Point-Separable Classes
of Simple Computable Planar Curves

A curve is a mathematical model which describes the “path (or locus) of a continuously moving point”. Therefore, a planar curve is defined in mathematics as the image of a continuous function $f:[0,1]\to\mathbb{R}^{2}$. Surprisingly, under this definition, a curve can be so complicated that it fills even a square (cf. [Peano1890, EHMoo1900]). In fact, as shown independently by Hahn and Mazurkiewicz in about 1913, a point set is a curve if and only if it is a locally connected continuum (we ignore the mathematical details in this paper which are not related to our discussion). However, if we are interested only in the curves which do not cross themselves (i.e., simple) and have finite length (i.e., rectifiable), then the curves defined in these ways coincide with our intuition about “curves” and they do have the “two-sidedness” and “thinness” (cf [Why42]). For rectifiable simple curves, the parametrizations can be required to be injective or length-normalized while the induced class of curves remains the same. Therefore, a rectifiable simple curve can be defined as any of the following: a point set of some special topological properties, the image of a continuous function, the image of an injective continuous function, or the image of a continuous function which is length-normalized.

If a point-movement is “algorithmically determined”, then its path (the curve) should be considered “computable”. More precisely, the notion of computable curves can be defined by the effectivization of the classical definition of curves. This naturally raises the question whether the effectivizations of these four definitions of curves mentioned above lead to the same notion of “computable curves”? Our answer is no, even in a very strong sense. Before we can explain our answer in a more precise way, let us recall first the basic idea of how to define computability of continuous objects in general.

In computable analysis, computability over various continuous structures is typically defined by the Turing-machine-based bit model (see [Ko91, Wei00, BC06]). In order to input a real number $x$ to such a Turing machine, it must be represented by effectively convergent sequences of rational numbers (the names of $x$). Here, a sequence $(x_{n})$ “converges effectively” means that $|x_{n}-x_{n+1}|\leq 2^{-n}$ for all $n$. A real number $x$ is computable if it has a computable name, i.e., there is a computable sequence of rational numbers which converges to $x$ effectively. Furthermore, a real function $f$ is computable if there is a Turing machine which computes $f$ in the sense that, after inputing any name of a real number $x$ in the domain of $f$, the machine outputs a name of $f(x)$. By the same principle, computability of other mathematical objects can be defined by introducing proper “naming systems”. For example, the computability of subsets of the Euclidean space [BW99], of semi-continuous functions [WZ97], of functional spaces [ZW03] are all defined in this way. This approach is also called the “effectivization” of classical mathematical definitions.

The same approach can be applied to curves as well. In this paper, we only consider plane curves. Curves in higher dimensions can be discussed in essentially the same way. Furthermore we will restrict ourselfes to rectifiable curves unless otherwise said, where a curve is rectifiable if it has a finite length. As mentioned above, a curve can be defined as a connected and one-dimensional compact subset. Based on this approach, we can define the computable curves by means of the computability of compact subsets of Euclidean space ([BW99]). Physically, a curve records the trace of a particle motion. If the particle moves according to some algorithmically definable laws, its trace should be regarded as computable. In mathematical terms, a curve is the range of a continuous function defined on a closed interval and this function is called a parametrization of the curve. Thus, it is also natural to call a curve computable if it has a computable parametrization (see e.g., [GLM06, GLM11]).

However, the parametrization of a curve may have various extra properties, if the curve is simple. Here a curve is called simple if it does not intersect itself, or if it has an injective parametrization. Of course, the parametrization $f$ of a simple curve $C$ is not necessarily injective. If $f$ is not injective, then $f$ retraces some parts of the curve $C$. If a curve $C$ is simple, then it has even an arc-length normalized parametrization. Here, a parametrization $f:[0,1]\to C$ is arc-length normalized roughly means that the function $f$ models a particle movement along the curve $C$ with a constant speed.

In this paper, four versions of computable curves are introduced by effectivizing the above four mathematical approaches to curves. We will see that these four versions of computable curves are all different. The difference of the curve classes defined by simple computable parametrizations and computable injective parametrizations was already shown by Gu, Lutz and Mayordomo in a recent paper [GLM11]. However, in this paper we will distinguish these four versions of computable curves in a much stronger sense. Namely, the sets of points covered by the four classes of computable curves are different. In other words, different versions of computable curves can be separated by the points they cover, or they are “point-separable” (see definition in Section LABEL:sec-point-separable).

Interestingly, the computability of the curve length plays an important role for the computability of curves. If we look only at curves of computable lengths, then the four effectivizations mentioned above are indeed equivalent. This means that the definition of computable curves is robust, at least, for curves of computable lengths. On the other hand, Gu, Lutz and Mayordomo constructed in [GLM11] a computable curve of non-computable length such that none of its computable parametrizations can be injective, although the curve does not intersect itself. As an open question, they asked whether there exists a point which lies on a computable curve of finite length, but not on any computable curve of computable length, i.e., if the class of computable curves of computable lengths is point-separable from the class of computable curves of finite lengths. A positive answer will be given in this paper.

Our paper is organized as follows. In Section 2 we will briefly recall some basic notions related to curves, give the precise definition of computable curves and then show some basic properties of computable curves. In Section LABEL:sec-point-separable, we discuss some basic facts of point-separable classes and show a technical lemma which will be used in the proof of the main theorems. Section LABEL:sec-n-comp-curve investigates the class of length-normalized computable curves and shows a significant difference between this class and the class of computable curves of computable lengths. Then it is shown that these two classes are point-separable. In the last Section LABEL:sec-main we prove that the four classes of computable curves mentioned above are all point-separable.

In mathematics, a plane curve is defined as a subset $C\subseteq\mathbb{R}^{2}$ which is the range of a continuous function $f:[0;1]\to\mathbb{R}^{2}$, i.e., $C={\rm range}(f)$. This continuous function $f$ is then called a parametrization of $C$. Here, we use, w.l.o.g., the unit interval $[0,1]$ instead of more general closed intervals of the form $[a,b]$. Obviously, any curve has infinitely many parametrizations. Geometrically, a curve records the path of a particle movement in the plane. If the particle never visits one position more than once, in other words, if the curve does not intersect itself (or it has an injective parametrization $f:[0;1]\to\mathbb{R}^{2}$), then the curve is called simple. The simple curves defined in this way are also called open, or Jordan arcs. If a curve $C$ has a parametrization $f$ which is injective on the interval $[0;1)$ and fulfills the condition that $f(0)=f(1)$, then the curve $C$ is traditionally also called simple, but it is closed, or a Jordan curve. Equivalently, a Jordan curve is the continuous image of the unit circle. In this paper we look only at the open simple curves. But all results are true for closed simple curves as well.

For open simple curves, their lengths can be defined by means of the lengths of polygons which approximate the curves according to Jordan [Jor1882]. More precisely, Let $C$ be a simple curve and let $f:[0;1]\to\mathbb{R}^{2}$ be an injective continuous parametrization of $C$. Then the length $L$ of the curve $C$ is defined by

where $|f(a_{i})-f(a_{i+1})|$ is the length of the straight line connecting the points $f(a_{i})$ and $f(a_{i+1})$, and the supremum is taken over all possible partitions $0=a_{0}<a_{1}<...<a_{n}=1$ of the unit interval $[0,1]$. The length of a curve $C$ is denoted by $l(C)$. Notice that we actually defined the length $l(f)$ of the function $f:[0;1]\rightarrow\mathbb{R}^{2}$. The length of a simple curve is then the length of an injective parametrization of that curve. It is well known that the length of a simple curve is independent from its (injective) representations. A curve of finite length is traditionally called rectifiable. Not every curve, even a simple curve, has finite length. As already mentioned above, we focus mainly on simple rectifiable curves; unless otherwise stated a curve is always meant to be simple and rectifiable in this paper.

If $C$ is a simple rectifiable curve of the length $l$, then there exists a bijective continuous function $g:[0,l]\to C$ such that the arc $g([0,s])$ has exactly the length $s$. That is, the arc-length $s$ is used as the argument of the function $g$. Let $f(t):=g(l\cdot t)$. Then the function $f:[0,1]\to C$ is a parametrization such that the curve segment $f([0,t])$ has the length $t\cdot l(C)$ for all $t\in[0,1]$. We call the parametrization $f$ of this property length-normalized or simply normalized. Thus, a simple rectifiable curve can have three different kinds of parametrizations—continuous, injective continuous and normalized. In addition, a curve can also be defined as a connected, one-dimensional, compact point set. By effectivizing all these approaches to curves, we can introduce four totally different versions of computable curves.

Remember that a real function $f:[0;1]\to\mathbb{R}$ is computable if there is a Turing machine $M$ which transfers any name of $x\in[0,1]$ to a name of $f(x)$. Equivalently, $f$ is computable iff there is a computable sequence $(p_{n})_{n\in\mathbb{N}}$ of computable rational polygon functions which converges uniformly and effectively to $f$ (see [PR89]). Naturally, a function $f:[0;1]\to\mathbb{R}^{n}$ is computable if all of its component functions are computable, or equivalently, if there is a Turing machine $M$ which transfers any name of $x\in[0,1]$ into a tuple $(\alpha_{1},\cdots,\alpha_{n})$ of names of $f_{1}(x),\cdots,f_{n}(x)$ respectively, where $f(x)=(f_{1}(x),\cdots,f_{n}(x))$. In this case, we simply say that $M$ computes the function $f$. Remember also that any computable function must be continuous.

In this paper, an $\varepsilon$-neighborhood $V_{\varepsilon}(z)$ of a point $z=(a,b)$ with Cartesian coordinates $(a,b)$ is the rectangle bounded by the lines $x=a\pm\varepsilon$ and $y=b\pm\varepsilon$. A neighborhood $V_{\varepsilon}(z)$ is called rational if $z$ is a rational point and $\varepsilon$ is a rational number. For a set $A\subseteq\mathbb{R}^{2}$, the $\varepsilon$-neighborhood of $A$ is defined by $V_{\varepsilon}(A):=\bigcup_{z\in A}V_{\varepsilon}(z)$. Then for any two point sets $A,B$, their Hausdorff distance is defined by $d_{H}(A,B)=\inf\{\varepsilon:A\subseteq V_{\varepsilon}(B)\ \&\ B\subseteq V_{\varepsilon}(A)\}$. Notice that, we always have $d_{H}(V_{\varepsilon}(z),z)\leq\sqrt{2}\varepsilon$.

Now we can define the different versions of computable curves as follows.

Let $C\subseteq\mathbb{R}^{2}$ be a simple, not necessarily rectifiable, planar curve.

1. (1)

   $C$ is called $K$-computable if there is a computable sequence $(Q_{n})$ of finite sets of rational neighborhoods such that

   $\displaystyle C\subseteq\bigcup Q_{n}$

   and

   $\displaystyle d_{H}\left(\bigcup Q_{n},\,\,C\right)<2^{-n}$

   (2)

   for all $n\in\mathbb{N}$, where $d_{H}$ denotes the Hausdorff distance.

2. (2)

   $C$ is called $R$-computable if there is a computable function $f:[0;1]\rightarrow\mathbb{R}^{2}$ such that $\mbox{\rm range}(f)=C$.

3. (3)

   $C$ is called $M$-computable if there is an injective computable function $f:[0;1]\rightarrow\mathbb{R}^{2}$ such that $\mbox{\rm range}(f)=C$.

4. (4)

   $C$ is called $N$-computable if $C$ has a computable parametrization $f:[0;1]\rightarrow\mathbb{R}^{2}$ such that the length of the curve segment $f([0,t])$ is equal to $t\cdot l(C)$ for all $t\in[0,1]$.

In item (2), the letter $R$ stands for $R$etracable because the parametrization $f$ of a $R$-computable curve $C$ can retrace the curve $C$. Namely, there might be some disjoint subintervals $I_{1},I_{2}\subset[0,1]$ such that $f(I_{1})=f(I_{2})$. In this case, $f$ traces the segment $f(I_{1})$ of $C$ more than once, or, we say that $f$ is retraceable.

If the parametrization of a curve $C$ is injective, then $C$ records the movement of a particle with a monotone direction. The letter $M$ in $M$-computability stands for $M$onotonically directed movement or $M$onotone paramatrization. Notice that, if we consider also closed simple curves, then the monotonicity has to exclude the endpoints of the unit interval.

Finally, if a parametrization $f:[0,1]\to\mathbb{R}^{2}$ satisfies the condition that the length of the curve segment $f([0,t])$ is proportional to $t$, i.e., $l(f([0,t]))=t\cdot l(C)$ for all $t\in[0,1]$, then it is normalized. Thus, $N$-computability stands for $N$ormalized parametrization.

By definition 2, any $N$-computable curve must be rectifiable. However, it is known that $M$-computable curves can have infinite lengths (see e.g [Ko98]). We will give a simple proof of this fact below by constructing a Koch curve, which is well known to be $M$-computable (see e.g. [Kam96]). The main reason for re-proving the following result is to introduce basic curve construction techniques which will be used throughout the more involved proofs in the next sections.