# Nonasymptotic Densities for Shape Reconstruction

Authors: Sharif Ibrahim; Kevin Sonnanburg; Thomas J. Asaki; Kevin R. Vixie
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

In this work, we study the problem of reconstructing shapes from simple nonasymptotic densities measured only along shape boundaries. The particular density we study is also known as the integral area invariant and corresponds to the area of a disk centered on the boundary that is also inside the shape. It is easy to show uniqueness when these densities are known for all radii in a neighborhood ofr=0, but much less straightforward when we assume that we only know the area invariant and its derivatives for only one r>0. We present variations of uniqueness results for reconstruction (modulo translation and rotation) of polygons and (a dense set of) smooth curves under certain regularity conditions.


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## Body

## 1. Introduction

This work discusses the integral area invariant introduced by Manay et al. [1], particularly with regard to reconstructability of shapes. This topic has been considered previously by Fidler et al. [2, 3] for the case of star-shaped regions. Recent results have shown local injectivity in the neighborhood of a circle [4] and for graphs in a neighborhood of constant functions [5].The present work does not assume a star-shaped condition but does make use of a tangent-cone graph-like condition which is local to the integral area circle. We also present an interpretation of the integral area invariant as a nonasymptotic density. This is based on a poster presented by the authors [6].Our tangentially graph-like and tangent-cone graph-like conditions (Definitions5 and 7 in Section 2) restrict our attention to shapes with boundaries that can locally (i.e., within radius r) be viewed as graphs of functions in a Cartesian plane in one particular orientation (in the case of tangentially graph-like) or a particular set of orientations (for tangent-cone graph-like). Intuitively, these conditions guarantee that the boundary does not turn too sharply within the given radius and that working locally in Euclidean space is the same as working locally on the boundary of our shapes (i.e., the shape boundary does not pass through any given invariant circle multiple times, Section 2.2). These simplifying assumptions allow us to explicitly analyze what happens when we move along the boundary and to work locally without worrying about global effects.We show that the tangent-cone graph-like property can be preserved when approximating a shape with a polygon (Section3) and discuss what the derivatives of these nonasymptotic densities represent (Section 4) and show that all tangentially graph-like boundaries can be reconstructed (modulo translations and rotations) given sufficient information about the nonasymptotic density and its derivatives (Section 5 and Appendix A).The main contribution of this paper is to show (under our tangent-cone graph-like condition) that all polygons (Theorem29 in Section 6) and a C1-dense set of C2 boundaries (Theorem 30 in Section 7) are reconstructible (modulo translations and rotations). We briefly discuss and sketch the proofs of these two theorems.Theorem 1.
For a polygonΩ which is tangent-cone graph-like with radius r, suppose that one has the integral area invariant g(s,r) where s is parameterized by arc length. Suppose that for all s one knows g(s,r) and its first derivatives with respect to r (disk radius) and s (position along the boundary). This information is sufficient to completely determine Ω up to translation and rotation; that is, one can recover the side lengths and angles of Ω.The proof of this theorem uses the discontinuities in thes derivative to determine the locations of vertices (and thus the side lengths between them). We combine the r derivative and the one-sided s derivative information when centered on a vertex to recover the angles at which the polygon enters and exits the circle (which might not be the polygon vertex angle if the circle contains another vertex). Doing this with the other one-sided s derivative gives the same thing but using the orientation determined by the other polygon side incident to the vertex. The combination of these yields the polygon's angle at each vertex.Theorem 2.
DefineG≡{γ∣γ is a C2 simple closed curve and tangentially graph-like for r=r^}. Suppose that, for r=r^, for all s∈[0,L], and for each γ∈G, one knows the first-, second-, and third-order partial derivatives of gγ(s,r). Then the set of reconstructible γ∈G is C1 dense in G where reconstructability is modulo reparametrization, translation, and rotation.The first part of the proof shows that the derivative information can be used to obtain the curvature. However, it is not the curvature at the boundary point where the circle is centered but rather the curvature at each of the points where the boundary enters and exits the circle. Although the Euclidean distance to these points is known, the arc length distances are not and can vary from point to point. Thus the sequences of curvatures we obtain also lose the arc length parameterization of our area invariant. The rest of the proof is concerned with finding the arc length distance from the center to the entry and exit points which effectively recovers the curvature for all points. This relies on matching up the unique features of exit angle sequences with each other which in turn relies on the existence of unique maxima and minima in these sequences. While this is not true in general, it can be arranged to be so by a suitable small perturbation of the boundary (which is why our result is one of density rather than for all shapes).This is a theoretical paper about a measure that is useful in applications: we do not pretend that the reconstruction techniques in our proofs are practically useful. In fact, the reconstructions we use to show uniqueness would be seriously disturbed by the noise that any practical application would encounter. We do, however, comment on some possible approaches to reconstruction (Section8) using the OrthoMads direct search algorithm [7] to successfully reconstruct shapes which are not predicted by our theory.


## 2. Notation and Preliminaries

Unless otherwise specified, we will be assuming throughout this paper thatΩ⊂R2 is a compact set with simple closed, piecewise continuously differentiable boundary ∂Ω of length L. Let γ:[0,L]→∂Ω be a continuous arc-length parameterization of ∂Ω (see Figure 1). We will adopt the convention that γ traverses ∂Ω in a counterclockwise direction so it always keeps the interior of Ω on the left (there is no compelling reason for this particular choice, but adopting a consistent convention allows us to avoid some ambiguities later). Note that γ(0)=γ(L) and that γ restricted to [0,L) is a bijection. Denote by D(p,r) the closed disk and C(p,r) the circle of radius r centered at the point p∈R2.Figure 1
Notation and basic setup.In geometric measure theory, them-dimensional density of a set A⊆Rn at a point p∈Rn is given by
(1)Θm(A,p)=lim⁡r↓0Hm(A∩D(p,r))αmrm,
where Hm is the m-dimensional Hausdorff measure and αm is the volume of the unit ball in Rm [8]. In the current context, the 2-dimensional density of Ω at γ(s) is simply
(2)Θ2(Ω,γ(s))=lim⁡r↓0Area(Ω∩D(γ(s),r))πr2.
While we can evaluate this for all s∈[0,L), just knowing the density at every point along the boundary is generally insufficient to reconstruct the original shape. If γ′(s) exists, then Area(Ω∩D(γ(s),r)) is approximated arbitrarily well for sufficiently small r by replacing ∂Ω with its tangent line (which gives us an area of exactly πr2/2). Hence, we have Θ2(Ω,γ(s))=1/2 at any point where γ is differentiable. That is, just knowing Θ2 (i.e., the limit) is insufficient to distinguish any two shapes with C1 boundary.Contrast this with the situation where we knowArea(Ω∩D(γ(s),r)) for every s∈[0,L) and r>0 (i.e., we have all of the values needed to compute the limit as well). This added information is sufficient to uniquely identify C2 curves by recovering their curvature at every point (see Appendix A).One natural question to ask (and the focus of the present work) is whether failing to pass to the limit (i.e., using some fixed radiusr instead of the limit or all r>0) and collecting the values for all points along the boundary preserves enough information to reconstruct the original shape. That is, can a nonasymptotic density (perhaps along with information about its derivatives) be used as a signature for shapes?


### 2.1. Definitions

Definition 3.
In the current context, the integral area invariant [1] is denoted by g:[0,L)×R+→R+ and given by
(3)g(s,r)=∫D(γ(s),r)∩Ω‍dx=Area(Ω∩D(γ(s),r)).Remark 4.
Note the lack of the normalizing factorπr2 in the definition of g(s,r). Since we presume that r is fixed and known for the situations we study, it is trivial to convert data between the forms g(s,r) and g(s,r)/πr2; we choose to leave out the normalizing factor in the definition of g(s,r) as it is the integral area invariant of Manay et al. [1] and this form proves useful when computing derivatives in Section 4.We introduce the tangentially graph-like condition as a simplifying assumption for the shapes we consider.Definition 5.
For a fixed radiusr, one says that ∂Ω is graph-like (GL) at a point p∈∂Ω (or graph-like on D(p,r)) if it is possible to impose a Cartesian coordinate system such that the set of points ∂Ω∩D(p,r) is the graph of some function f in this coordinate system. Without loss of generality, one adopts the convention that p is the origin so that f(0)=0. One defines tangentially graph-like (TGL) in the same way but further requires that ∂Ω be continuously differentiable and f′(0)=0 (noting that f is C1 because ∂Ω is). This is illustrated in Figure 2(a). Without loss of generality (and in keeping with our convention that γ traverses ∂Ω counterclockwise), one assumes that the interior of Ω is “up” in the circle (i.e., (0,ϵ)∈Ω for sufficiently small ϵ>0). If ∂Ω is (tangentially) graph-like on D(p,r) for all p∈∂Ω, one says that ∂Ω is (tangentially) graph-like for radius r.(a) Tangentially and (b) tangent-cone graph-like.
(a)
(b)It is instructive to consider what isnot considered graph-like or tangentially graph-like. Violations of the graph-like condition are generally due to a radius that is too large (certainly, choosing a radius so large that all of Ω is in the disk will do it). For example, a unit side length square is not graph-like with radius (1/2)+ϵ for any ϵ>0 (position the circle at the center of a side; see Figure 3(a)). Notice that the same square is graph-like with any radius 1/2 or below. A shape can fail to be tangentially graph-like while still being graph-like if it fails to be a graph in the required orientation but works in some other (see Figure 3(b)).(a) The square is not graph-like with the indicated radius (no orientation makes it a graph). (b) The rounded rectangle is graph-like but not tangentially graph-like with the indicated center and radius.
(a)
(b)We would like to consider shapes with corners but our tangentially graph-like condition requires that the boundary be differentiable everywhere. The following definitions allow us to generalize the tangentially graph-like condition to this situation by using one-sided derivatives.Definition 6.
Given a piecewiseC1 function γ:[0,L]→R2, one defines the tangent cone of γ at a point s (which is denoted by Tγ(s)) in terms of the one-sided derivatives. In particular, one lets Tγ(s)={αΓ-+βΓ+∣α,β≥0,α+β>0} where Γ-=lim⁡t↑sγ′(t) and Γ+=lim⁡t↓sγ′(t).Definition 7.
One extends the tangentially graph-like notion to boundaries that are piecewiseC1 by defining ∂Ω to be tangent-cone graph-like (TCGL) at a point γ(s)∈∂Ω if it is graph-like at γ(s) for every orientation in the tangent cone of ∂Ω at s. More precisely, for every w∈Tγ(s) and every pair of distinct points u,v∈∂Ω∩D(p,r), one has 〈w,u-v〉≠0. See Figure 2(b).Remark 8.
It is clear thatTγ(s) in Definition 6 is a convex cone. The tangent cone is dependent on the direction in which γ traverses ∂Ω (which by convention was counterclockwise) since an arc-length traversal γ^(s,r)=γ(L-s,r) would have different tangent cones (namely, w∈Tγ(s) iff -w∈Tγ^(s)). However, these differences are irrelevant to the application of Definition 7.Remark 9.
Note that when∂Ω is C1, there is only one direction in Tγ(s) for each s (i.e., the tangent to ∂Ω at γ(s)). Thus, the definitions of tangentially graph-like and tangent-cone graph-like coincide when ∂Ω is C1 and every tangentially graph-like boundary is tangent-cone graph-like.


### 2.2. Two-Arc Property

The graph-like condition implies (in proof of the following lemma) thatΩ will never be entirely contained in the disk, no matter where on the boundary we center it. That is, some part of Ω lies outside of D(p,r) for every p∈∂Ω.Lemma 10.
Letr∈R+ and p∈∂Ω. If ∂Ω is graph-like on D(p,r), then |∂Ω∩C(p,r)|≥2.Proof.
Suppose by way of contradiction that|∂Ω∩C(p,r)|<2. Since ∂Ω is a simple closed curve, we have ∂Ω⊆D(p,r). As ∂Ω is graph-like at p with radius r, there exists some orientation for which ∂Ω∩D(p,r)=∂Ω is the graph of a well-defined function. However, ∂Ω is a simple closed curve so it is not the graph of a function in any orientation, yielding a contradiction.The next result is the reason we find the tangent-cone graph-like condition useful. It says that if∂Ω is tangent-cone graph-like with radius r, then, for every p∈∂Ω, the disk D(p,r) has only two points of intersection with ∂Ω and these are transverse. In other words, this means that when working locally in the disk D(p,r), we need only to consider a single piece of ∂Ω.Theorem 11.
If∂Ω is tangent-cone graph-like with radius r∈R+ at p∈∂Ω, then |∂Ω∩C(p,r)|=2 and ∂Ω crosses C(p,r) transversely at these points. As a result, for every q1,q2∈∂Ω∩D(p,r), there is a unique arc along ∂Ω between them in D(p,r).Proof.
By Lemma10, we have that |∂Ω∩C(p,r)|≥2. Note that ∂Ω contains an interior point (p) and at least two boundary points of the disk D(p,r) (since |∂Ω∩C(p,r)|≥2). As ∂Ω is connected and simply closed, there must exist an arc of ∂Ω within the disk going from some point on C(p,r) through p to another point on C(p,r).
Suppose|∂Ω∩C(p,r)|>2; that is, there are other points of intersection. Letting q denote one of these, there are two cases to consider (illustrated in Figure 4).
(a)
∂
Ω
Does Not Cross
C
(
p
,
r
)
at
q. As ∂Ω is tangent-cone graph-like at q, then ∂Ω∩C(q,r) is a graph in every orientation in the tangent cone of ∂Ω at q. In particular, note that the tangent line to C(p,r) at q is in this cone. However, the line from p to q is normal to this line and thus ∂Ω∩C(q,r) is not graph-like in this orientation, a contradiction. Therefore, this case cannot occur. This argument applies to all points in ∂Ω∩C(p,r) so we immediately have the result that ∂Ω always crosses C(p,r) transversely.
(b)
∂
Ω
Crosses
C
(
p
,
r
)
at
q. There exists q′∈∂Ω∩C(p,r) such that there is a path along ∂Ω in D(p,r) from q to q′. That is, there exist s1,s2∈[0,L) (without loss of generality, s1<s2) such that γ(s1)=q, γ(s2)=q′, and the image of [s1,s2] under γ is contained in D(p,r) (but does not include p, since it is on another arc and ∂Ω is simple). Thus, γ enters C(p,r) at s1 and exits at s2.
If we can finds∈[s1,s2] and w in the tangent cone of ∂Ω at γ(s) satisfying 〈w,p-γ(s)〉=0, we will contradict that ∂Ω is tangent-cone graph-like.
Definev:[s1,s2]→R2 by
(4)v(s)={lim⁡t↓s1γ′(s),s=s1,lim⁡t↑sγ′(s),s∈(s1,s2].
Note that v(s) is in the tangent cone of ∂Ω at γ(s) so that ∂Ω∩D(γ(s),r) is graph-like using the orientation given by v(s).
Defineϕ(s):[s1,s2]→R by ϕ(s)=〈v(s),p-γ(s)〉. Note that from γ(s1) both v(s1) and p-γ(s1) are directions pointing into the circle so ϕ(s1)>0. Similarly, v(s2) points out and p-γ(s2) points in so that ϕ(s2)<0.
Observe thatv (and therefore ϕ) is piecewise continuous since γ is piecewise C1. By a piecewise continuous analogue of the intermediate value theorem, there exists s¯∈[s1,s2] such that
(5)lim⁡t→s¯-ϕ(t)≤0≤lim⁡t→s¯+ϕ(t).
By continuity of the inner product and γ, we have
(6)lim⁡t→s¯-ϕ(t)=〈lim⁡t→s¯-γ′(t),p-γ(s¯)〉.
Similarly, lim⁡t→s¯+ϕ(t)=〈lim⁡t→s¯+γ′(t),p-γ(s¯)〉.
Ifγ is differentiable at s¯, then ϕ(s¯)=lim⁡t→s¯ϕ(t)=0 and we have our contradiction. Otherwise, let w1=lim⁡t→s¯-γ′(t) and w2=lim⁡t→s¯+γ′(t). As both w1 and w2 are in the convex tangent cone of ∂Ω at γ(s¯), any positive linear combination of them is as well. Letting ψ(λ)=λw1+(1-λ)w2, we have
(7)〈ψ(0),p-γ(s¯)〉≤0≤〈ψ(1),p-γ(s¯)〉.
Noting that ψ is continuous in λ, we apply the intermediate value theorem to obtain λ¯∈(0,1) such that 〈ψ(λ¯),p-γ(s¯)〉=0. Letting w=ψ(λ¯), we obtain our contradiction.
Therefore, there are no other points of intersection and|∂Ω∩C(p,r)|=2.Figure 4
Additional points of intersection violate the TCGL condition.Definition 12.
One says thatΩ has the two-arc property for a given radius r if, for every point p∈∂Ω, one has that D(p,r) divides ∂Ω into two connected arcs: ∂Ω∩D(p,r) and ∂Ω∖D(p,r). Instead of considering how D(p,r) divides ∂Ω, one can equivalently frame the definition in terms of how ∂Ω divides C(p,r). That is, Ω has the two-arc property if the circle C(p,r) is divided into two connected arcs by ∂Ω for every p∈∂Ω.Corollary 13.
IfΩ is tangent-cone graph-like for some radius r, then it has the two-arc property.Proof.
This is a trivial consequence of Theorem11.Corollary 14.
IfΩ is tangentially graph-like for some radius r, then it has the two-arc property for radius r.Remark 15.
While the assumption of the two-arc property for disks of radiusr=r^ does not imply the two-arc property for all r<r^ (see Figure 5), it is the case that TGL for r=r^ does imply that γ is TGL for all 0<r<r^. The fact that γ is TGL for all 0<r<r^ follows easily from the definition of TGL and the fact that D(p,r)⊊D(p,r^).Figure 5
The two-arc property forr=r^ does not imply that it holds for all r<r^.


### 2.3. Notation

Suppose that∂Ω is tangent-cone graph-like with radius r and we have some s∈[0,L) such that ∂Ω is tangentially graph-like at γ(s) with radius r. Since ∂Ω is TGL at γ(s), it has two points of intersection with C(γ(s),r) by Theorem 11. In the orientation forced by the TGL condition, one of these points of intersection must be on the right side of the circle and one must be on the left side.With reference to Figure6 we define s+(s) and s-(s)∈[0,L) so that γ(s+(s)) is the point of intersection on the right and γ(s-(s)) is the point of intersection on the left. The notation is motivated by the fact that 0<s-(s)<s<s+(s)<L in general due to our convention that γ traverses ∂Ω counterclockwise. The only case where this is not true is when γ(L)=γ(0) is in the disk but even then it will hold for a suitably shifted γ^ that starts at some point outside the current disk.Figure 6
Notation and conventions.The quantitiesθ1(s) and θ2(s) are the angles that the rays from the origin to the right and left points of intersection, respectively, make with the positive x axis. We can assume θ1(s)∈(-π/2,π/2) and θ2(s)∈(π/2,3π/2).We defineν1(s) as the angle between the vector γ(s+(s))-γ(s) and the vector lim⁡t↓s+(s)γ′(t), the one-sided tangent to ∂Ω at the point of intersection on the right.That is, we are measuring the angle between the outward normal to the disk at the point of intersection and the actual directionγ is going as it exits the disk. We define ν2(s) similarly. We have ν1,ν2∈(-π/2,π/2) due to the fact that all circle crossings are transverse by Theorem 11.When the propers to use is implied by context, we will often simply write s+, s-, θ1, θ2, ν1, and ν2 in place of s+(s),s-(s), and so forth.


### 2.4. Calculus on Tangent Cones

The following result is a version of the intermediate value theorem for elements of the tangent cones.Lemma 16.
Suppose that∂Ω is tangent-cone graph-like on D(γ(s),r) and s1<s2 such that γ(s1),γ(s2)∈D(γ(s),r). Further suppose that w1∈Tγ(s1), w2∈Tγ(s2), and α∈(0,1), and let w′=αw1+(1-α)w2. Then, there exists s′∈[s1,s2] such that either w′ or -w′ is in Tγ(s′).Proof.
Letn be a unit vector in R2 with n⊥(αw1+(1-α)w2). We have α〈n,w1〉=-(1-α)〈n,w2〉. It suffices to consider only 〈n,w1〉≤0≤〈n,w2〉 as the argument is identical in the other case. Note that since 0≤〈n,w2〉=c1〈n,lim⁡t↑s2γ′(t)〉+c2〈n,lim⁡t↓s2γ′(t)〉 for some nonnegative constants c1,c2 not both zero, at least one of the inner products on the right is nonnegative. Using the notation of Definition 6, we define M2=argmaxΓ∈{Γ+,Γ-}〈n,Γ〉 and have 〈n,M2〉≥0. We similarly define M1 with respect to w1 such that 〈n,M1〉≤0.
Define(8)v(t)={Mi,t=si,i=1,2lim⁡t↑tγ′(s)
and ϕ(t)=〈n,v(t)〉. Since ϕ(s1)≤0≤ϕ(s2), the argument proceeds as in Theorem 11 to yield s¯∈[s1,s2] and w¯∈Tγ(s¯) such that 〈n,w¯〉=0. Thus, w¯=kw′ for some k≠0. In particular, w′=(1/k)w¯ so either w′∈Tγ(s¯) or -w′∈Tγ(s¯) (depending on the sign of k).In addition to the intermediate value theorem, we have an analogous mean value theorem for tangent cone elements.Lemma 17.
Suppose thatγ:[a,b]→R2 is a simple, arc-length parameterized curve with piecewise continuous derivative defined on (a,b) except possibly on finitely many points. Further suppose that the image of γ has no cusps. Then there exists c in (a,b) such that either γ(b)-γ(a) or -(γ(b)-γ(a)) is in Tγ(c).Proof.
Letn be a unit vector with 〈γ(b)-γ(a),n〉=0. Consider ψ(t)=〈γ(t),n〉 and note that ψ′(t)=〈γ′(t),n〉 is defined wherever γ(t) is differentiable. We have ∫ab‍ψ′(t)=ψ(b)-ψ(a)=〈γ(b)-γ(a),n〉=0. Thus, either ψ′(t)=0 everywhere it is defined or it takes on both positive and negative values. In particular, there exists a point c∈(a,b) such that either ψ′(c)=0 or lim⁡t↑cψ′(t)≤0≤lim⁡t↓cψ′(t).
Ifψ′(c)=0, then we have 〈γ′(c),n〉=0 so that γ′(c)=k(ϕ(b)-ϕ(a)) for some k≠0. As γ′(c)∈Tγ(c), we have (k/|k|)(ϕ(b)-ϕ(a))∈Tγ(c) which gives us our conclusion.
Iflim⁡t↑cψ′(t)≤0≤lim⁡t↓cψ′(t), there exists α∈(0,1) such that 0=αlim⁡t↑cψ′(t)+(1-α)lim⁡t↓cψ′(t). Note that lim⁡t↑cψ′(t)=〈w1,n〉 and lim⁡t↓cψ′(t)=〈w2,n〉 for some w1,w2∈Tγ(c) and let w′=αw1+(1-α)w2.
By the convexity ofTγ(c), we have w′∈Tγ(c) with 〈w′,n〉=0 which follows as in the previous case.The following lemma tells us that the tangent-cone graph-like condition is sufficient to apply Lemma17.Lemma 18.
If∂Ω is tangent-cone graph-like for some radius r, then ∂Ω has no cusps.Proof.
Suppose∂Ω has a cusp at γ(s). Then, using the terminology of Definition 6 and the fact that γ is arc length parameterized, we have Γ+=-Γ-. We let w=0 and note that w=Γ++Γ-∈Tγ(s). Letting u,v∈∂Ω∩D(γ(s),r) with u≠v, we have 〈w,u-v〉=0, contradicting the fact that ∂Ω is tangent-cone graph-like. Therefore, ∂Ω has no cusps.


### 2.5. TCGL Boundary Properties

The following technical lemmas allow us to bound various distances and areas encountered in tangent-cone graph-like boundaries.Lemma 19.
Suppose that∂Ω is tangent-cone graph-like with radius r and points p1,p2∈∂Ω with d(p1,p2)<r. Then one of the arcs (call it P) along ∂Ω between p1 and p2 is such that, for any two points q1,q2∈P, one has d(q1,q2)<r.Proof.
Note thatp2∈D(p1,r) so that there is an arc along ∂Ω from p1 to p2 which is fully contained in the interior of D(p1,r) by Theorem 11. We will call this arc P.
For allx on P, let Px denote the subpath of P from p1 to x (so P=Pp2). We claim that Px is contained in D(x,r) for all x on P (thus, P is contained in D(p2,r)). Indeed, if this were not the case, then there must be some x^ on P such that Px^ is contained in D(x^,r) but C(x^,r)∩Px^ is nonempty (i.e., we can move the disk along P until some part of the subpath hits the boundary). That is, the subpath Px^ has a tangency with the disk D(x^,r) which is impossible because of Theorem 11.
Letq1∈P and note that since Px is contained in D(x,r) for all x on P, we have that P is contained in D(q1,r). Therefore, d(q1,q2)<r for all q1,q2∈P as desired.Lemma 20.
Ifq1=γ(s1), q2=γ(s2)∈P where P is as in the previous lemma, then the arc length between q1 and q2 along P is at most 2d(q1,q2).Proof.
SinceΩ is tangentially graph-like, for any w1∈Tγ(s1), w2∈Tγ(s2), the angle between w1 and w2 is at most π/2. Since this is true for all q∈P, there is a point q′=γ(s′)∈P and w′∈Tγ(s′) such that the angle between w′ and tangent vectors for any other point q∈P is at most π/4.
This means thatP is the graph of a Lipschitz function g of rank 1 in the orientation defined by w′. This does not necessarily imply that D(q′,r)∩∂Ω, D(p1,r)∩∂Ω, or D(p2,r)∩∂Ω is the graph of a Lipschitz function; we explore a Lipschitz condition for the disks in Section 3. Let x1,x2∈[-r,r] with p1=(x1,g(x1)), p2=(x2,g(x2)). Then the arclength from p1 to p2 is given by
(9)∫x1x2‍1+g′(x)2dx≤∫x1x2‍2dx=2(x2-x1)≤2d(p1,p2).Lemma 21.
Ifγ is tangent-cone graph-like with radius r and 0≤s1≤s2<L with d(γ(s1),γ(s2))=δ<r, then the image of [s1,s2] together with the straight line from γ(s1) to γ(s2) encloses a region with O(δ2) area.Proof.
By Lemma20, we have that the image of [s1,s2] under γ has arc length s2-s1≤2δ. Therefore, the region of interest has perimeter at most (2+1)δ so by the isoperimetric inequality it has area at most ((2+1)2/4π)δ2 from which the conclusion follows.


## 2.1. Definitions

Definition 3.
In the current context, the integral area invariant [1] is denoted by g:[0,L)×R+→R+ and given by
(3)g(s,r)=∫D(γ(s),r)∩Ω‍dx=Area(Ω∩D(γ(s),r)).Remark 4.
Note the lack of the normalizing factorπr2 in the definition of g(s,r). Since we presume that r is fixed and known for the situations we study, it is trivial to convert data between the forms g(s,r) and g(s,r)/πr2; we choose to leave out the normalizing factor in the definition of g(s,r) as it is the integral area invariant of Manay et al. [1] and this form proves useful when computing derivatives in Section 4.We introduce the tangentially graph-like condition as a simplifying assumption for the shapes we consider.Definition 5.
For a fixed radiusr, one says that ∂Ω is graph-like (GL) at a point p∈∂Ω (or graph-like on D(p,r)) if it is possible to impose a Cartesian coordinate system such that the set of points ∂Ω∩D(p,r) is the graph of some function f in this coordinate system. Without loss of generality, one adopts the convention that p is the origin so that f(0)=0. One defines tangentially graph-like (TGL) in the same way but further requires that ∂Ω be continuously differentiable and f′(0)=0 (noting that f is C1 because ∂Ω is). This is illustrated in Figure 2(a). Without loss of generality (and in keeping with our convention that γ traverses ∂Ω counterclockwise), one assumes that the interior of Ω is “up” in the circle (i.e., (0,ϵ)∈Ω for sufficiently small ϵ>0). If ∂Ω is (tangentially) graph-like on D(p,r) for all p∈∂Ω, one says that ∂Ω is (tangentially) graph-like for radius r.(a) Tangentially and (b) tangent-cone graph-like.
(a)
(b)It is instructive to consider what isnot considered graph-like or tangentially graph-like. Violations of the graph-like condition are generally due to a radius that is too large (certainly, choosing a radius so large that all of Ω is in the disk will do it). For example, a unit side length square is not graph-like with radius (1/2)+ϵ for any ϵ>0 (position the circle at the center of a side; see Figure 3(a)). Notice that the same square is graph-like with any radius 1/2 or below. A shape can fail to be tangentially graph-like while still being graph-like if it fails to be a graph in the required orientation but works in some other (see Figure 3(b)).(a) The square is not graph-like with the indicated radius (no orientation makes it a graph). (b) The rounded rectangle is graph-like but not tangentially graph-like with the indicated center and radius.
(a)
(b)We would like to consider shapes with corners but our tangentially graph-like condition requires that the boundary be differentiable everywhere. The following definitions allow us to generalize the tangentially graph-like condition to this situation by using one-sided derivatives.Definition 6.
Given a piecewiseC1 function γ:[0,L]→R2, one defines the tangent cone of γ at a point s (which is denoted by Tγ(s)) in terms of the one-sided derivatives. In particular, one lets Tγ(s)={αΓ-+βΓ+∣α,β≥0,α+β>0} where Γ-=lim⁡t↑sγ′(t) and Γ+=lim⁡t↓sγ′(t).Definition 7.
One extends the tangentially graph-like notion to boundaries that are piecewiseC1 by defining ∂Ω to be tangent-cone graph-like (TCGL) at a point γ(s)∈∂Ω if it is graph-like at γ(s) for every orientation in the tangent cone of ∂Ω at s. More precisely, for every w∈Tγ(s) and every pair of distinct points u,v∈∂Ω∩D(p,r), one has 〈w,u-v〉≠0. See Figure 2(b).Remark 8.
It is clear thatTγ(s) in Definition 6 is a convex cone. The tangent cone is dependent on the direction in which γ traverses ∂Ω (which by convention was counterclockwise) since an arc-length traversal γ^(s,r)=γ(L-s,r) would have different tangent cones (namely, w∈Tγ(s) iff -w∈Tγ^(s)). However, these differences are irrelevant to the application of Definition 7.Remark 9.
Note that when∂Ω is C1, there is only one direction in Tγ(s) for each s (i.e., the tangent to ∂Ω at γ(s)). Thus, the definitions of tangentially graph-like and tangent-cone graph-like coincide when ∂Ω is C1 and every tangentially graph-like boundary is tangent-cone graph-like.


## 2.2. Two-Arc Property

The graph-like condition implies (in proof of the following lemma) thatΩ will never be entirely contained in the disk, no matter where on the boundary we center it. That is, some part of Ω lies outside of D(p,r) for every p∈∂Ω.Lemma 10.
Letr∈R+ and p∈∂Ω. If ∂Ω is graph-like on D(p,r), then |∂Ω∩C(p,r)|≥2.Proof.
Suppose by way of contradiction that|∂Ω∩C(p,r)|<2. Since ∂Ω is a simple closed curve, we have ∂Ω⊆D(p,r). As ∂Ω is graph-like at p with radius r, there exists some orientation for which ∂Ω∩D(p,r)=∂Ω is the graph of a well-defined function. However, ∂Ω is a simple closed curve so it is not the graph of a function in any orientation, yielding a contradiction.The next result is the reason we find the tangent-cone graph-like condition useful. It says that if∂Ω is tangent-cone graph-like with radius r, then, for every p∈∂Ω, the disk D(p,r) has only two points of intersection with ∂Ω and these are transverse. In other words, this means that when working locally in the disk D(p,r), we need only to consider a single piece of ∂Ω.Theorem 11.
If∂Ω is tangent-cone graph-like with radius r∈R+ at p∈∂Ω, then |∂Ω∩C(p,r)|=2 and ∂Ω crosses C(p,r) transversely at these points. As a result, for every q1,q2∈∂Ω∩D(p,r), there is a unique arc along ∂Ω between them in D(p,r).Proof.
By Lemma10, we have that |∂Ω∩C(p,r)|≥2. Note that ∂Ω contains an interior point (p) and at least two boundary points of the disk D(p,r) (since |∂Ω∩C(p,r)|≥2). As ∂Ω is connected and simply closed, there must exist an arc of ∂Ω within the disk going from some point on C(p,r) through p to another point on C(p,r).
Suppose|∂Ω∩C(p,r)|>2; that is, there are other points of intersection. Letting q denote one of these, there are two cases to consider (illustrated in Figure 4).
(a)
∂
Ω
Does Not Cross
C
(
p
,
r
)
at
q. As ∂Ω is tangent-cone graph-like at q, then ∂Ω∩C(q,r) is a graph in every orientation in the tangent cone of ∂Ω at q. In particular, note that the tangent line to C(p,r) at q is in this cone. However, the line from p to q is normal to this line and thus ∂Ω∩C(q,r) is not graph-like in this orientation, a contradiction. Therefore, this case cannot occur. This argument applies to all points in ∂Ω∩C(p,r) so we immediately have the result that ∂Ω always crosses C(p,r) transversely.
(b)
∂
Ω
Crosses
C
(
p
,
r
)
at
q. There exists q′∈∂Ω∩C(p,r) such that there is a path along ∂Ω in D(p,r) from q to q′. That is, there exist s1,s2∈[0,L) (without loss of generality, s1<s2) such that γ(s1)=q, γ(s2)=q′, and the image of [s1,s2] under γ is contained in D(p,r) (but does not include p, since it is on another arc and ∂Ω is simple). Thus, γ enters C(p,r) at s1 and exits at s2.
If we can finds∈[s1,s2] and w in the tangent cone of ∂Ω at γ(s) satisfying 〈w,p-γ(s)〉=0, we will contradict that ∂Ω is tangent-cone graph-like.
Definev:[s1,s2]→R2 by
(4)v(s)={lim⁡t↓s1γ′(s),s=s1,lim⁡t↑sγ′(s),s∈(s1,s2].
Note that v(s) is in the tangent cone of ∂Ω at γ(s) so that ∂Ω∩D(γ(s),r) is graph-like using the orientation given by v(s).
Defineϕ(s):[s1,s2]→R by ϕ(s)=〈v(s),p-γ(s)〉. Note that from γ(s1) both v(s1) and p-γ(s1) are directions pointing into the circle so ϕ(s1)>0. Similarly, v(s2) points out and p-γ(s2) points in so that ϕ(s2)<0.
Observe thatv (and therefore ϕ) is piecewise continuous since γ is piecewise C1. By a piecewise continuous analogue of the intermediate value theorem, there exists s¯∈[s1,s2] such that
(5)lim⁡t→s¯-ϕ(t)≤0≤lim⁡t→s¯+ϕ(t).
By continuity of the inner product and γ, we have
(6)lim⁡t→s¯-ϕ(t)=〈lim⁡t→s¯-γ′(t),p-γ(s¯)〉.
Similarly, lim⁡t→s¯+ϕ(t)=〈lim⁡t→s¯+γ′(t),p-γ(s¯)〉.
Ifγ is differentiable at s¯, then ϕ(s¯)=lim⁡t→s¯ϕ(t)=0 and we have our contradiction. Otherwise, let w1=lim⁡t→s¯-γ′(t) and w2=lim⁡t→s¯+γ′(t). As both w1 and w2 are in the convex tangent cone of ∂Ω at γ(s¯), any positive linear combination of them is as well. Letting ψ(λ)=λw1+(1-λ)w2, we have
(7)〈ψ(0),p-γ(s¯)〉≤0≤〈ψ(1),p-γ(s¯)〉.
Noting that ψ is continuous in λ, we apply the intermediate value theorem to obtain λ¯∈(0,1) such that 〈ψ(λ¯),p-γ(s¯)〉=0. Letting w=ψ(λ¯), we obtain our contradiction.
Therefore, there are no other points of intersection and|∂Ω∩C(p,r)|=2.Figure 4
Additional points of intersection violate the TCGL condition.Definition 12.
One says thatΩ has the two-arc property for a given radius r if, for every point p∈∂Ω, one has that D(p,r) divides ∂Ω into two connected arcs: ∂Ω∩D(p,r) and ∂Ω∖D(p,r). Instead of considering how D(p,r) divides ∂Ω, one can equivalently frame the definition in terms of how ∂Ω divides C(p,r). That is, Ω has the two-arc property if the circle C(p,r) is divided into two connected arcs by ∂Ω for every p∈∂Ω.Corollary 13.
IfΩ is tangent-cone graph-like for some radius r, then it has the two-arc property.Proof.
This is a trivial consequence of Theorem11.Corollary 14.
IfΩ is tangentially graph-like for some radius r, then it has the two-arc property for radius r.Remark 15.
While the assumption of the two-arc property for disks of radiusr=r^ does not imply the two-arc property for all r<r^ (see Figure 5), it is the case that TGL for r=r^ does imply that γ is TGL for all 0<r<r^. The fact that γ is TGL for all 0<r<r^ follows easily from the definition of TGL and the fact that D(p,r)⊊D(p,r^).Figure 5
The two-arc property forr=r^ does not imply that it holds for all r<r^.


## 2.3. Notation

Suppose that∂Ω is tangent-cone graph-like with radius r and we have some s∈[0,L) such that ∂Ω is tangentially graph-like at γ(s) with radius r. Since ∂Ω is TGL at γ(s), it has two points of intersection with C(γ(s),r) by Theorem 11. In the orientation forced by the TGL condition, one of these points of intersection must be on the right side of the circle and one must be on the left side.With reference to Figure6 we define s+(s) and s-(s)∈[0,L) so that γ(s+(s)) is the point of intersection on the right and γ(s-(s)) is the point of intersection on the left. The notation is motivated by the fact that 0<s-(s)<s<s+(s)<L in general due to our convention that γ traverses ∂Ω counterclockwise. The only case where this is not true is when γ(L)=γ(0) is in the disk but even then it will hold for a suitably shifted γ^ that starts at some point outside the current disk.Figure 6
Notation and conventions.The quantitiesθ1(s) and θ2(s) are the angles that the rays from the origin to the right and left points of intersection, respectively, make with the positive x axis. We can assume θ1(s)∈(-π/2,π/2) and θ2(s)∈(π/2,3π/2).We defineν1(s) as the angle between the vector γ(s+(s))-γ(s) and the vector lim⁡t↓s+(s)γ′(t), the one-sided tangent to ∂Ω at the point of intersection on the right.That is, we are measuring the angle between the outward normal to the disk at the point of intersection and the actual directionγ is going as it exits the disk. We define ν2(s) similarly. We have ν1,ν2∈(-π/2,π/2) due to the fact that all circle crossings are transverse by Theorem 11.When the propers to use is implied by context, we will often simply write s+, s-, θ1, θ2, ν1, and ν2 in place of s+(s),s-(s), and so forth.


## 2.4. Calculus on Tangent Cones

The following result is a version of the intermediate value theorem for elements of the tangent cones.Lemma 16.
Suppose that∂Ω is tangent-cone graph-like on D(γ(s),r) and s1<s2 such that γ(s1),γ(s2)∈D(γ(s),r). Further suppose that w1∈Tγ(s1), w2∈Tγ(s2), and α∈(0,1), and let w′=αw1+(1-α)w2. Then, there exists s′∈[s1,s2] such that either w′ or -w′ is in Tγ(s′).Proof.
Letn be a unit vector in R2 with n⊥(αw1+(1-α)w2). We have α〈n,w1〉=-(1-α)〈n,w2〉. It suffices to consider only 〈n,w1〉≤0≤〈n,w2〉 as the argument is identical in the other case. Note that since 0≤〈n,w2〉=c1〈n,lim⁡t↑s2γ′(t)〉+c2〈n,lim⁡t↓s2γ′(t)〉 for some nonnegative constants c1,c2 not both zero, at least one of the inner products on the right is nonnegative. Using the notation of Definition 6, we define M2=argmaxΓ∈{Γ+,Γ-}〈n,Γ〉 and have 〈n,M2〉≥0. We similarly define M1 with respect to w1 such that 〈n,M1〉≤0.
Define(8)v(t)={Mi,t=si,i=1,2lim⁡t↑tγ′(s)
and ϕ(t)=〈n,v(t)〉. Since ϕ(s1)≤0≤ϕ(s2), the argument proceeds as in Theorem 11 to yield s¯∈[s1,s2] and w¯∈Tγ(s¯) such that 〈n,w¯〉=0. Thus, w¯=kw′ for some k≠0. In particular, w′=(1/k)w¯ so either w′∈Tγ(s¯) or -w′∈Tγ(s¯) (depending on the sign of k).In addition to the intermediate value theorem, we have an analogous mean value theorem for tangent cone elements.Lemma 17.
Suppose thatγ:[a,b]→R2 is a simple, arc-length parameterized curve with piecewise continuous derivative defined on (a,b) except possibly on finitely many points. Further suppose that the image of γ has no cusps. Then there exists c in (a,b) such that either γ(b)-γ(a) or -(γ(b)-γ(a)) is in Tγ(c).Proof.
Letn be a unit vector with 〈γ(b)-γ(a),n〉=0. Consider ψ(t)=〈γ(t),n〉 and note that ψ′(t)=〈γ′(t),n〉 is defined wherever γ(t) is differentiable. We have ∫ab‍ψ′(t)=ψ(b)-ψ(a)=〈γ(b)-γ(a),n〉=0. Thus, either ψ′(t)=0 everywhere it is defined or it takes on both positive and negative values. In particular, there exists a point c∈(a,b) such that either ψ′(c)=0 or lim⁡t↑cψ′(t)≤0≤lim⁡t↓cψ′(t).
Ifψ′(c)=0, then we have 〈γ′(c),n〉=0 so that γ′(c)=k(ϕ(b)-ϕ(a)) for some k≠0. As γ′(c)∈Tγ(c), we have (k/|k|)(ϕ(b)-ϕ(a))∈Tγ(c) which gives us our conclusion.
Iflim⁡t↑cψ′(t)≤0≤lim⁡t↓cψ′(t), there exists α∈(0,1) such that 0=αlim⁡t↑cψ′(t)+(1-α)lim⁡t↓cψ′(t). Note that lim⁡t↑cψ′(t)=〈w1,n〉 and lim⁡t↓cψ′(t)=〈w2,n〉 for some w1,w2∈Tγ(c) and let w′=αw1+(1-α)w2.
By the convexity ofTγ(c), we have w′∈Tγ(c) with 〈w′,n〉=0 which follows as in the previous case.The following lemma tells us that the tangent-cone graph-like condition is sufficient to apply Lemma17.Lemma 18.
If∂Ω is tangent-cone graph-like for some radius r, then ∂Ω has no cusps.Proof.
Suppose∂Ω has a cusp at γ(s). Then, using the terminology of Definition 6 and the fact that γ is arc length parameterized, we have Γ+=-Γ-. We let w=0 and note that w=Γ++Γ-∈Tγ(s). Letting u,v∈∂Ω∩D(γ(s),r) with u≠v, we have 〈w,u-v〉=0, contradicting the fact that ∂Ω is tangent-cone graph-like. Therefore, ∂Ω has no cusps.


## 2.5. TCGL Boundary Properties

The following technical lemmas allow us to bound various distances and areas encountered in tangent-cone graph-like boundaries.Lemma 19.
Suppose that∂Ω is tangent-cone graph-like with radius r and points p1,p2∈∂Ω with d(p1,p2)<r. Then one of the arcs (call it P) along ∂Ω between p1 and p2 is such that, for any two points q1,q2∈P, one has d(q1,q2)<r.Proof.
Note thatp2∈D(p1,r) so that there is an arc along ∂Ω from p1 to p2 which is fully contained in the interior of D(p1,r) by Theorem 11. We will call this arc P.
For allx on P, let Px denote the subpath of P from p1 to x (so P=Pp2). We claim that Px is contained in D(x,r) for all x on P (thus, P is contained in D(p2,r)). Indeed, if this were not the case, then there must be some x^ on P such that Px^ is contained in D(x^,r) but C(x^,r)∩Px^ is nonempty (i.e., we can move the disk along P until some part of the subpath hits the boundary). That is, the subpath Px^ has a tangency with the disk D(x^,r) which is impossible because of Theorem 11.
Letq1∈P and note that since Px is contained in D(x,r) for all x on P, we have that P is contained in D(q1,r). Therefore, d(q1,q2)<r for all q1,q2∈P as desired.Lemma 20.
Ifq1=γ(s1), q2=γ(s2)∈P where P is as in the previous lemma, then the arc length between q1 and q2 along P is at most 2d(q1,q2).Proof.
SinceΩ is tangentially graph-like, for any w1∈Tγ(s1), w2∈Tγ(s2), the angle between w1 and w2 is at most π/2. Since this is true for all q∈P, there is a point q′=γ(s′)∈P and w′∈Tγ(s′) such that the angle between w′ and tangent vectors for any other point q∈P is at most π/4.
This means thatP is the graph of a Lipschitz function g of rank 1 in the orientation defined by w′. This does not necessarily imply that D(q′,r)∩∂Ω, D(p1,r)∩∂Ω, or D(p2,r)∩∂Ω is the graph of a Lipschitz function; we explore a Lipschitz condition for the disks in Section 3. Let x1,x2∈[-r,r] with p1=(x1,g(x1)), p2=(x2,g(x2)). Then the arclength from p1 to p2 is given by
(9)∫x1x2‍1+g′(x)2dx≤∫x1x2‍2dx=2(x2-x1)≤2d(p1,p2).Lemma 21.
Ifγ is tangent-cone graph-like with radius r and 0≤s1≤s2<L with d(γ(s1),γ(s2))=δ<r, then the image of [s1,s2] together with the straight line from γ(s1) to γ(s2) encloses a region with O(δ2) area.Proof.
By Lemma20, we have that the image of [s1,s2] under γ has arc length s2-s1≤2δ. Therefore, the region of interest has perimeter at most (2+1)δ so by the isoperimetric inequality it has area at most ((2+1)2/4π)δ2 from which the conclusion follows.


## 3. TCGL Polygonal Approximations

IfΩ is tangent-cone graph-like with radius r, it can sometimes be nice to know that there is an approximating polygon to Ω which is also tangent-cone graph-like. The following lemmas explore this idea.Lemma 22.
If∂Ω is TCGL with radius r, then, for each ϵ∈(0,r), there exists a polygonal approximation to ∂Ω that is TCGL with radius r-ϵ and such that every point on ∂Ω is within distance ϵ/6 of the polygon.Proof.
First, choose a finite number of points along the boundary such that the arc length alongγ between any two neighboring points is no more than ϵ/3. These will be the vertices of our polygon. Similarly to γ, we let ϕ be an arclength parameterization of this polygon so that they both encounter their common points in the same order.
The fine spacing between vertices guarantees that we obtain theϵ/6 bound. Indeed, given any point p∈∂Ω and its neighboring vertices v1 and v2, the arc length along ∂Ω from v1 to p plus that from p to v2 is at most ϵ/3 by assumption. Since Euclidean distance is bounded above by arc length, we have d(p,v1)+d(p,v2)≤ϵ/3. This bound in turn implies that at least one of d(p,v1) and d(p,v2) is bounded above by ϵ/6.
Consider a pointp=ϕ(t) on a side of the polygon (i.e., not a vertex) and its neighboring vertices v1=ϕ(t1)=γ(s1) and v2=ϕ(t2)=γ(s2) (chosen with t1<t<t2 and s1<s2). By Lemma 17, there exists s∈(s1,s2) such that v2-v1∈Tγ(s). Note that this is the only member of Tϕ(t) up to positive scalar multiplication.
Combining the arcs alongγ and ϕ between v1 and v2, we obtain a closed curve with total length at most 2ϵ/3, so that the distance between any two points on the curve is at most ϵ/3. That is, for any s′∈[s1,s2] and t′∈[t1,t2], we have d(γ(s′),ϕ(t′))≤ϵ/3.
Letx∈D(ϕ(t),r-ϵ). Then d(x,γ(s))≤d(x,ϕ(t))+d(ϕ(t),γ(s))≤r-(2ϵ/3) so that D(ϕ(t),r-ϵ) is contained in D(γ(s),r-(2ϵ/3)).
Leta,b be distinct points on the polygon in D(ϕ(t),r-ϵ) and consider the line connecting them. This line also intersects a′,b′ on γ such that we have a′≠b′, d(a,a′)≤ϵ/3, and d(b,b′)≤ϵ/3 so that a′,b′∈∂Ω∩D(γ(s),r). As a-b=c(a′-b′) for some scalar c>0, we have
(10)〈v2-v1,a-b〉=c〈v2-v1,a′-b′〉≠0,
since γ is TCGL at γ(s) with radius r and v2-v1∈Tγ(s). Thus, ϕ is TCGL at p with radius r-ϵ.
The case wherep=ϕ(t) is a vertex is similar but we must consider an arbitrary vector w∈Tϕ(t) in the inner product. We wish to show that, for every w∈Tϕ(t), there is a s′ such that either w or -w∈Tγ(s′) and d(p,γ(s′))≤ϵ/3, after which the proof follows as in the first case with w (or -w) in place of v2-v1. We let γ(s)=ϕ(t)=p and let v1=ϕ(t1)=γ(s1) and v2=ϕ(t2)=γ(s2) be the neighboring vertices (so t1<t<t2 and s1<s<s2).
As above, there exists1′,s2′ such that s1≤s1′≤s≤s2′≤s2, γ(s)-γ(s1)∈Tγ(s1′), and γ(s2)-γ(s)∈Tγ(s2′). Note that Tϕ(t) is exactly the set of positive linear combinations of these vectors. By Lemma 16, for every w∈Tϕ(t), there is a s′∈[s1′,s2′] such that w∈Tγ(s′). As d(p,γ(s′))<ϵ/3, the proof is complete.Definition 23.
One says thatΩ is tangentially graph-like and Lipschitz (TGLL) with radius r if Ω is tangentially graph-like with radius r and there is some constant 0<K<∞ such that, for every p∈∂Ω, the arc D(p,r)∩∂Ω is the graph of a Lipschitz function (in the same orientation used by the tangentially graph-like definition) and that the Lipschitz constant is at most K.Remark 24.
Note that tangentially graph-like does not imply tangentially graph-like and Lipschitz: takingγ to be a square with side length 5 whose corners are replaced by quarter circles of radius 1 and then considering disks of radius 2 centered on γ yields one example.
Becauseγ is arclength parameterized by s, ∥γ′(s)∥=1 for all s. Since γ is assumed C1 on its compact domain [0,L], γ′ is uniformly continuous: for any ϵ>0, there is a δϵ such that if |s2-s1|<δϵ, then ∥γ′(s2)-γ′(s1)∥<ϵ.
We will use the fact thatγ always crosses ∂D transversely to prove that γ is in fact TGLL on slightly bigger disks of radius r+δ as long as one takes a somewhat bigger Lipschitz constant K^. It is then an immediate result of Lemma 22 that we can find an approximating polygon that is TCGL with radius r.Lemma 25.
Ifγ is TGLL with radius r, then it is TGLL with radius r+δ for some δ>0 and there is an approximating polygon Pγ which is TCGL with radius r.Proof.
Step  1. Show that the quantities ν1 and ν2 are continuous as a function of s∈[0,L] (see Figure 6).
DefineR2(s,t)≡∥γ(s)-γ(t)∥2. Taking the derivative, we get
(11)DR=[〈γ(s)-γ(t)R(s,t),γ′(s)〉,〈γ(t)-γ(s)R(s,t),γ′(t)〉].
Because ν1 and ν2 are both less than π/2 and γ is graph-like in the disk, we have that both elements of this derivative are nowhere zero. By the implicit function theorem, we get that s-(s) and s+(s) are continuous functions of s. From this it follows that ν1 and ν2 are continuous on [0,L].
Step  2. From the previous step and the compactness of [0,L], we get that ν1(s) and ν2(s) are both bounded by Mν<π/2. We define ϵν≡π/2-Mν>0. Fix a t∈[0,L]. Define ρ^(s) by ρ^2(s)=R2(s,t)=∥γ(s)-γ(t)∥2. Then ρ^˙(s)=〈(γ(s)-γ(t))/ρ^,γ′(s)〉=〈nt(s),γ′(s)〉 where nt(s)=(γ(s)-γ(t))/∥γ(s)-γ(t)∥=(γ(s)-γ(t))/ρ^, the external normal to ∂D(γ(t),ρ^) at γ(s) (see Figure 7). On any interval in s where ρ^˙(s)>0 we have that ρ^(s) is one to one and strictly increasing. Define s*≡s+(t) and s*≡s-(t). We showed above that ρ^˙(s*)=〈nt(s*),γ′(s*)〉≥cos⁡(Mν)>0.
For〈nt(s),γ′(s)〉=0, nt(s) and γ′ will have to have together turned by at least π/2-Mν radians. And until they have turned this far, 〈nt(s),γ′(s)〉>0. But n˙t(s)≤1/ρ≤1/rmin⁡ for some rmin⁡>0. (Choosing rmin⁡=r/2 works.) And γ′ is uniformly continuous on [0,L]. Therefore, there is a δs such that on [s*,s*+δs], nt(s), and γ′ both turn by less than ϵν/3. Therefore, for s∈[s*,s*+δs], we have that 〈nt(s),γ′(s)〉>cos⁡(π/2-ϵν/3) and γ([s*,s*+δs)) intersects C=∂D(γ(t),ρ) once for each ρ∈[r,r+δr], where δr≡δscos⁡(π/2-ϵν/3).
A completely analogous argument works to show thatγ([s*-δs,s*]) intersects C=∂D(γ(t),ρ) once for each ρ∈[r,r+δr].
Defined(t) to be the distance from D(γ(t),r) to γ∖γ([s*-δs,s*+δs]). Since γ is TGL, d(t) is greater than zero for all t and is continuous in t. Therefore, there is a smallest distance δd such that d(t)≥δd for all t. Define δγo=min⁡(δd/2,δr/2).
Therefore,∂D(γ(t),ρ) intersects γ exactly twice for ρ∈[r,r+δγo] for any t∈[0,L].
A similar argument shows that∂D(γ(t),ρ) intersects γ exactly twice for ρ∈[r-δγi,r] for any t∈[0,L]. Defining δγ≡min⁡(δγi,δγo), we get that ∂D(γ(t),ρ) intersects γ exactly twice for ρ∈[r-δγ,r+δγ], with the additional fact that 〈nt(s),γ′(s)〉>cos⁡(π/2-ϵν/3) at all those intersections.
Step  3. TGLL implies that there is a constant K<∞ such that γ∩D(γ(t),r) is the graph of a function whosex-axis direction is parallel to γ′(t) and this function is Lipschitz with Lipschitz constant K.
Sinceγ′ is uniformly continuous, there will be a δ1 such that if |u-v|<δ1, then ∠(γ′(u),γ′(v))<arctan2K-arctanK. Define δK,s=min⁡(δs,δ1). Define δK,r=min⁡(δγ,δK,scos⁡(π/2-ϵν/3)). Then γ∩D(γ(t),r+δK,r) is the graph of a Lipschitz function with Lipschitz constant at most 2K when γ′(t) is used as thex-axis direction. That is, for all t, γ is TGLL with Lipschitz constant 2K for disks of radius r+δK,r. The result follows by Lemma 22.Figure 7
TGLL implies TCGL: notation.


## 4. Derivatives ofg(s,r)

Lemma 26.
Using the notation of Figure6, we have (∂/∂r)g(s,r)=(θ2-θ1)r. That is, the derivative exists and equals the length of the curve C(γ(s),r)∩Ω.Proof.
We have the following (see Figure8):
(12)∂∂rg(s,r)=lim⁡Δr→0Area(Ω∩D(γ(s),r+Δr))-Area(Ω∩D(γ(s),r))Δr.
This difference of areas can be modeled by the difference in the circular sectors of D(γ(s),r+Δr) and D(γ(s),r) with angle θ1-θ2. The actual area depends on the image of γ outside of D(γ(s),r), but this correction will be a subset of the circular segment of D(γ(s),r+Δr) which is tangent to D(γ(s),r) at the point γ exits. This has area O(Δr2) by Lemma 21.
Thus we have(13)∂∂rg(s,r)=lim⁡Δr→0(θ1-θ2)rΔr+(1/2)(θ1-θ2)Δr2+O(Δr2)Δr=(θ1-θ2)r.Figure 8
Deriving∂g/∂r as the arclength of the circular segment.Lemma 27.
Using the notation of Figures6 and 9, one has (∂/∂s)g(s,r)=h2-h1=rsin(θ2)-rsin(θ1).Figure 9
Deriving∂g/∂s as the difference in heights of the entry and exit points.Proof.
We have(14)∂∂sg(s,r)=lim⁡Δs→0Area(Ω∩D(γ(s+Δs),r))-Area(Ω∩D(γ(s),r))Δs.The situation is illustrated in Figure 9 where we can see that the area being added as we go from s to s+Δs is the shaded region on the right with height r-h1 and, considering first-order terms only, uniform width Δs so has area (r-h1)Δs. Similarly, we are subtracting the area (r-h2)Δs on the left. Therefore, we have
(15)∂∂sg(s,r)=lim⁡Δs→0(r-h1)Δs-(r-h2)ΔsΔs=h2-h1.


## 5. Reconstructing Shapes from T-Like Data

In this section, we consider the case where nonasymptotic densities and first derivatives are known along a T-shaped set (i.e., for alls with a fixed radius r^ and for all r≤r^ with a fixed s^). We show that this information is sufficient to guarantee reconstructability modulo reparametrizations, translations, and rotations.Lemma 28.
Assume thatγ is TGL for r^ (and thus all r≤r^). Then if one knows g(s,r), gs(s,r)=∂g(s,r)/∂s, and gr(s,r)=∂g(s,r)/∂r for (s,r)∈([0,L]×{r^})∪({s^}×(0,r^]), one can reconstruct γ(s)∈R2 for all s∈[0,L] modulo reparametrizations, translation, and rotations (see Figure 10).Figure 10
T-like data: we restrict the domain ofg(s,r) to a fixed radius r^ plus any vertical segment from r=0 to r=r^.Proof.
As was shown in Section4, gr gives us the length of the arc ∂D(s,r^)∩Ω and gs tells us precisely what position this arc is along ∂D(s,r^) with respect to the direction γ′(s). The assumption of TGL for r=r^ implies TGL for 0<r<r^ (see Remark 15) and this implies that γ has the 2-arc property and transverse intersections with ∂D(s,r) for all disks corresponding to (s,r)∈([0,L]×{r^})∪({s^}×[0,r^]). Since we care only about reconstructing a curve γ isometric to the original curve, we choose γ(s^)=(0,0)∈R2 and γ′(s^)=(1,0). Taken together, gs(s^,r) and gr(s^,r) locate both points in ∂D(s^,r)∩γ for all r∈[0,r^]. This yields γ∩D(γ(s^),r^). Now, simply increase s, sliding the center of a disk of radius r^ along γ∩D(γ(s^),r^), using gr(s,r^) to find the element of γ∩D(γ(s),r^) outside D(γ(s^),r^), and using the fact that the other element of γ∩D(γ(s),r^) is inside D(γ(s^),r^) and known. This process can be continued until the entire curve is traced out in R2.


## 6. TCGL Polygon Is Reconstructible fromgr and gs without Tail

Theorem 29.
For a tangent-cone graph-like polygonΩ, knowing g(s,r), gr(s,r), and gs(s,r) for all s∈[0,L) and a particular r for which ∂Ω is tangent-cone graph-like is sufficient to completely determine Ω up to translation and rotation; that is, one can recover the side lengths and angles of Ω.Proof.
For a givens and r where gr and gs exist, we can use them to obtain r(θ2-θ1) as the length of the circular arc between the entry and exit points by Lemma 26 and r(sinθ2-sinθ1) as the difference in heights of the entry and exit points by Lemma 27.
We wish to recoverθ1 and θ2 from these quantities. Note that if (θ1,θ2)=(ϕ1,ϕ2) is one possible solution, then so is (θ1,θ2)=(2π-ϕ2,2π-ϕ1) and so solutions always come in pairs.
We can imagine placing a circular arc with anglegr/r on our circle and sliding it around until the endpoints have the appropriate height difference, yielding θ1 and θ2. Note that since Ω is tangent-cone graph-like, one endpoint must be on the left side of the circle and the other must be on the right and we cannot slide either endpoint to or beyond the vertical line through the center of the circle.
Therefore, as we slide the right endpoint down, the left endpoint slides up so that the height difference as a function of the slide is strictly monotonic. Therefore, the slide that gives usθ1 and θ2 is unique for a given starting arc placement. However, there are two starting arc placements: the first calls the angle for the right endpoint θ1 and the left endpoint θ2 (so the interior of Ω is “up” in the circle) and the second swaps these (so the interior of Ω is “down”). Since we have adopted the convention that ∂Ω is traversed in a counterclockwise direction (so the interior of Ω is up in the circles), we therefore pick the first option; this gives us a unique solution for θ1 and θ2.
This procedure works whenevergr and gs exist which is certainly true whenever the density disk does not touch a vertex of Ω either at its center or on its boundary because if we avoid these cases, then there is only one graph-like orientation to deal with and ∂Ω is C∞ for all the points that enter into the computation. In fact, with a moment's thought, we can make a stronger statement than this: gr always exists and gs exists as long as the center of the density disk is not a vertex of the polygon.
We can identify thes values at which gs(s,r) does not exist to obtain the arc length positions of the vertices (and therefore obtain side lengths). For a given s corresponding to a vertex, we can find gr and the one-sided derivatives gs- and gs+. These correspond to the graph-like orientations required by the polygon sides adjacent to the current vertex.
Referring to Figure11, the one-sided derivatives along with the argument at the beginning of the proof yield the angles θ1, θ2, ϕ1, and ϕ2. Thus, we can calculate ψ=θ1-ϕ1 which means that the polygon vertex at s has angle π-ψ.
Doing this for alls corresponding to vertices, we can determine all of the angles of the polygon. With the side lengths identified earlier, this completely determines the polygon Ω up to translation and rotation.Figure 11
Usinggs- and gs+ to obtain the polygon angle at s.


## 7. Simple Closed Curves Are Generically Reconstructible Using Fixed Radius Data

We will assume thatγ is TGL for the radius r^. We will also assume that we know the first, second, and third derivatives of g(s,r) for r=r^. Under these assumptions, γ is generically reconstructible. By generic we mean theadmittedly weak condition of density—reconstructible curves are C1 dense in the space of C2 simple closed curves.Theorem 30.
DefineG≡{γ∣γ as a C2 simple closed curve and TGL for r=r^}. Suppose that, for r=r^, for all s∈[0,L], and for each γ∈G, one knows the first-, second-, and third-order partial derivatives of gγ(s,r). Then the set of reconstructible γ∈G is C1 dense in G where reconstructability is modulo reparametrization, translation, and rotation.Proof.
In Section4 we showed that ∂g(s,r)/∂r=r(θ2-θ1) and ∂g(s,r)/∂s=r(sin(θ2)-sin(θ1)), where the notation is as in Figure 6. Because γ is TGL, we can solve for θ1 and θ2 from these two derivatives as in the proof of Theorem 29.Claim 1.
The following equations hold:∂2g(s,r)/∂r2=θ2-θ1+r(∂θ2/∂r-∂θ1/∂r) and ∂2g(s,r)/∂r∂s=sin(θ2)-sin(θ1)+r(cos⁡(θ2)(∂θ2/∂r)-cos⁡(θ1)(∂θ1/∂r)).Proof of Claim1.
Simply differentiate the expressions we already have for∂g(s,r)/∂r and ∂g(s,r)/∂s.We wish to express this in terms ofν1 and ν2. Note that if we expand the circle radius by Δr, the right exit point s+(s) moves approximately (i.e., considering first-order terms only) a distance of k≡Δrsec(ν1) (so ∂k/∂r=secν1, a fact we will use later to compute curvature). Therefore,
(16)∂θ1∂r=lim⁡Δr→0((arctan(rsinθ1+ksin(θ1+ν1)rcos⁡θ1+kcos⁡(θ1+ν1))-θ1)×(Δr)-1(rsinθ1+ksin(θ1+ν1)rcos⁡θ1+kcos⁡(θ1+ν1))).
Straightforward techniques yield ∂θ1/∂r=tanν1/r and a similar calculation shows that ∂θ2/∂r=tanν2/r.Therefore, rewriting the second derivatives ofg(s,r) in terms of ν1 and ν2, we get
(17)∂2g(s,r)∂r2=θ2-θ1+tan(ν2)-tan(ν1),∂2g(s,r)∂r∂s=sin(θ2)-sin(θ1)+cos⁡(θ2)tan(ν2)-cos⁡(θ1)tan(ν1).
Using these 2 derivatives, together with the previous two, we can solve for ν1=arctan(r(∂θ1/∂r)) and ν2=arctan(r(∂θ1/∂r)) whenever cos⁡(θ1)≠cos⁡(θ2). Since we are assuming that the curve is a simple closed curve, cos⁡(θ1)≠cos⁡(θ2) is always true.Claim 2.
Knowing∂3g(s,r)/∂r3 and ∂3g(s,r)/∂r2∂s gives us κ(s+(s)) and κ(s-(s)), the curvatures of γ at s+(s) and s-(s).Proof of Claim2.
Computing, we get(18)∂3g(s,r)∂r3=∂θ2∂r-∂θ1∂r+sec2(ν2)∂ν2∂r-sec2(ν1)∂ν1∂r∂3g(s,r)∂r2∂s=cos⁡(θ2)∂θ2∂r-cos⁡(θ1)∂θ1∂r-sin(θ2)∂θ2∂rtan(ν2)+sin(θ1)∂θ1∂rtan(ν1)+cos⁡(θ2)sec2(ν2)∂ν2∂r-cos⁡(θ1)sec2(ν1)∂ν1∂r.
Since ν2′≡∂ν2/∂r and ν1′≡∂ν1/∂r are the only unknowns, we end up having to invert
(19)[1-1cos⁡(θ2)cos⁡(θ1)]
again and this is always nonsingular, giving us ν1′ and ν2′ as a function of s, the coordinate of the center of the disk.
Relative to the horizontal, the angle of the curve ats+(s) is θ1+ν1 so the rate of change in angle as we expand the circle is (∂θ1/∂r)+ν1′. Recalling that rate of movement of this exit point as we expand the circle is given by ∂k/∂r=secν1, we have that the curvature is given by κ(s+(s))=(∂k/∂r)(∂θ1/∂r+ν1′)=secν1(∂θ1/∂r+ν1′). Similarly, κ(s-(s))=sec(ν2)(∂θ2/∂r+ν2′).Claim 3.
Generically, we can deduces+(s) from knowledge of ν1(s), ν2(s), θ1(s), and θ2(s).Proof.
We outline the proof without some of the explicit constructions that follow without much trouble from the outline. We have thatθ1(s-(s))+ν1(s-(s))=π-θ2(s)-ν2(s) and θ1(s)+ν1(s)=π-θ2(s+(s))-ν2(s+(s)). All four of these quantities (the left- and right-hand sides of each of the 2 equations) are the turning angles between the tangent to the curve at the center of the disk and the tangent to the curve at a point r away from the center of the disk.
Now we use this correspondence between theθ+ν curves to solve for s-(s) and s+(s). But these curves can differ by a homeomorphism of the domain. Thus, we can only find the correspondence if there is a distinguished point on those curves as well as no places where the values attained are constant. The turning angle curves, having isolated critical points and a unique maximum or minimum, are enough.
To get isolated extrema, start by approximating the curveγ with another one γ^ that agrees in C1 at a large but finite number of points {si}i=1N (i.e., agrees in tangent direction as well as position) and has isolated critical points in the derivative of the tangent direction. Now perturb γ^ to one that is C1 close (but not C2 close) by using oscillations about the curve so that the 2nd and 3rd derivatives are never simultaneously below the bounds on the 2nd and 3rd derivatives of the curve we started with. We do this in a way that alternates around the curve. See Figure 12. In a bit more detail, suppose that max⁡{d2γ^/ds2,d3γ^/ds3}<L1. Choose a starting point on the curve; s=0 works. Now begin perturbing γ^ at the point sr^ in the positive s direction such that |γ^(sr^)-γ^(0)|=r^. We name the newly perturbed curve γ^^ and we keep L1<max⁡{d2γ^^/ds2,d3γ^^/ds3}<L2. We continue perturbing until we have reached s2r defined by |γ^(s2r^)-γ^(sr^)|=r^. We begin perturbing again when we reach s3r^. Continue in this fashion around γ^. The last piece, shown in green in the figure, will require a perturbation that is distinct in size due to the fact that it will interact with the perturbation that starts at sr^. On this last piece, we enforce L2<max⁡{d2γ^^/ds2,d3γ^^/ds3}<L3. All these perturbations can be chosen with isolated singularities in derivatives, thus giving us θ+ν curves that are monotonic between isolated singularities. (In fact, we might as well choose all perturbations to be piecewise polynomial perturbations. This immediately gives us the isolated singularities and monotonicity that we want.)
Finally, if there is not a distinct maximum, we can choose one of the maxima and add a small twist to the curve at that point. See Figure13. The idea is that a small twist, applied to leading edge of the tangents we are comparing to get the turning angle, will increase the angle most at the center of the twist. If this corresponds to a nonunique global maximum, we end up with a unique global maximum.
Now the correspondence scheme works. That is, we know that the global maximums must match, and because the turning angle curves are monotonic between isolated critical points, we can find the homeomorphisms ins that move the turning angle curves into correspondence.Figure 12
In this schematic figure, we illustrate the alternating perturbation around the curve, keeping the curveC1 close to and messing with the second and third derivatives to eliminate any critical points other than isolated maxima and minima. Here, the perturbation is of course greatly exaggerated.Figure 13
A twist perturbation. Notice that if the twist is applied precisely at a global max of the turning angle (as measured by the tangent here and the one lagging it ins), we will increase the turning angle there and will end up with a unique global maximum.Taken together, the last two claims give us the curvature as a function of arclength. This determinesγ up to translations and rotations.


## 8. Numerical Experiments

In this section, we consider a numerical curve reconstruction for the situation in whichg(s,r) is known for a given radius r but no derivative information is available. This reconstruction is more strict than the scenarios of Sections 5–7. Our motivation is to explore whether any γ can be uniquely and practically reconstructed with this limited information.We considerγa(s¯)∈PN, the set ofsimple polygons of N ordered vertices {(x1,y1),…,(xN,yN)} parameterized by the set {s¯k}k=1N with s¯k=k/N, as
(20)xk=∑j=0m-1‍a1,jcos⁡(2πjs¯kN)+a2,jsin(2πjs¯kN),yk=∑j=0m-1‍a3,jcos⁡(2πjs¯kN)+a4,jsin(2πjs¯kN),
for some coefficients ai,j∈R. In this way, the polygon γ is a discrete approximation of a C∞ curve. The sides of γa(s¯) are not necessarily of equal length.We take the vector signaturega(s¯,r)∈RN to be the discrete area densities of γa(s¯) computed at each vertex. Given such a signature for fixed radius r and fixed partition s¯, we seek a* satisfying
(21)a*∈argmin⁡b∈R4m∥gb(s¯,r)-ga(s¯,r)∥22s.t.γb∈PN.
Equation (21) represents a nonlinearly constrained optimization problem with continuous nonsmooth objective. The constraint ensures that polygons are simple though any optimal reconstruction γa* is not expected to lie on the feasible region boundary except in cases of noisy signatures. This approach to reconstructing curves seeks a polygon that matches a given discrete signature, rather than an analytic sequential point construction procedure.We use the direct search OrthoMads algorithm [7] to solve this problem. Mads class algorithms do not require objective derivative information [7, 9] and converge to second-order stationary points under reasonable conditions on nonsmooth functions [10]. We implement our constraint using the extreme barrier method [11] in which the objective value is set to infinity whenever constraints are not satisfied. We utilize the standard implementation with partial polling and minimal spanning sets of 4m+1 directions.We performed a series of numerical tests using the synthetic shamrock curve shown in black in the upper portion of Figure14. This curve is given as a polygon in P256 with discretization coefficients a∈R4×20 (m=20). A sequence of reconstructions was performed with all integer values 8≤m≤20. The m=8 reconstruction begins with initial coefficients, ai,j, which determine a regular 256-gon with approximately the same interior area as the shamrock (as determined by the signature ga(s¯,r). In particular, the value(s) ai,j supplied initially are those which define the best fit circle (m=1), which can be computed directly. That is, only a1,0 and a4,0 are nonzero. Subsequent reconstructions begin with initial coefficients optimal to the previous relatively coarse reconstruction. Curve reconstructions for m=12 (blue) and m=18 (red) are compared to the shamrock in the upper portion of Figure 14. Reconstructions for m≥20 are visually indistinguishable from the actual curve and are not shown. Corresponding area density signatures are shown in the lower portion of Figure 14. A representative disk of radius r is shown in green along with corresponding location in the signature; note that the shamrock is not tangent-cone graph-like with this radius.Figure 14
Shamrock reconstruction: comparing the original curve with those found form=12 and m=18. Curves for m≥20 are visually indistinguishable from the original curve. The shape signatures are given at the bottom.When comparing and interpreting the shamrock curves, it is important to note that the scale of the curves is determined entirely by the fit parametersai,j. On the other hand, as the density signature is independent of curve rotation, the rotation is eyeball adjusted for easy visual comparison. Also note that the two-arc property does not hold for this example so our reconstructability results do not apply. The accuracies of both the curve reconstruction and area density signature fit suggest that somewhat more general reconstructability results hold. In particular, we speculate that general simple polygons may be reconstructible from g(s,r) for fixed r and no derivative information.


## 9. Conclusions

We have studied the integral area invariant with particular emphasis on the tangent-cone graph-like condition. In particular, we have shown that all TCGL polygons and aC1-dense set of C2 TGL curves are reconstructible using only the integral area invariant for a fixed radius along the boundary and its derivatives.We also showed that TCGL boundaries can be approximated by TCGL polygons, determined what the derivatives represented, and commented on other sets of data sufficient for reconstruction (namely, both T-like and all radii in a neighborhood of 0).These reconstructions are all modulo translations, rotations, and reparametrizations. The arc length parameterization plays a special role here since any two such parameterizations of a boundary will differ only by a shift and can easily be placed into correspondence. The situation becomes more complicated in higher dimensions as boundaries are no longer canonically parameterized by a single variable which is a fundamental assumption of our results and methods. It is not immediately obvious how to resolve the issues created by higher dimensions except that it may be possible to modify some of the machinery to work with star convex regions which restore some semblance of canonical representation.Another space which is open for further development is that of reconstruction algorithms. This is doubly true since our theoretical reconstructions are unstable and the numerical examples in the present work do not have guaranteed reconstruction. However, even without these guarantees, the numerical examples hint at more expansive reconstructability results.



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