inscribed if all its vertices but the last form a chain and its first and last vertice are equal.

• Let N be a positive integer and $x_0, x_1, ..., x_{N}$ points of $\mathbb R^2$. The polygonal line PL = $[x_0x_1\cdots x_{N}]$ can be considered as the image of the parametrized curve $pl : [0, N] \mapsto \mathbb {R}^2$ such that $pl(t) = x_{\lfloor t \rfloor } (t-\lfloor t \rfloor )+ (1-t+\lfloor t \rfloor )x_{\lfloor t \rfloor +1}$ where for $r \in \mathbb {R}$, $\lfloor r \rfloor$ in the integer part of the real r. In other words, for any integer i between 0 and N, if $t\in [i, i+1)$, then $pl(t) = (t-i)x_i+(1-t+i)x_{i+1}$, and thus $pl([i, i+1])$ is the segment $[x_i,x_{i+1}]$ of $\mathbb {R}^2$. A polygonal line is simple if it is simple for the previous parametrization and thus a simple polygon is a Jordan curve.

• Given a curve $\mathcal C$ and two points a, b on $\mathcal C$ ($a\ne b$), we write $\mathcal C_a^b$ for the arc ending at a and b if $\mathcal C$ is not closed. If $\mathcal C$ is closed, $\mathcal C_a^b$ and $\mathcal C_b^a$ stand for the two arcs of $\mathcal C$ ending at a and b.

• The angle between two vectors ${\varvec{u}}$ and ${\varvec{v}}$ is denoted by $({\varvec{u}}, {\varvec{v}})$ ($({\varvec{u}}, {\varvec{v}})\in \mathbb R/2\pi \mathbb Z)$. The geometric angle between two vectors ${\varvec{u}}$ and ${\varvec{v}}$, denoted by $({\varvec{u}}\hat{,} {\varvec{v}})$, or two directed straight lines oriented by ${\varvec{u}}$ and ${\varvec{v}}$, is the absolute value of the reference angle taken in $(-\pi ,\pi ]$ between the two vectors. Thus, $({\varvec{u}}\hat{,} {\varvec{v}})\in [0,\pi )$.

Definition 1

(Turn).

• The turn $\kappa (L)$ of a polygonal line $L=[x_i]_{i=0}^{N}$ is defined by:

  $$\begin{aligned} \kappa (L):=\sum _{i=1}^{N-1}\widehat{\left( x_{i-1}x_i, x_i x_{i+1} \right) }. \end{aligned}$$

• The turn $\kappa (P)$ of a polygon $P=[x_i]_{i=0}^{N}$ (where $x_{N}=x_0$) is defined by:

  $$\begin{aligned} \kappa (P):=\sum _{i=1}^{N}{\widehat{\left( x_{i-1}x_i, x_i x_{i+1} \right) }}. \end{aligned}$$

• The turn $\kappa (\mathcal C)$ of a simple curve $\mathcal C$ is the upper bound of the turn of its inscribed polygonal lines

• The turn $\kappa (\mathcal C)$ of a Jordan curve $\mathcal C$ is the upper bound of the turn of its inscribed polygons.

It should be noticed that the turn does not depend on the orientation of the curve. Indeed, it is well-known that $({\varvec{u}}, {\varvec{v}})=(-{\varvec{u}}, -{\varvec{v}})=-(-{\varvec{v}}, -{\varvec{u}})$. Thus $\kappa \left( [x_i]_{i=0}^{N}\right) =\kappa \left( [x_i]_{i=N}^{0}\right)$.

Furthermore, since the turn of a polygon is equal to the upper bound of the turn of the polygonal lines inscribed in it (cf Corollary p. 119 [1]), the turn of the polygon seen as a closed curve is equal to the turn of the polygon. Hence the notation $\kappa$ is well defined (Fig. 1).

Fig. 1.

The turn of the polygon is the sum of the green angles. (Color figure online)

In the same way that we estimate the length of a curve, the following proposition makes it possible to calculate the turn thanks to multiscale samplings. Given a curve $\mathcal C$, we denote by $\mathcal L(\mathcal C)$ the length of $\mathcal C$.

Proposition 1

(Convergence of the length and turn of a sequence of polygonal lines [1], p. 23, 30, 121, 122). Let $\mathcal C$ be a simple curve and $(L_m)_{m \in \mathbb N}$ a sequence of polygonal lines inscribed in $\mathcal C$ and with same endpoints as $\mathcal C$. If $\lim _{m \rightarrow + \infty } \lambda _m =0$, where $\lambda _m$ is the maximal Euclidean distance between two consecutive vertices of $L_m$, then

$$\begin{aligned} \lim _{m \rightarrow + \infty } \mathcal L(L_m) = \mathcal L(\mathcal C) \end{aligned}$$

and

$$\begin{aligned} \lim _{m \rightarrow + \infty } \kappa (L_m) = \kappa (\mathcal C). \end{aligned}$$

Moreover, if $\kappa (\mathcal C)$ is finite, then $\mathcal L(\mathcal C)$ is also finite, that is $\mathcal C$ is rectifiable.

In Proposition 1, if we assume that the sequence $(L_m)$ is increasing ($L_m$ is inscribed in $L_{m+1}$), then the sequences $(\mathcal L(L_m))$ and $(\kappa (L_m))$ are both increasing [1, Lemma 5.1.1].

Proposition 2

(Turn for regular curves [1], p. 133). Let $\gamma :[0,\ell ]\rightarrow \mathbb R^2$ be a parametrization by arc length of a simple curve $\mathcal C$. Assume that $\gamma$ is of class $C^2$ and denote the curvature at the point $\gamma (s)$ by k(s). Then,

$$\begin{aligned} \kappa (\gamma )=\int _0^{\ell } k(s)\mathrm ds. \end{aligned}$$

For regular curves, therefore, the turn corresponds to the integral of the curvature (with respect to an arc-length parametrization).

The following theorem gives a lower bound of the turn for closed curves.

Theorem 1

(Fenchel’s Theorem, [1] Theorem 5.1.5 p. 125). For any Jordan curve $\mathcal C$, $\kappa (\mathcal C) \ge 2 \pi$. Moreover $\kappa (\mathcal C)= 2 \pi$ if and only if the interior of $\mathcal C$ is convex.

Proposition 3

Let $\mathcal C$ be a oriented simple curve from a to b. Let $\mathcal C_a^{x_2}$ and $\mathcal C_{x_1}^{b}$ be two arcs of $\mathcal C$ that overlap with $a< x_1< x_2 <b$. Then,

$$\begin{aligned} \kappa (\mathcal C) \le \kappa (\mathcal C_a^{x_2})+\kappa (\mathcal C_{x_1}^{b}). \end{aligned}$$

Proof

Let $(L_m)$ be a sequence of polygonal lines $[l_{m,i}]_{i=0}^{N_m}$ inscribed in $\mathcal C$ such that

$$l_{m,0}=a,\quad l_{m,N_m}=b,$$

$$\begin{aligned} {\Vert l_{m,i+1} - l_{m,i}\Vert }_2 \le \frac{1}{m}, \end{aligned}$$

and for each m there exists j such that

$$\begin{aligned} x_1< l_{m,j} <x_2. \end{aligned}$$

For each m, let us denote by $l_{m, i_2}$ the largest $l_{m,i}$ such that $l_{m,i}$ lies in $\mathcal C_a^{x_2}$ and $l_{m,i_1}$ the smallest $l_{m,i}$ that lies in $\mathcal C_{x_1}^{b}$. Then,

$$\begin{aligned} l_{m,i_1} \le l_{m,j} \le l_{m,i_2}. \end{aligned}$$

Therefore, from Definition 1, we derive that

$$\begin{aligned} \kappa ([l_m]_{i=0}^{N_m}) \le \kappa ([l_m]_{i=0}^{i_2})+\kappa ([l_m]_{i=i_1}^{N_m}). \end{aligned}$$

Then, by Proposition 1,

$$\begin{aligned} \kappa (\mathcal C) \le \kappa (\mathcal C_a^{x_2})+\kappa (\mathcal C_{x_1}^{b}). \end{aligned}$$

   $\square$

Without the hypothesis of an overlapping non reduced to a singleton, Proposition 3 is false in presence of angular points as it is illustrated in Fig. 2.

Fig. 2.

Counterexample: the turn of the arcs $\mathcal C_{a_1}^{a_2}$ and $\mathcal C_{a_2}^{a_3}$ are zero but the turn of the arc $\mathcal C_{a_1}^{a_3}$ is nonzero. This is why the arcs are required to be overlapping in Proposition 3.

Remark 1

The turn is stable under homothetic maps. Indeed, obviously, the turn is invariant by any conformal map, in particular by the homotheties.

3 Locally Turn-Bounded Curves

Definition 2

Let $\theta \ge 0$. The $\theta$-turn step $\sigma (\theta )$ of a Jordan curve $\mathcal C$ is the infimum of the (Euclidean) distances between two points a and b in $\mathcal C$ such that the turns of $\mathcal C$ between a and b are both greater than $\theta$:

$$\begin{aligned} \sigma (\theta ) = \inf \{||a-b||_2 \mid \kappa (\mathcal C_a^b)> \theta \wedge \kappa (\mathcal C_b^a) > \theta \} . \end{aligned}$$

(1)

Let $\theta \ge 0$, $\delta \ge 0$. A Jordan curve $\mathcal C$ is locally turn-bounded with parameters $(\theta ,\delta )$ if $\delta \le \sigma (\theta )$.

The Fig. 3 illustrates the definition of the turn step with different curves.

Fig. 3.

For the chosen value of $\theta$, the corresponding $\theta$-turn-step $\sigma (\theta )$.

On a locally turn-bounded curve with parameters $(\theta ,\delta )$ the smallest turn of the curve $\mathcal C$ between two points of $\mathcal C$ a and b such that $||b-a||_2 < \delta$ is less than or equal to $\theta$.

The turn step function $\theta \mapsto \sigma (\theta )$ is increasing. Indeed, the value of the $\theta$-turn step is an infimum of a set that decreases (for the inclusion order) in function of $\theta$. If the turn of $\mathcal C$ is finite, there exists a value $\theta _{\max }$ above which this set is empty and the $\theta _{\max }$-turn step is then infinite.

Examples

• The 0-turn step of a Jordan curve is 0.

• The $\pi$-turn step of a convex Jordan curve is $+\infty$ (see Theorem 1).

• The $\theta$-turn step of a circle with radius r is $2r\sin (\theta /2)$ if $\theta \le \pi$ and is infinite for $\theta >\pi$.

• The $\theta$-turn step of a polygon having vertices with interior angles strictly less than $\pi - \theta$ is zero. Indeed, the turn of an arc joining two points arbitrary close, located before and after such a vertex is greater than $\theta$.

• The turn between two points on a polygonal curve is a finite sum of geometric angles, then the turn step function is a step function.

Remark 2

Local turn-boundedness is scale invariant: let $\mathcal C$ be a locally turn-bounded Jordan curve with parameters $(\theta , \delta )$. Then, the curve $k\,\mathcal C$, $k\,{>}\,0$, is locally turn-bounded with parameters $(\theta , k\delta )$. It is a direct consequence of Remark 1.

The next proposition makes it possible to localize a locally turn-bounded curve from a sufficiently tight sampling. Figure 4 illustrates the proposition.

Proposition 4

Let $\mathcal C$ be a simple curve locally turn-bounded with parameters $(\theta \in (0,\pi ), \delta )$. Let $a<b$ be two points on $\mathcal C$ such that $||a-b||_2 < \delta$. Then, one of the arc of $\mathcal C$ between a and b is included in the union of the two truncated closed disks where the line segment [a, b] is seen from an angle greater than or equal to $\pi -\theta$.

Proof

Since the Euclidean distance between a and b is less than the $\theta$-turn step, the turn of one of the arcs of $\mathcal C$ between a and b is less than or equal to $\theta$. Denote this arc by $\mathcal C_0$. Let c be a point on $\mathcal C_0$. By definition, the turn of the polygonal line [a, c, b] is less than or equal to the turn of $\mathcal C_0$ Then the geometric angle $\widehat{(a-c, b-c)}$ is greater than or equal to $\pi -\theta$. We conclude the proof by invoking the inscribed angle theorem.    $\square$

Fig. 4.

Illustration of Proposition 4 for three values of the parameter $\theta$: $\pi /3$, $\pi /2$, $2\pi /3$. Given two points $a,b\in \mathcal C$ such that $||a-b||_2 < \sigma (\theta )$, then one of the arcs of $\mathcal C$ between a and b belongs to the grey area.

Locally turn-bounded curves for angles $\theta \le \pi /2$ are locally connected subsets of the Euclidean plane.

Proposition 5

Let $\mathcal C$ be a Jordan curve locally turn-bounded with parameters $(\theta \in (0,\pi /2],\delta )$ and $a\in \mathcal C$. Then the intersection of $\mathcal C$ with the open disk $B(a,\epsilon )$ is path-connected whenever $\epsilon \le \delta$.

Proof

Let $b\in \mathcal C\cap B(a,\epsilon )$. Then, by the very definition of $\delta$, the turn of one of the arc of $\mathcal C$ between a and b is less than or equal to $\theta$. So, from Proposition 4 and for $\theta \le \pi /2$, this arc is included in the disk with diameter [a, b] which is itself included in $B(a,\epsilon )$. Hence, $\mathcal C\cap B(a,\epsilon )$ is path-connected.    $\square$

For $\pi /2<\theta <\pi$, in particular for polygons with acute angles, Proposition 5 does not hold (the intersection of the curve with a ball near an acute angle may have two connected components).

Proposition 6

A locally turn-bounded Jordan curve with parameters $(\theta \in (0,\pi /2],\delta )$ has a finite turn and is thus rectifiable.

Proof

Let $\mathcal C$ be a locally turn-bounded curve with parameters $(\theta ,\delta )$. The open balls $B(a,\delta /2)$, $a\in \mathcal C$, cover the compact set $\mathcal C$. Then, there exists a finite subset of $\mathcal C$, $\{a_0,\cdots , a_m\}$ such that $\bigcup _{i=0}^m B(a_i,\delta /2)$ covers $\mathcal C$. By Proposition 5, for each i, $\mathcal C\cap B(a_i,\delta /2)$ is an arc of $\mathcal C$.

Since the balls are open and thus overlaps, by Proposition 3 $\kappa (\mathcal C)\le \sum _{i=0}^m\kappa (\mathcal C\cap B(a_i,\delta /2))$. Besides, by hypothesis, $\kappa (\mathcal C\cap B(a,\delta /2))\le \theta$. Therefore, $\kappa (\mathcal C)\le (m+1)\theta$.    $\square$

The next technical lemma is used in the proofs of Propositions 7 and 8.

Lemma 1

Let $\mathcal C$ be a curve with endpoints a, b such that the line segment [a, b] does not intersect the curve $\mathcal C$. Let P be a polygonal line from a to b lying in the interior of the Jordan curve $\mathcal C\cup [a,b]$ and such that $P\cup [a,b]$ is convex. Then $\kappa (\mathcal C) \ge \kappa (P)$.

Proof

Let c be any point in (a, b) and $Q=[aq_1\cdots q_{m}b]$ the polygonal line obtained by projecting $P=[ap_1\cdots p_mb]$ on $\mathcal C$ from c (see Fig. 5). Then, $\kappa (\mathcal C) \ge \kappa (Q)$ by definition of $\kappa (\mathcal C)$. Besides, $\kappa (Q\cup [b,a])\ge \kappa (P\cup [b,a])$, since $P\cup [b,a]$ is convex, and $\kappa ([q_{m}baq_1])\le \kappa ([p_{m}bap_1])$ by construction of Q. Since $\kappa (Q\cup [b,a]) = \kappa (Q) + \kappa ([q_{m}baq_1])$ and $\kappa (P\cup [b,a]) = \kappa (P) + \kappa ([p_{m}bap_1])$, the result holds.    $\square$

Fig. 5.

Thick, blue: the curve $\mathcal C$ and the line segment [a, b]. Black: the polygonal line $P=[ap_1p_2b]$. Black, dashed: the projection of $p_1$ and $p_2$ on $\mathcal C$ yields the points $q_1$ and $q_2$. Red: the polygonal line $Q=[aq_1q_2b]$. (Color figure online)

The digitization process may digitize several continuous curves in a single digitization. The information on the turn makes it possible to select the continuous curves staying close to the digitization points as illustrated in Fig. 6.

Proposition 7

(Hausdorff distance between the curve and a pixel). Assuming an n-regular tiling of the plane with edge length h ($n\in \{3,4,6\}$), let $\mathcal C$ be a locally turn-bounded Jordan curve with parameters $(\theta < {2\pi }/ {n}, \delta > h \sqrt{n-2})$. Let T be a tile crossed by $\mathcal C$ and a, b be respectively the infimum and the supremum of $\mathcal C\cap T$ ($\mathcal C$ is ordered by some parametrization). Then, the Hausdorff distance between T and one of the segments of $\mathcal C$ bounded by a and b is less than $h(1-\cos (\theta ))/2\sin (\theta )$.

Proof

Let $\mathcal C_1$, $C_2$ be the arcs of $\mathcal C$ bounded by a and b. We assume $\kappa (\mathcal C_1)\le \kappa (\mathcal C_2)$. As the diameter of T is $\sqrt{n-2}h$, by the hypothesis $\delta \ge \sqrt{n-2}h$, $\kappa (\mathcal C_1) \le \theta$. If the arc $\mathcal C_1$ leaves T, it intersects the border of T on a point c. By the definition of a and b there exists $d \ne c$ such that d is on the border of T. The point d belongs to the same edge as c. Indeed, if it wasn’t the case, there would be a polygonal line included in the border of T containing at least one vertex and separating an arc segment of $\mathcal C_1$ from the line segment [c, d]. Then, by Lemma 1, the turn of $\mathcal C_1$ would be greater than or equal to ${2\pi }/{n}> \theta$. Hence, c and d belongs to the same edge. Therefore, by Proposition 4, we derive that $\mathcal C_1$ lies in the union of T with the interior of n truncated circles whose Hausdorff distance to T is $h(1-\sin \theta )/2\cos \theta$.    $\square$

Fig. 6.

Gray: a tile T with edge length h. Blue, thick: a locally turn-bounded curve arc passing through T. Red: the union of n circle arcs from which the edges of T are viewed from an angle $2\pi - \theta$. The Hausdorff distance between the circle arcs and the pixel T is e. (Color figure online)

The next proposition makes a link between the well-composedness of the digitized object and a local turn-boundedness hypothesis. Let us first define the Gauss digitization and the well-composedness. Let $h>0$ be a sampling grid step, the Gauss digitization of a shape K is defined as $K \cap (h\mathbb Z)^2$. By abuse of language, given a Jordan curve $\mathcal C$ which is the border of the compact shape K, we define its Gauss digitization —we write $\mathrm {Dig}_h(\mathcal C)$— as the border of the union of the squares ${p} \oplus [-h/2,h/2]\times [-h/2,h/2]$ where $\oplus$ denotes the Minkowski sum and $p\in K \cap (h\mathbb Z)^2$. The Gauss digitization of $\mathcal C$ is well-composed if it is a Jordan curve.

Proposition 8

Let $\mathcal C$ be a locally turn-bounded Jordan curve with parameters $(\theta \in (0,\pi /2], \delta )$ where $\delta$ is less than the diameter of $\mathcal C$. Then, the Gauss digitization of $\mathcal C$ for a grid step $h<\delta /\sqrt{2}$ is almost surely a disjoint union of Jordan curves (the digitization of the shape is well-composed). More precisely, given a Jordan curve $\mathcal C$, the set of translations of the grid for which the digitization is not well-composed (in bijection with a subset of the Euclidean plane) is of Lebesgue measure zero.

Proof

The proof is made by contradiction. So, let a be a double point on $\mathrm {Dig}_h(\mathcal C)$ traveled counterclockwise. Since the interior points of $(h\mathbb Z)^2$ are on the left of the discrete curve, there is only one configuration modulo rotations and symmetries depicted in Fig. 7. Furthermore, we can assume almost surely $\mathcal C\cap (h\mathbb Z)^2=\emptyset$. Indeed, the set of translations for which a given point $x\in (h\mathbb Z)^2$ lies on the translation of $\mathcal C$ is $\{x-p\mid p\in \mathcal C\}$. In other words, it is a translation of $-\mathcal C$ (in the translation parameter space). Then, since $\mathcal C$ is rectifiable, this set is one-dimensional. Therefore, the set S of translations for which there exists a point of $(h\mathbb Z)^2$ on the translation of $\mathcal C$ is a countable union of one-dimensional sets. Thus, the Lebesgue measure of S is zero.

Fig. 7.

Blue: the curve $\mathcal C$. Red: the line segments $[a_0a]$, $[aa_1]$, $[a_2a]$, $[aa_3]$ are edges of $\mathrm {Dig}_h(\mathcal C)$. Black: the points $E_1$, $E_2$, $I_1$, $I_2$ are in $(h\mathbb Z)^2$, $E_1$, $E_2$ are exterior to $\mathcal C$ while $I_1$, $I_2$ are interior to $\mathcal C$. (Color figure online)

With the notation of the figure, we claim that on each edge of the square $T=[I_1,E_1,I_2,E_2,I_1]$ lies a point of the curve $\mathcal C$. Indeed, each edge links an interior point and an exterior point. Furthermore, thanks to the assumptions that $\mathcal C\cap (h\mathbb Z)^2=\emptyset$, these four points actually lie on the open edges of T. Let b, d be respectively the infimum and the supremum of $\mathcal C\cap T$ and $[bc_0c_1c_2c_3d]$ be the inscribed polygon formed by these six points ($\mathcal C$ is ordered by some parametrization). We denote by $\mathcal C_b^d$ the oriented arc of $\mathcal C$ from b to d. On the one hand, the arc $\mathcal C_b^d$ contains the four points $c_0$, $c_1$, $c_2$ and $c_3$. Thus, whatever the geometric configuration ($[c_0c_1c_2c_3]$ simple or not), the turn of $\mathcal C_b^d$ is greater than $\pi /2$. Nevertheless, since $\delta >h\sqrt{2}$, the points b, $c_0$, $c_1$, $c_2$, $c_3$, d are included in the ball $B(a,\delta )$. Thus, from Proposition 5, the arc $\mathcal C_b^d$ is included in the ball $B(a,\delta )$. On the other hand, since $\kappa (C_b^d)\ge \pi /2$ and $\delta >h\sqrt{2}$, the turn of $\mathcal C\setminus \mathcal C_b^d$ is less than $\theta$ and a fortiori less than $\pi /2$. Then, from Proposition 4, $\mathcal C\setminus \mathcal C_b^d$ is included in the disk with diameter [b, d] which is itself included in the disk $B(a,\delta )$. Thereby, the whole curve $\mathcal C$ is included in $B(a,\delta )$ which contradicts the assumption on its diameter.    $\square$

We began the proof of Proposition 8 without any restriction on the localization of the curve (omitting “almost surely”). But there are many special cases to consider and it can not be given as part of a short article.

4 Conclusion

In this article, the notion of local turn boundedness which adapted to both regular curves and polygons having large enough interior angles, has been developed to have control on curves without smoothness assumption. Under this assumption on a curve, it has been shown that the domain where an arc of this curve is lying is bounded. Then the well-composedness of the curve Gauss-digitization has been deduced almost surely. A bound on the Hausdorff distance between one pixel and an arc crossing it has also been exhibited.

The local turn boundedness also needs to be related to others notions used in discrete geometry for similar purposes. To this end, we will prove in an upcoming article that a par(r)-regular curve of class $\mathcal C^1$ is locally turn-bounded with parameters $(\theta ,2r\sin (\theta /2))$ where $\theta$ is any angle in $[0,\pi )$. Besides, we expect that these results will make it possible to uniquely associate the vertices of a Gauss digitization to points on the continuous curve without smoothness assumption.