[

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Abstract

Equal-volume polygons are obtained from adequate discretizations of curves in -space, contained or not in surfaces. In this paper we explore the similarities of these polygons with the affine arc-length parameterized smooth curves to develop a theory of discrete affine invariants. Besides obtaining discrete affine invariants, equal-volume polygons can also be used to estimate projective invariants of a planar curve. This theory has many potential applications, among them evaluation of the quality and computation of affine invariants of silhouette curves.

Darboux vector field, Affine arc-length parameterization, Affine evolute, Projective length, Discrete affine geometry
\theoremstyle

definition \theoremstyleremark \numberwithinequationsection

Affine geometry of polylines ] Affine geometry of equal-volume polygons
in -space

M.Craizer]Marcos Craizer

S.Pesco]Sinesio Pesco

1\subjclass

53A15, 53A20

1 Introduction

We say that a smooth curve in -space is parameterized by affine arc-length if , where denotes the determinant of vectors. For polygons, the corresponding condition is that the area of the triangle determined by three consecutive vertices is constant. Planar polygons satisfying this last condition are called equal-area and the affine geometry of these polygons has been recently studied ([2],[11]). In this paper we generalize this study to polygons in -space by considering the concept of equal-volume polygons. Since we obtain discrete counterparts of known objects of the smooth theory, our results clearly belong to the field of Discrete Differential Geometry.

For a smooth curve contained in a surface , we say that the parameterization is adapted to M if

(1)

where denotes the determinant of vectors and is the parallel Darboux vector field of ([3]). In centro-affine geometry, we consider curves in -space together with a distinguished origin , and we say that is parameterized by centro-affine arc-length with respect to if

(2)

([6]). A smooth curve in -space is parameterized by affine arc-length if

(3)

([5],[9]). We observe that, in all these contexts, the basic condition is the constancy of some volume. In this paper, we describe polygons in -space whose corresponding volumes are constant, and we call them equal-volume polygons. We obtain affine invariant measures only for these equal-volume polygons, but we also describe a simple algorithm that, by re-sampling an arbitrary polygon, obtain an equal-volume one.

We say that a smooth curve contained in a surface is non-degenerate if its osculating plane does not coincide with the tangent plane of at any point. For such curves, there exists a vector field tangent to and transversal to such that its derivative is tangent to . The direction defined by is unique and is called the Darboux direction of . Moreover, there exists a vector field in the Darboux direction such that is tangent to the curve , i.e.,

(4)

for some scalar function . This vector field is unique up to a multiplicative constant and is called the parallel Darboux vector field. It turns out that is a silhouette curve with respect to some point if and only if is constant ([3]).

As a discrete model for curves contained in surfaces, consider a polyhedron whose faces are planar quadrilaterals and let be vertices of a polygon whose sides are connecting opposite edges of a face of . The edges of containing vertices correspond to Darboux directions and we can choose a vector field in this direction such that the difference is parallel to the corresponding side of the polygon . This vector field is unique up to a multiplicative constant, and is called the parallel Darboux vector field. We can write a discrete counterpart of equation \eqrefeq:DefineSigma, namely

(5)

for some scalar function , where we are replacing derivatives by differences. We prove that is a silhouette polygon for the polyhedron if and only if is constant. This result may be used as a measure of quality of a silhouette polygon.

A non-degenerate curve admits a parameterization satisfying equation \eqrefeq:AffineArcLengthDarboux, unique up to a translation. The plane generated by is called the affine normal plane, while the envelope of these affine normal planes is a developable surface called the affine focal set of the pair ([3],[4]). For silhouette curves relative to , equation \eqrefeq:AffineArcLengthDarboux reduces to equation \eqrefeq:CentroAffineArcLength.

We say that the polygon contained in the polyhedron is equal-volume if

(6)

for all . Note that equation \eqrefeq:DiscEqualVolumeDarboux is a discrete counterpart of equation \eqrefeq:AffineArcLengthDarboux. For such polygons, define the affine normal plane as the plane generated by and the affine focal set as a discrete envelope of these affine normal planes. For silhouette polygons relative to , equation \eqrefeq:DiscEqualVolumeDarboux reduces to

(7)

which is a discrete counterpart of equation \eqrefeq:CentroAffineArcLength.

The smooth curves whose affine focal set reduces to a single line were characterized in [4]. Consider a smooth planar curve parameterized by affine arc-length and denote by the affine distance or support function of with respect to some point ([1]). Then the affine focal set of the silhouette curve reduces to a single line and conversely, if is a single line, then is a silhouette curve obtained by this construction for some planar curve and some . We prove in this paper a discrete counterpart of this characterization for equal-volume polygons contained in a polyhedron.

Consider a smooth curve in -space parameterized by affine arc-length, i.e., satisfying equation \eqrefeq:AffineArcLength. The planes through parallel to are called affine rectifying planes and the envelope of the affine rectifying planes is called the intrinsic affine binormal developable ([9]). The characterization of curves such that is cylindrical is easily obtained from the characterization of curves whose affine focal set is a single line ([4],[9]). A polygon in -space is said to be equal-volume if

(8)

for all , which is equivalent to say that the difference polygon is equal-volume with respect to the origin. Although it is not clear how to obtain a discrete version of the intrinsic affine binormal developable, we can obtain interesting consequences of the discrete characterization of polygons whose affine focal set is a single line.

We can also apply the equal-volume model in a projective setting. Given a smooth planar curve , there exists a projectively equivalent curve in -space satisfying equation \eqrefeq:CentroAffineArcLength with equal the origin. From this curve, we can define the projective length (see [7]). For a planar polygon , we can also obtain a projectively equivalent equal-volume polygon in -space and, from this polygon, we obtain two definitions for the projective length, and , that unfortunately do not coincide. Nevertheless, we prove that if the polygon is obtained from a dense enough sampling of a smooth curve, both the discrete projective length and are close to the projective length of the smooth curve.

The paper is organized as follows: In section 2 we review the smooth results for affine geometry of curves contained in surfaces, affine geometry of curves in -space and projective geometry of planar curves. In section 3 we calculate affine invariants of equal-volume polygons contained in polyhedra. In section 4, we apply the results of section 3 to compute affine invariants for equal-volume polygons in -space. In section 5 we discuss the projective length of a planar polygon.

2 Affine geometry of smooth curves in -space

2.1 Curves contained in surfaces

Let be a curve contained in a surface and a vector field tangent to and transversal to . We shall assume that is non-degenerate, i.e., the osculating plane of does not coincide with the tangent plane of at any point. Under this hypothesis, there exists a vector field , unique up to scalar (non-constant) multiple, such that is tangent to , for any . The vector field determines a unique direction tangent to , which is called the Darboux direction along . In the Darboux direction, there exists a vector field , unique up to a constant multiple, such that is tangent to , for any . We call this vector field the parallel Darboux vector field. The parallel Darboux vector field satisfies equation \eqrefeq:DefineSigma, for some scalar function .

The envelope of tangent planes is the developable surface

This surface is called the Osculating Tangent Developable Surface of along and will be denoted ([3],[10]). The surface is a cone if and only if is constant. In this case, the vertex of the cone is given by and the curve is a silhouette curve from the point of view of .

Under the non-degeneracy hypothesis, there exists a parameterization of , unique up to a translation, such that equation \eqrefeq:AffineArcLengthDarboux holds. The plane generated by is called the affine normal plane of . Condition \eqrefeq:AffineArcLengthDarboux is equivalent to tangent to . Thus we can write

(9)

for some scalar functions and .

There exists a basis of the affine normal plane with parallel, i.e., tangent to . In fact, define the vector field by , where . Taking , we obtain the equation

(10)

which in particular says that is parallel. The affine focal set , or affine evolute, is the envelope of affine normal planes. It is the developable surface generated by the lines passing through and . The affine focal set reduces to a single line if and only if and are constant ([4]).

If is contained in a plane , then for any and conversely, if for any , then is planar. Denote by a euclidean unitary normal to and let be the vector field in the Darboux direction such that , where denotes the usual inner product. Then is a parallel Darboux vector field. In this case, the adapted parameter corresponds to the affine arc-length parameter and is the affine curvature of . For planar curves, and so is parallel. Then the set coincides with the affine evolute of the planar curve ([8]).

For silhouette curves, and so equation \eqrefeq:AffineArcLengthDarboux becomes \eqrefeq:CentroAffineArcLength, i.e., is parameterized by centro-affine arc-length. Assuming equals the origin, equation \eqrefeq:Frenet1 becomes

(11)

Moreover , which implies .

Curves whose affine focal set reduces to a single line

The affine focal set reduces to a single line if and only if and are constant. Since is constant, is necessarily a silhouette curve from the point of view of , that we shall assume to be the origin. The condition constant can be written as . In this case equation \eqrefeq:FrenetCA becomes , which is equivalent to

for some constant vector . Assuming that and writing , this equation becomes

(12)

Consider a convex planar curve . Assume that is parameterized by affine arc-length, i.e., , and let denotes the affine curvature of , i.e., , . Let and denote by the affine distance, or support function, of with respect to a point ([1]).

The following proposition was proved in [4]:

Proposition 2.1

The affine focal set of the curve is a single line, and conversely, any curve whose affine focal set is a single line can be obtained as above, for some planar curve and some point .

2.2 Curves in -space

Consider now a curve in the -space, without being contained in a given surface . We say that a parameterization of is by affine arc-length if formula \eqrefeq:AffineArcLength holds. This condition implies that belongs to the plane generates by and and thus we obtain equation

(13)

for some scalar functions and . Writing , we observe that equation \eqrefeq:FrenetSp reduces to \eqrefeq:FrenetCA.

The plane passing through and generated by is called affine rectifying plane and the envelope of the affine rectifying planes is called the intrinsic affine binormal developable of . It is proved in [9] that is cylindrical if and only if .

Curves with constant

The condition constant is equivalent to . Consider a convex planar curve parameterized by affine arc-length and let

Then represents the area of the planar region bounded by , , and the segments and . The following proposition is a direct consequence of proposition 2.1:

Proposition 2.2

For the curve , is constant, and conversely, any curve in -space with constant is obtained by this construction, for some convex planar curve and some point .

2.3 Projective invariants

Consider a parameterized planar curve , , without inflection points. Any curve of the form is projectively equivalent to and is called a representative of . It turns out that there exists a representative of satisfying formula \eqrefeq:CentroAffineArcLength with equal to the origin. Then belongs to the space generated by and equation \eqrefeq:FrenetCA holds, for some scalar functions and .

The quantity is projectively invariant and if and only if is contained in a quadratic cone. In fact,

(14)

is the projective length of (see [7]).

3 Affine geometry of equal-volume polygons contained in polyhedra

In this section, we obtain discrete counterparts of the results of section 2.1. The derivatives are replaced by differences, and so for a function , we denote

and so on.

3.1 Basic model

Consider a polyhedron whose faces are planar quadrilaterals and let be a polygonal line such that each of its sides are connecting opposite edges of a face of . We shall denote by , , the vertices of such polygon and by a vector in the direction of the edges of containing . Edges of that don’t intersect are not important in our model (see Figure 1).

Figure 1: Polygonal line contained in polyhedron .

We shall denote by the face of that contains the side . By the planar quadrilaterals hypothesis, the vectors , and belong to , which is a discrete counterpart of the Darboux condition tangent to . It is clear that there exists , unique up to a multiplicative constant, such that is parallel to . This vector field is the parallel Darboux vector field of and equation \eqrefeq:DiscSigma holds, for some scalar function (see Figure 2).

Figure 2: Parallel Darboux vectors field . The segments connecting the endpoints of the vectors are parallel to the sides of .

3.2 Osculating developable polyhedron

The line , , is the support line of the edge of the polyhedron that contains . Thus and are co-planar and denote by the intersection point of these lines. We have that

(15)

The osculating developable polyhedron is the polyhedron whose face is the region of bounded by and and containing the side of (see Figure 3).

Figure 3: Osculating developable polyhedron

We have the following proposition:

Proposition 3.1

The following statements are equivalent:

  1. does not depend on .

  2. The point does not depend on .

  3. is a cone.

{proof}

Observe that

(16)

Thus (1) is equivalent to (2). The equivalence between (2) and (3) is obvious.

The polygons for which reduces to a point is the object of study of the centro-affine geometry. In this case, the polygon can be thought as a silhouette polygon of from the point of view of .

3.3 Equal-volume polygons

We say that the polygon contained in the polyhedron is equal-volume if equation \eqrefeq:DiscEqualVolumeDarboux holds.

Lemma 3.2

The polygon is equal-volume if and only if

{proof}

Equation \eqrefeq:DiscEqualVolumeDarboux is equivalent to

for each , which is equivalent to

By the parallel Darboux condition, the first parcel is zero and thus the above condition is equivalent to

which is clearly equivalent to belongs to .

We shall assume along the paper that the polygon is equal-volume. By the above lemma we can write

(17)

for some scalar functions , and satisfying the compatibility equation

(18)

Equations \eqrefeq:DiscFrenet1 are discrete counterparts of equation \eqrefeq:Frenet1.


Remark 3.3

Starting from a general polygon , we may obtain an equal-volume polygon by the following inductive algorithm (see Figure 4):

  1. Let , for .

  2. Given the pair at , and , consider a plane parallel to through and let be the intersection of this plane with the polygonal line .

  3. The direction of the vector is obtained by linear interpolation of and , where is the index of the side of containing . Thus .

Figure 4: Algorithm to construct an equal-volume polygon

3.4 Discrete affine focal set

Take any satisfying and define

(19)

Define also

(20)
Lemma 3.4

The following discrete counterpart of equation \eqrefeq:EtaParallel holds:

(21)

In particular, is parallel.

{proof}

We have that

thus proving the lemma.

Define

(22)

and denote by the line connecting and , where is defined by equation \eqrefeq:DefineO.

The discrete affine focal set is the polyhedron with edges , , and faces contained in , , bounded by and containing the segments and (see Figure 5).

Figure 5: Discrete affine focal set
Proposition 3.5

The following statements are equivalent:

  1. and are constant.

  2. The points and are fixed.

  3. The discrete affine focal set reduces to a single line.

{proof}

By proposition 3.1, constant is equivalent to fixed. From equation \eqrefeq:DefineQ we obtain

(23)

which implies that is constant if and only if is fixed. Thus (1) and (2) are equivalent. It is obvious that (2) implies (3) and so it remains to prove that (3) implies (2). If and were not both fixed, then equations \eqrefeq:DerivO and \eqrefeq:DerivQ say that or are not changing in the direction of . Thus would not be a single line.

3.5 Planar polygons

Lemma 3.6

A polygon is contained in a plane if and only if .

{proof}

Observe that if and only if the points , , and are co-planar.

Denote by a euclidean unitary normal to and let be the vector field in the Darboux direction such that , where denotes the usual inner product. Then is a parallel Darboux vector field. In this case equation \eqrefeq:DiscEqualVolumeDarboux can be written as

where denotes determinant in the plane . Thus is an equal-area polygon and is its discrete affine curvature ([2],[11]). The set is exactly the discrete affine evolute of the planar equal-area polygon ([2]) (see Figure 6).

Figure 6: For a planar curve , the set coincides with the discrete affine evolute of .

3.6 Silhouette polygons

Assume that is a silhouette polygon from the point of view of , that we assume to be the origin. In this case, equation \eqrefeq:DiscEqualVolumeDarboux becomes equation \eqrefeq:EqualVolumeCA. By Lemma 3.2, this condition is equivalent to belongs to the plane generated by and .

Since , we have that , . The Frenet equations \eqrefeq:DiscFrenet1 reduce to

(24)

while equation \eqrefeq:DefineMu becomes

(25)

We have also that

(26)

3.7 Polygons whose discrete affine focal set reduces to a line

By proposition 3.5, reduces to a single line if and only if and are constant. Since is constant, is a silhouette polygon. By formula \eqrefeq:DiscMuLinha, the condition constant is equivalent to .

Assume constant. Then equation \eqrefeq:DiscEtaParallel implies that

for some constant vector . Assume and write . Then, using equation \eqrefeq:DefineEta we obtain

and so

(27)

Observe that

and so , for some constant . By rescaling we may assume that .

Denote by a polygon such that . Then

and so is an equal-area polygon with discrete affine curvature . The affine distance or support function of with respect to a point is given by

(28)

(see Figure 7, left).

Figure 7: A planar equal-area polygon and its support function (left). The corresponding spatial curve and its affine focal set (right).
Proposition 3.7

The polygonal line (see Figure 7, right) satisfies equation \eqrefeq:Difference, and conversely, any solution of the difference equation \eqrefeq:Difference is obtained by this construction, for some planar polygon and some point .

{proof}

Observe first that

Thus

thus proving that satisfies equation \eqrefeq:Difference. Since has two degrees of freedom, this is the general solution of the second order difference equation \eqrefeq:Difference.

4 Polygons in -space

In this section, we obtain discrete counterparts of the results of section 2.2. Consider a polygon in -space, without being contained in any polyhedron . The polygon is equal-volume, i.e., satisfies equation \eqrefeq:EqualVolume, if and only if the difference polygon is equal-volume with respect to the origin.

4.1 Frenet equations

For equal-volume polygons , Frenet equations \eqrefeq:DiscFrenetCA1 are written as

(29)

Defining by equation \eqrefeq:DiscMu, equation \eqrefeq:DiscMuLinha still holds. It is not clear how to define a discrete version of the intrinsic affine binormal developable.

4.2 Polygons with constant

Consider an equal area planar polygon and let be given by

where is given by equation \eqrefeq:Discz, for some point . Then represents the area of the planar region bounded by , , and the segments and (see Figure 8). In this context, Proposition 3.7 can be written as follows:

Proposition 4.1

The polygon has constant , and conversely, any equal-volume polygon with constant is obtained by this construction, for some planar polygonal line and some point .

Figure 8: A planar equal-area polygon and the area
represented by .

5 Projective polygons

In this section, we obtain discrete counterparts of the results of section 2.3.

Consider a planar polygon , . Assume that

(30)

5.1 Equal-volume representative

Any polygon in of the form , , is a projective representative of .

Lemma 5.1

There exists a projective representative of such that equation \eqrefeq:EqualVolumeCA holds with equal to the origin.

{proof}

Observe first that

So we need to choose , such that

(31)

for some constant . Since by the hypothesis \eqrefeq:DiscConvex , given and we can find unique , such that \eqrefeq:ab holds.

Assume that is a representative of such that equation \eqrefeq:EqualVolumeCA holds with equal to the origin (Figure 9). Then, by lemma 3.2, belongs to the plane generated by . So we can use equations \eqrefeq:DiscFrenetCA1 to define , and .

Figure 9: Two views of a planar projective polygon and its equal-volume representative .

5.2 Projective length

We would like to define the projective length of as

(32)

or

(33)

but unfortunately these two definitions do not coincide. Nevertheless, if the polygonal line is obtained from a dense enough sampling of a smooth curve, both of these formulas are close to projective length of the smooth curve given by equation \eqrefeq:ProjectiveLength. Denote by any quantity such that , for any .

Lemma 5.2

Assume that the polygonal line , , is obtained from , , by uniform sampling. Then, for , , we have

A similar result holds for .

{proof}

It is standard in numerical analysis that and . Thus equation \eqrefeq:FrenetCA can be written as

We conclude that and . This last equation implies that . Thus we conclude that

which proves the lemma.

From this lemma we can obtain the following convergence result:

Corollary 5.3

The discrete projective lengths given by equations \eqrefeq:DiscProjectiveLength1 and \eqrefeq:DiscProjectiveLength2 converge to the smooth projective length given by \eqrefeq:ProjectiveLength when .

Example 1

Consider

Then is projectively equivalent to , which satisfies equation \eqrefeq:CentroAffineArcLength with equal the origin. Straightforward calculations show that , and

We have done some experiments considering uniform samplings of this curve with points. Table 1 presents the results for . Observe that both and get closer to as decreases.

N h
10 0.62831 4.26627 3.55522
100 0.06283 6.87572 6.80410
1000 0.00628 7.13407 7.12691
Table 1: Experimental results of example 1.

Footnotes

  1. thanks: The first author thanks CNPq for financial support during the preparation of this paper.

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