[
Abstract
Equalvolume 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 arclength parameterized smooth curves to develop a theory of discrete affine invariants. Besides obtaining discrete affine invariants, equalvolume 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.
definition \theoremstyleremark \numberwithinequationsection
Affine geometry of polylines ]
Affine geometry of equalvolume polygons
in space
M.Craizer]Marcos Craizer
S.Pesco]Sinesio Pesco
53A15, 53A20
1 Introduction
We say that a smooth curve in space is parameterized by affine arclength 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 equalarea 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 equalvolume 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 centroaffine geometry, we consider curves in space together with a distinguished origin , and we say that is parameterized by centroaffine arclength with respect to if
(2) 
([6]). A smooth curve in space is parameterized by affine arclength 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 equalvolume polygons. We obtain affine invariant measures only for these equalvolume polygons, but we also describe a simple algorithm that, by resampling an arbitrary polygon, obtain an equalvolume one.
We say that a smooth curve contained in a surface is nondegenerate 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 nondegenerate 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 equalvolume 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 arclength 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 equalvolume polygons contained in a polyhedron.
Consider a smooth curve in space parameterized by affine arclength, 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 equalvolume if
(8) 
for all , which is equivalent to say that the difference polygon is equalvolume 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 equalvolume 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 equalvolume 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 equalvolume polygons contained in polyhedra. In section 4, we apply the results of section 3 to compute affine invariants for equalvolume 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 nondegenerate, 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 (nonconstant) 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 nondegeneracy 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 arclength 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 centroaffine arclength. 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 arclength, 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 arclength 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 arclength 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 equalvolume 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).
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).
3.2 Osculating developable polyhedron
The line , , is the support line of the edge of the polyhedron that contains . Thus and are coplanar 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).
We have the following proposition:
Proposition 3.1
The following statements are equivalent:

does not depend on .

The point does not depend on .

is a cone.
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 centroaffine geometry. In this case, the polygon can be thought as a silhouette polygon of from the point of view of .
3.3 Equalvolume polygons
We say that the polygon contained in the polyhedron is equalvolume if equation \eqrefeq:DiscEqualVolumeDarboux holds.
Lemma 3.2
The polygon is equalvolume if and only if
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 equalvolume. 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 equalvolume polygon by the following inductive algorithm (see Figure 4):

Let , for .

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

The direction of the vector is obtained by linear interpolation of and , where is the index of the side of containing . Thus .
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.
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).
Proposition 3.5
The following statements are equivalent:

and are constant.

The points and are fixed.

The discrete affine focal set reduces to a single line.
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 .
Observe that if and only if the points , , and are coplanar.
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 equalarea polygon and is its discrete affine curvature ([2],[11]). The set is exactly the discrete affine evolute of the planar equalarea polygon ([2]) (see Figure 6).
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 equalarea 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).
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 .
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 equalvolume, i.e., satisfies equation \eqrefeq:EqualVolume, if and only if the difference polygon is equalvolume with respect to the origin.
4.1 Frenet equations
For equalvolume 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 equalvolume polygon with constant is obtained by this construction, for some planar polygonal line and some point .
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 Equalvolume 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.
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 .
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 .
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 
Footnotes
 thanks: The first author thanks CNPq for financial support during the preparation of this paper.
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