Equivalence detection of rational curves and surfaces / submitted by Michael Hauer

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Submitted by Michael Hauer Submitted at Institut f¨ur Angewandte Geometrie Supervisor and First Examiner

Univ.-Prof. Dr. Bert J¨uttler Second Examiner Univ.-Prof. Dr. Laureano Gonzalez-Vega January 2018 JOHANNES KEPLER UNIVERSITY LINZ Altenbergerstraße 69 4040 Linz, ¨Osterreich www.jku.at DVR 0093696

Equivalence detection

of rational curves and

surfaces

Doctoral Thesis

to obtain the academic degree of

Doktor der technischen Wissenschaften

in the Doctoral Program

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Eidesstattliche Erkl¨

arung

Ich erkläre an Eides statt, dass ich die vorliegende Dissertation selbstständig und ohne fremde Hilfe verfasst, andere als die angegebenen Quellen und Hilfsmittel nicht benutzt bzw. die wörtlich oder sinngemäß entnommenen Stellen als solche kenntlich gemacht habe.

Die vorliegende Dissertation ist mit dem elektronisch ¨ubermittelten Textdokument identisch

Linz, am 17. Januar 2018

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Abstract

Rational parameterizations of curves and surfaces are frequently used in Computer Aided Geometric Design (CAGD) and Algebraic Geometry, where the most common representations are based on the power basis and the Bernstein-B´ezier basis. We can concentrate on one of them, since these two representations are closely related by a projective transformation of the parameter domain. By considering rational parameter-izations in projective space one can avoid the use of rational functions and work with polynomials instead.

The problem of detecting symmetries and equivalences of curves and surfaces at-tracted substantial attention since it is an essential problem in Pattern Recognition, Computer Graphics and Computer Vision. Knowledge about symmetries helps analyz-ing pictures and is used for compression and shape completion. Equivalence detection is used to identify a given object with objects in a database. We note here, that the detection of symmetries is a special case of equivalence detection.

It is known, that proper parameterizations of rational curves in reduced form are unique up to bilinear reparameterizations, i.e., projective transformations of the param-eter domain. This observation has been used in a series of papers by Alc´azar et al. to formulate algorithms for detecting Euclidean equivalences as well as similarities (which include also some scaling) for rational planar and space curves. In this work we generalize this approach in several directions.

In a first step we consider projective and affine equivalences of curves in arbitrary dimensions. Equivalences with respect to the group of projective transformations are the most general of those ones mentioned above and we can treat Euclidean equivalences, similarities and affine equivalences as special cases. Moreover, the freedom of considering arbitrary dimensions is an advantage of our approach.

As a second step we state, that for proper, base point free rational surfaces a sim-i

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ilar property can be shown, i.e., these representations are unique up to a projective transformation of the parameter domain, which we identify with the projective plane. Furthermore, again we use this insight to detect projective equivalences of surfaces in arbitrary dimension.

We use these observations about rational curves and surfaces to characterize equiv-alences and symmetries by a polynomial system of equations in the variables describing the linear rational reparameterization. We solve this system using the Gr¨obner basis implementation of Maple, which is one of the standard computer algebra systems. We provide a substantial number of examples to illustrate our method. Among other re-sults, this allows us to verify known results about the classifications of quadratically parameterized surfaces in a simple way.

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Zusammenfassung

Rationale Parametrisierungen, vor allem in einer Darstellung bezüglich der Monom-oder der Bernstein-Bézier-Basis, werden in vielen mathematischen Gebieten, wie zum Beispiel in Computer Aided Geometric Design (CAGD) oder in der algebraischen Geome-trie, zur Beschreibung von Kurven und Flächen verwendet. Da diese zwei Darstellungs-formen mittels einer projektiven Transformation des Parameterbereichs, welche einer linearen rational Umparametrisierung entspricht, und einer einfachen Skalierung inein-ander überführt werden können, werden wir uns auf die Monomdarstellung konzentrieren. Indem wir die rationalen Parametrisierungen in homogenen Koordinaten betrachten, ist es uns möglich mit polynomialen anstelle von rationalen Funktionen zu arbeiten.

Die Erkennung von Symmetrien und Äquivalenzen von Kurven weckte nicht zuletzt durch vielfältige Anwendungsbereiche in Pattern Recognition, Computer Graphics und Computer Vision großes Interesse. Das Wissen über Symmetrien von Objekten hilft dabei Bilder zu analysieren, den Speicherbedarf zu verringern sowie fehlende bzw. ver-lorene Informationen zu vervollständigen. Die Erkennung von Äquivalenzen bietet die Möglichkeit beliebige Objekte mit bekannten Objekten in einer Datenbank zu verglei-chen und zu identifizieren. Die Symmetrieerkennung kann als Sonderfall der ¨ Aquivalenz-erkennung angesehen werden.

Bekannterweise sind propere Parametrisierungen rationaler Kurven eindeutig bis auf lineare rationale Umparametrisierungen. Diese Beobachtung wurde von Alcázar et al. in einer Reihe von Veröffentlichungen dazu benutzt, euklidische Äquivalenzen und ¨ Ahnlich-keiten (welche auch eine Skalierung beinhalten können) von rationalen ebenen Kurven und rationalen Raumkurven zu erkennen. Diese Dissertation beschreibt eine Verallge-meinerung davon in mehreren Bereichen.

Zuerst betrachten wir projektive und affine ¨Aquivalenzen von Kurven beliebiger Di-mension. Von allen bisher genannten Transformationen sind projektive die allgemeinsten

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und affine Transformationen, Ähnlichkeiten sowie euklidische Transformationen können als Spezialfälle aufgefasst werden. Außerdem bietet unsere Methode die Freiheit, Kurven in beliebiger Raumdimension zu betrachten.

Als zweiten Schritt beschreiben wir eine Verallgemeinerung unserer Methode auf Fl¨ a-chen. Dazu zeigen wir, dass propere, basispunktfreie rationale Parametrisierungen wie-der eindeutig bis auf eine projektive Transformation des Parameterbereichs sind. Somit können wir projektive Äquivalenzen von Flächen in beliebiger Dimension berechnen.

Sowohl bei Kurven als auch bei Flächen erzeugen wir ein polynomiales Gleichungs-system in den vier bzw. neun Variablen, welche die lineare rationale Umparametrisierung beschreiben. Wir lösen dieses Gleichungssystem mittels der Gr¨ obnerbasisimplementier-ung von Maple, einem bekannten Computer Algebra System. Mittels zahlreicher Bei-spiele, sowohl für Kurven als auch für Flächen veranschaulichen wir unsere Methode die es uns unter anderem auch erlaubt, bekannte Ergebnisse über die Klassifizierung von quadratisch parametrisierbaren Flächen auf eine einfache Weise nachzuvollziehen.

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Acknowledgements

My first and biggest gratitude goes to my supervisor, Professor Dr. Bert J¨uttler, who offered me the possibility to write this thesis at the Institute of Applied Geometry at the Johannes Kepler University in Linz. With his big knowledge, creativity and patience in many scientific discussions he provided the perfect conditions for guiding me through the process of producing this work and offered me constant support.

Second I would like to thank Professor Dr. Josef Schicho whose expertise on surfaces is irreplaceable. In particular, the algebraic background and the proof of Theorem 24 is his merit. Moreover, his friendly way made it really easy to work with him. It is a special honour to have Professor Dr. Laureano Gonzalez-Vega as a referee of this work. I gratefully acknowledge that this research was partially supported by the Austrian Science Fund (FWF): W1214-N15, project DK3.

If you define the quality of your working environment by the social climate among you and your co-workers, the Institute of Applied Geometry would be top rated. I am very grateful for all colleagues who used to be rather friends than co-workers. In particular, I would like to mention Monika Bayer for her help with administrative tasks, Thomas Takacs and Mario Kapl for their guidance in my teaching, Andrea Bressan and Dominik Mokri˘s for climbing with me and Michael Haberleitner for many out-of-office activities. Also during my studies (Roland Wagner, Stephan Zweckinger, Claudia Mühlberger, Eva-Maria Infanger, Gerald Infanger, Paul Gschöpf), school time (Andreas Birlmüller, Stefan Wampl, Martin Oberroither, Johannes Reisenberger, Markus Kofler) and leasure time (Martin Reiter, Lisa Murauer, Christine Murauer, Alexander Pühringer, Daria Liebensteiner) I got to know great people. Thank you for sharing so many nice memories! I could not have achieved any of this without a caring family who supported me through all my life. My parents, Franz and Elfriede, offered me the best possible educa-tion and my sister, Johanna, was a role model from my childhood on. Thank you!

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Chapter 1 Introduction

In this chapter we relate the results of this thesis to some related work and discuss the context of possible applications. Finally we present the outline and the structure of this work. The main results have been published by Hauer and J¨uttler (2017) and Hauer et al. (2017).

1.1 Rational parameterizations

In many of the application-oriented tasks in Geometric Modeling and Simulation the rational parameterization of curves and surfaces is a standard form to represent geomet-ric objects. It has been observed that many of the results from Algebraic Geometry, Symbolic Computation and Computer Aided Geometric Design are directly useful to adress the problems of analysing and optimizing shape and design of the objects given by rational parameterizations.

The investigation of singularities, which is one of the classical topics in algebraic geometry, is clearly important for geometric design. For instance, several authors de-signed methods for detecting singularities of rational planar curves (Chen et al., 2008; P´erez-D´ıaz, 2007) as well as of rational space curves (Shi et al., 2013; Shi and Chen, 2010; Rubio et al., 2009) from their parametric representations, using resultants and µ-bases. A similar task for surfaces is even more challenging but can also be adressed using µ-bases. We refer to the survey paper of Jia et al. (2018) for an overview.

Another interesting question concerning planar rational curves and surfaces in three space is the computation and the study of their offsets. Some properties of the offsets

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of curves, such as the degree and the topological behaviour, have been investigated in Segundo and Sendra (2009) and Alc´azar (2008). The computation of offsets for special classes of surfaces was discussed in several publications, see e.g. Bastl et al. (2008); Aigner et al. (2009); Segundo and Sendra (2008) to note only a few of them.

Another line of research is devoted to properties of the parameterization. Among numerous papers on this topic, P´erez-D´ıaz (2006) addressed the issue of reparameteri-zation to generate proper parameterireparameteri-zations, and Tabera (2011) explored the generation of optimal parameterizations in the sense of minimal coefficients.

The transformation between an implicit and a parametric representation is one of the most basic questions in Algebraic Geometry. While any rationally parameterized curve or surface is an algebraic curve or surface and hence can be represented by an implicit equation, the reverse is not always possible. As exact methods for implicitization, which rely on µ-bases (Chen and Wang, 2002; Jia et al., 2018), Gröbner-bases (Kalkbrener, 1990; Cox et al., 1992) or resultants (Goldman and Sederberg, 1985) are usually com-putationally slow, also approximative methods are used (Dokken and Thomassen, 2003; Shalaby and Jüttler, 2008) and analysed (Schicho and Szilágyi, 2005). The parame-terizations problem was considered for example in Schicho (1998). Note, that due to the numerous publications on this topic, we only mentioned one or two representative references for each of these approaches.

1.2 Symmetries and equivalences

The detection of symmetries and equivalences of geometric objects is of interest in the fields of Computer Graphics, Computer Vision and Pattern Recognition and here several types of input data have been considered. The input data can be discrete, coming from point sets, polygons and meshes or describe manifolds such as curves and surfaces by implicit or parametric representations. Symmetry detection is a special case of equivalence detection, where twice the same input is used.

For discrete input data (point sets, polygons and meshes) of moderate size, the ques-tion of equivalence detecques-tion was investigated already thirty years ago in the field of Computational Geometry. The detection is well understood and several efficient algo-rithms are available.

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1.2. SYMMETRIES AND EQUIVALENCES 3 involutional symmetries in O(n log(n)) for unstructured point sets. Alt et al. (1988) con-sider exact and approximate Euclidean transformations between two point sets. Huang and Cohen (1996) detect affine transformations and apply it to the problem of classi-fying silhouettes of aircrafts using B-spline moments approximated from a sample of points. Braß and Knauer (2004) propose another approach for Euclidean symmetries of discrete 3-dimensional objects and suggest to apply their method to the control polygon of B´ezier curves and surfaces. This can be seen as one of the first ideas that combines the discrete data with an algebraic representation. However, they do not consider any reparameterizations.

In recent years, research in Geometry Processing has focused on efficient algorithms for finding approximate congruences and symmetries of large point sets generated by 3D scans. The interested reader may consult the survey article by Mitra et al. (2013) for further information.

On the other hand, the computation of symmetries and equivalences of algebraic curves, in particular of rational ones, experienced an increase of interest in the last years. These curves (and also surfaces) are important in Geometric Modeling, i.e., they are often used as a standard representation.

Using the implicit representation of planar curves, Lebmeir and Richter-Gebert (2008) formulate a method for detecting congruences and symmetries. For curves of genus at least 2, Hess (2004) describes an algorithm for computing abstract isomorphisms (i.e., not necessarily projective ones) between algebraic curves based on their function fields. This algorithm is implemented in the computer algebra system Magma (Bosma et al., 1997).

Symmetries in the sense of biregular automorphisms of algebraic surfaces are well understood theoretically, see Koitabashi (1988); Zhang (2001). In the context of Al-gebraic Geometry, one considers nonsingular surfaces in projective spaces of arbitrary dimension. For surfaces in projective 3-space, which often possess self-intersections and other singular curves, one may (theoretically) use the existence of a canonical resolution process that can be used to prove that the projective automorphism group of a singular surface in 3-space is a subgroup of the group of abstract automorphisms of its resolution. This approach, however, does not make it easy to compute projective symmetries or to decide projective equivalence. For this reason, we concentrate on parametric surfaces with known parameterization.

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1.3 Equivalences of rationally parameterized objects

Due to the practical importance of these two fields of research, it seems quite natural to combine them, i.e., to ask how to find equivalences of rationally parameterized ob-jects. Most of the work in this direction is based on a well known result from Algebraic Geometry, which states that two proper parameterizations of the same rational curve are related by a linear rational reparameterization, cf. Lemma 4.17 by Sendra et al. (2008). Recently Alcázar and his co-authors (Alcázar, 2014; Alcázar et al., 2014a,b, 2015b; Alcázar et al., 2016) published a series of papers on the detection of Euclidean equiv-alences and similarities of rationally parametrized planar and space curves. The first two publications make use of a coefficient-based method in the complex plane, whereas the latter ones employ invariants from differential geometry, such as the curvature and the torsion (possibly scaled for the detection of similarities). Sánchez-Reyes (2015) used the Bernstein-Bézier basis to detect symmetries of polynomial curve segments using the control points.

A first approach to symmetries of parametric surfaces has been presented by Alc´azar and Hermoso (2016), dealing with involutions of polynomially parametrized surfaces. There, the goal is to find involutions in the Euclidean group preserving the given surface. In Alc´azar et al. (2017) they investigate symmetries of canal surfaces and Dupin cyclides. Here the authors employ a representation by a spine curve and use its symmetries.

Most of these publications, both for discrete data and algebraic representations (im-plicite and parametric), concentrate on Euclidean and similarity transformations. We propose an approach that deals with the general group of projective transformations for both rationally parametrized curves and surfaces and consider Euclidean, similarity and affine transformations as special cases.

1.4 Outline

In Chapter 2 we recall some basic geometric concepts, such as homogeneous coor-dinates, projective transformations and the relation to affine, similarity and Euclidean transformations, we define projective equivalences and clearify some notation.

In Chapter 3 we compare two representations of rationally parameterized curves and surfaces, which are the most commonly used, i.e., the power basis and the

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Bernstein-1.4. OUTLINE 5 B´ezier form. Furthermore we see, that it suffices to consider only one of these represen-tations, as they are closely related.

The main result of this thesis is presented in the following Chapters 4 and 5. In Chapter 4 we derive a method for the detection of projective equivalences of rationally parameterized curves in arbitrary dimension. Our method is based on Lemma 4.17 by Sendra et al. (2008), that states that two proper parameterizations of one curve are related by a projective transformation α of the parameter domain. Using this Lemma and homogeneous coordinates for both the parameter and the image domain we derive a polynomial system of equations in the four unknowns describing α, the (d + 1)2

un-knowns for the projective transformation and an additional unknown u that ensures the regularity of these transformations. In an additional step we reduce this system to the five unknowns in α and u. Each solution of this reduced system characterizes a projective equivalence. We propose some simplifications for the special case of affine equivalences of polynomial curves. Finally we present some examples and compare the methods of the reduced and the original system.

In Chapter 5 we generalize this approach to surfaces. In particular we show that Lemma 4.17 by Sendra et al. (2008) admits a generalization to proper birational and base-point-free parametrizations of surfaces (Theorem 24). We use this result to derive a similar method for the detection of projective equivalences of surfaces. In the end of the chapter we apply this method first to quadratically parameterizable surfaces, for which a full classification is known, and then we provide some examples of higher degree. Finally in Chapter 6 we conclude the thesis and present an outlook to possible future work.

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Chapter 2 Preliminaries

In this propaedeutic chapter we recall basic geometric and algebraic concepts such as homogeneous coordinates, homogenization of polynomials, transformations (projective, affine and Euclidean ones and similarities) of finite dimensional spaces as well as equiva-lences and symmetries of lower dimensional manifolds given by point sets. Furthermore, we define our notation on these concepts.

Unless otherwise stated we consider the field of real numbers, i.e., all coefficients of the surfaces and all variables describing the transformation and reparameterization are given as real numbers. Note that our results, in particular those in Chapter 5, are valid for other fields such as complex numbers as well, but in applications the real case is more interesting and we restrict ourselves to it, although the generalization to complex numbers would rather simplify the derivation and computation.

2.1 Coordinates

We consider algebraic varieties in the Euclidean d-space ¯Ed_{, which has been}

projec-tively closed (indicated by the bar) by adding points at infinity. Its points are represented by homogeneous coordinate vectors

x = (x0 : x1 : · · · : xd)T ∈ Rd+1\{(0, . . . , 0)} ' Pd = Pd(R).

Linearly dependent pairs of homogeneous coordinate vectors represent the same point, and this relation will be denoted by '. More precisely, we write x ' y if and only if

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there exists µ 6= 0 such that x = µy.

Homogeneous coordinate vectors with x0 = 0 represent points at infinity, and the

collection of these points forms the hyperplane at infinity. All other points can be represented by Cartesian coordinates x = (x₁, . . . , x_d)T = (x1/x0, . . . , xd/x0)T. Note

that we use bold characters whenever we have a vector or a vector valued function. In particular we investigate those varieties, that can be given by rational functions in Cartesian coordinates and hence have a polynomial parameterization in homogeneous coordinate vectors. They are frequently used since rational B´ezier curves and surfaces, which are one of the standard representations in Computer Aided Geometric Design (CAGD), belong to this class of varieties. Another common form, for instance in Alge-braic Geometry, is the monomial representation. For m free variables (t1, . . . , tm) in the

rational (respectively polynomial) parameterization describing the variety, these func-tions describe a m-dimensional manifold M in ¯Ed. When referring to the manifold M without giving a parameterization we consider it as a set of points.

In Chapter 4 we consider 1-dimensional manifolds, that means curves C, whereas in Chapter 5 we focus on surfaces S (2-dimensional manifolds). The generalization to arbitrary dimensions is possible but the notation then becomes quite nasty and for the sake of simplicity we restrict ourselves to these two cases, which are the most interesting ones in application. Besides, the practical use for higher dimensions is questionable. Due to the growth of the emerging system, it is not solvable in a reasonable time on computers available today.

In the prevailing part of this thesis we work with homogeneous polynomials in mono-mial representation, that means, the parameter domain is also considered to be a projec-tive space, i.e., the projecprojec-tive line P1_{(R) for curves and the projective plane P}2_{(R) for} surfaces. Polynomials of maximum degree n in standard (i.e., non-homogeneous) form are homogenized by replacing each monomial

tj1 1 · · · t jm m with t (n−Pm i=1ji) 0 t j1 1 · · · t jm m.

Similar to the use of homogeneous coordinates, this provides an easy way of including the extraordinary cases, when the parameter reaches a value at infinity.

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2.2. PROJECTIVE TRANSFORMATIONS 9

2.2 Projective transformations

We recall that using homogeneous coordinates allows to represent any regular pro-jective transformation f by a matrix multiplication

f : ¯Ed → ¯Ed: x 7→ f (x) = M x,

where M = (mij)i,j=0,...,d is a non-singular real matrix. If the elements of the first row

satisfy

m00 6= 0 and m01= · · · = m0d = 0, (2.1)

then f is an affine transformation. The class of affine transformations include trans-lations, rotations, uniform and non-uniform scaling, reflections and shears. Projective transformations further include transformations that do not necessarily preserve parallel lines but collinearity and incidence. If additionally the matrix

A = mij m00

i,j=1,...,d

,

is orthogonal, we have an Euclidean transformation, i.e., the composition of a rotation, a translation and possibly an inversion. If AT_{A = λI with λ ∈ R}+ additionally some scaling is involved and this is called a similarity transformation. All of these cases are special cases of projective transformations and we consider pairs of manifolds, that are related by projective transformations.

Definition 1. Two manifolds M and M0 are said to be projectively (affinely) equivalent if there exists a regular projective (affine) transformation f such that M0 = f (M). Furthermore, M is said to possess a projective (affine) symmetry if there exists a regular projective (affine) transformation f , different from the identity, such that M = f (M).

Note that the prime symbol 0 does not denote a differentiation but is used instead to distinguish between the two manifolds. This should not lead to any misunderstanding, as we do not perform any differentiation in the whole thesis.

Definition 2. Two projectively equivalent manifolds are said to be similar if the trans-formation f describes a similarity and congruent if f is given by an Euclidean transfor-mation. Moreover, we say that M possesses a self-similarity if its projective symmetry

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is characterized by a similarity transformation and in the case of a Euclidean transfor-mation it is simply called a symmetry.

If M0 is projectively equivalent to M, then M is also projectively equivalent to M0, as the projective transformation f is assumed to be regular. Moreover, each manifold is projectively equivalent to itself by the identity map. The transitivity of the relation is implied by the group structure of regular projective mappings. Therefore, the projective equivalence defines an equivalence relation. The same argumentation holds for affine equivalences, similarities and congruences, that is Euclidean equivalences, as well.

Table 2.1: Projective equivalence of manifolds and its special cases

M0_{= f (M)} _{M = f (M),} _{f 6= id}

M regular M and M0 _{are projectively equivalent} _{M has the projective symmetry f}

A regular M and M0 _{are affinely equivalent} _{M has the affine symmetry f}

AT_{A = λI} _{M and M}0 _{are similar} _{M has the self-similarity f}

AT_{A = I} _{M and M}0 _{are congruent} _{M has the symmetry f}

Figure 2.1: Dependencies between the different types of equivalence relations projective equ. M is regular ← affine equ. M =m00 0 b m00A , A is regular (nonuniform scaling, shear) ← similarity M =m00 0 b m00A , AT_{A = λI} (uniform scaling) ← Euclidean equ. M =m00 0 b m00A , AT_{A = I} (rotations, trans-lations, reflections)

Table 2.1 summarizes the different notions of equivalences and symmetries, whereas Figure 2.1 provides a visualization for planar curves. The arrow “←” in this figure indicates that projective equivalences are the most general among these different types of equivalence relations and that congruences are a special case of all the others.

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2.3. NOTATION 11

2.3 Notation

When considering surfaces, we employ multi-indices given by a vector (identified by bold font), which in our case normally consist of three indices

i = (i0, i1, i2),

but a further generalization is straight forward. We also use the usual abbreviatory notations

|i| = i0+ i1+ i2

for the total degree,

In _{= {i ∈ N}3₀ | |i| = n} for the index set,

ti = ti0 0t i1 1t i2 2

for the power basis and

|i| i

:= |i|! i0!i1!i2!

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Chapter 3 Relation between power basis and

Bernstein-B´

ezier representation

Most rational curves and surfaces are given either in power basis representation or in a Bernstein-Bézier representation, which are the two standard forms in Algebraic Geom-etry and CAGD. These two ways of representing rationally parameterized manifolds can be transformed one into the other by changing the basis functions. The corresponding coefficient vectors, resp. control points, are computed by evaluating the Bézier curves and surfaces for the one and using blossoms (see Ramshaw, 1987) for the other direction. This change of the basis is nothing else but some scaling combined with a linear pro-jective transformation of the parameter domain, for which we will determine a closed form for the coefficients of curves in Lemma 8 and the coefficients of surfaces in Lemma 25, respectively. Hence when we are not considering a segment in a closed parameter interval but the curve parameterized by the whole projective line, the monomial coefficients and the scaled Bézier control points define the same curve for the power basis and similarly for surfaces.

We will derive this result for curves in Section 3.1 and for triangular surfaces in Sec-tion 3.2 and we will compare the effort for this translaSec-tion with the usual transformaSec-tion between the power basis representation and Bernstein-B´ezier form. Moreover, we will briefly discuss the tensor product case, which unfortunately is not base point free in the triangular power basis.

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3.1 Curves

Given the control points bi, a rational B´ezier curve has the form

q(u) =

n

X

i=0

B_in(u)bi

where the blending functions

B_in(u) =n i

ui(1 − u)n−i

denote the Bernstein polynomials. A rational curve in the power basis representation is defined by the coefficients ci and has the form

p(t) =

n

X

i=0

citi.

The control points and the coefficients of these two representations are closely related. Lemma 3. If the control points bi of the Bernstein-B´ezier representation q(u) and the

coefficients ci of the power basis representation p(t) fulfil

ci =

n i

bi i = 0, . . . , n, (3.1)

then q(u) and p(u) describe the same rational curve, for u, t ∈ R.

Proof. We note, that the affine curve parameter u and the projective curve parameter u = (u0, u1) are related by

u = u1 u0

,

and similarly for t and t. Using the projective transformation

r : t0 t1 ! = 1 −1 0 1 ! | {z } α u0 u1 ! = u0− u1 u1 ! ,

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3.1. CURVES 15 a short computation confirms that

q(u) = n X i=0 n i ui(1 − u)n−ibi ' n X i=0 n i ui₁(u0− u1)n−ibi = n X j=0 n i bitn−i0 t i 1 ' n X j=0 n i biti = p(t).

Note, that the parameterizations do not coincide but only the images, which are defined by them, coincide. In the proof we also give the transformation of the parameter domain, which we have to apply in order to obtain the coefficients from the control points, when we are interested in getting the same parameterization for both representations as well. For the other direction we simply have to use its inverse

r−1 : u0 u1 ! = 1 1 0 1 ! | {z } β t0 t1 ! = t0+ t1 t1 !

combined with the reciprocal of the scaling. We clarify this relations by a short example. Example 4. The reparameterization given by the matrix α is a projective transformation that maps for example the projective points

u0 =1 0 , u00 = 1 1/2 and u000 =1 1

to the projective points

t0 =1 0 , t00 =1/2 1/2 and t000 =0 1 .

Note, that the latter point t000 is a point at infinity.

We consider the lemniscate, which is given by the parameterization

p(u) =    1 + 6u2_{+ u}4 1 − u4 2u − 2u3   .

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Figure 3.1: Reparameterization relating power basis and Bernstein-B´ezier representa-tion u-axis t-axis center of projection z }| { t000 u0 = t0 u00 u000 t00

This representation gives the coefficients

c0 =    1 1 0   , c1 =    0 0 2   , c2 =    6 0 0   , c3 =    0 0 −2    and c4 =    1 −1 0   .

Using Lemma 3 we easily obtain the control-points for the Bernstein-B´ezier representa-tion of a different parameterizarepresenta-tion of the same curve

b0 = c0 =    1 1 0   , b1 = c1 4 =    0 0 1/2   , b2 = c2 6 =    1 0 0   , b3 = c3 4 =    0 0 −1/2    and b4 = c4 =    1 −1 0   .

In Figure 3.2 we illustrate the relations between the power basis and the Bernstein-B´ezier representation and their parameterizations.

♦ Hence, there are several ways to convert the power basis and the Bernstein-B´ezier representation of a rational curve into each other. While in the classical one also the

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3.1. CURVES 17 Figure 3.2: Reparameterization relating power basis and Bernstein-B´ezier representa-tion of curves α monomial B´ezier 1 0 1 1/2 1 1 1 0 1 1 0 1 p(0) = q(0) p(1) = q(1/2) p(∞) = q(1) u-axis t-axis

parameterization coincides, we propose a different approach by Lemma 3. Now we com-pare the computational effort of the two methods. For the classical method we compute the coefficients ci, for i ∈ {0, . . . , n}, from the control-points bj, for j ∈ {0, . . . , n}, by

the formula ci = n X j=0 n i i j (−1)i−jbj = n X j=0 n (j, i − j, n − i) (−1)i−jbj, (3.2)

which can be found for example in Prautzsch et al. (2002). Considering the two conver-sions as a matrix vector multiplication, i.e. Equation (3.1) writes as

         c0 c1 c2 .. . cn          =          n 0 0 0 · · · 0 0 n₁ 0 · · · 0 0 0 n₂ . .. ... .. . ... . .. ... 0 0 0 · · · 0 n_n                   b0 b1 b2 .. . bn         

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and Equation (3.2) writes as          c0 c1 c2 .. . cn          =          n 0 0 0 0 0 · · · 0 − n₁ 1 0 n 1 1 1 0 · · · 0 n 1 2 0 − n₁ 2 1 n 1 2 2 . .. .. . .. . ... . .. . .. 0 (−1)n n n n 0 (−1)n−1 n n n 1 · · · (−1) n_n n n−1 n n n n                   b0 b1 b2 .. . bn          ,

we easily conclude, that the effort for the conversion using of first version is linear in the degree of the curve, while the classical conversion is quadratic in the degree and analogously for the inverse, where the classical version is given by

bi = n X j=0 cj i j n j , whereas we arive at bi = 1 n i ci.

3.2 Surfaces

The two main B´ezier representations for surfaces are the triangular and the tensor-product form. We start with triangular B´ezier surfaces as they are more similar to the triangular power basis representation, which we use in the remainder of this thesis.

3.2.1 Triangular B´

ezier surfaces

Triangular B´ezier surfaces

q(u, v) =

i+j≤n

X

i,j≥0

B_i,jn(u, v)bi,j

possess the Bernstein basis functions

B_i,jn(u, v) = n (i, j, n − i − j) uivj(1 − u − v)n−i−j.

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3.2. SURFACES 19 The parameter form of a rational triangular surface in power basis representation writes as p(s, t) = i+j≤n X i,j≥0 ci,jsitj.

Analogously to curves the coefficients and control points are closely related.

Lemma 5. If the control points bi,j of the Bernstein-B´ezier representation q(u, v) and

the coefficients ci,j of the power basis representation p(s, t) fulfil

ci,j = n (i, j, n − i − j) bi,j for i, j ≥ 0, i + j ≤ n,

then q(u, v) and p(s, t) describe the same rational surface for (u, v) and (s, t) ∈ R2. Proof. Using the projective transformation of the projective plane

r :    t0 t1 t2   =    1 −1 −1 0 1 0 0 0 1    | {z } α    u0 u1 u2   =    u0− u1− u2 u1 u2   ,

a short computation confirms that

q(u, v) =

i+j≤n

X

i,j≥0

B_i,jn (u, v)bi,j

' i+j≤n X i,j≥0 n (i, j, n − i − j) ui₁uj₂(u0− u1− u2)n−i−jbi,j = i+j≤n X i,j≥0 n (i, j, n − i − j) bi,jti1t j 2t n−i−j 0 = p(t) ' p(s, t).

Similar to the case of curves, relating the coefficients and the control points by Lemma 5 includes a reparameterization. For completeness we also give the relation, that leaves the parameterization invariant. The classical version to obtain the

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coeffi-cients from the control points is ci= X j∈In n (i0, i1− j1, i2− j2, j1, j2) (−1)i1+i2−j1−j2_b j

and the inverse relation, i.e. to obtain the control points from the power basis coefficents, is given by bi = X j∈In 1 n i j0 (i0, i1− j1, i2− j2) cj.

The effort for the conversion using Lemma 5 is O n+2₂ = O (n2_{), whereas the}

classical method takes O n+2₂ 2

= O (n4_{) basic operations. Again we illustrate this}

relation by a short example.

Example 6. We consider the Steiner surface given by the quadratic parameter repre-sentation p(s, t) =       1 + s2_{+ t}2 st s t      

The coefficients with respect to the power basis are

c(2,0,0) = c(0,2,0) = c(0,0,2)=       1 0 0 0       , c(1,1,0)=       0 0 1 0       , c(1,0,1)=       0 0 0 1       , c(0,1,1) =       0 1 0 0       .

Due to Lemma 5 we have the control points

b(2,0,0) = c(2,0,0)=       1 0 0 0       , b(0,2,0)= c(0,2,0) =       1 0 0 0       , b(0,0,2) = c(0,0,2) =       1 0 0 0      

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3.2. SURFACES 21 b(1,1,0) = c(1,1,0) 2       0 0 1/2 0       , b(1,0,1) = c(1,0,1) 2       0 0 0 1/2       , b(0,1,1) = c(0,1,1) 2       0 1/2 0 0       .

Figure 3.3: Reparameterization relating power basis and Bernstein-B´ezier representa-tion of surfaces α monomial B´ezier   1 0 0     1 1/2 0     1 1 0     1 0 1/2     1 1/2 1/2     1 0 1     1 0 0     1 1 0     0 1 0     1 0 1     0 0 1     0 1 1   u-axis t-axis

Both representations map the projective plane to the same surface, but we need two different parameter domains in order to represent the same segments. _♦

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3.2.2 Tensor product surfaces

For the conversion between the power basis

p(s, t) =

i+j≤n

X

i,j≥0

ci,jsitj.

and the tensor product B´ezier representation

q(u, v) = h X i=0 m X j=0 bi,jBih(u)B m j (v)

one uses the tensor product structure and the reparameterization given in Section 3.1 for curves in each direction, see Prautzsch et al. (2002) for more details. This leads to a representation, where each direction of the parameter domain is considered as the projective line, in particular, (u, v) ∈ P (R) × P (R), which is structurally different to the projective plane P (R2_{), that we use in the triangular case. For the conversion between}

the tensor product Bernstein-B´ezier control points and the triangular ones, we refer to Brueckner (1980) and Goldman and Filip (1987).

In Chapter 5 we use in order to compute the projective equivalences that the repa-rameterization relating two surfaces is a linear projective transformation of the projective plane. To guarantee this property we use that the parameterizations are base point free over the algebraic closure of the domain, i.e., they do not possess a common root in P (C2_{). Unfortunately, tensor product surfaces are not base point free. For the sake of}

completeness we provide a short proof of this well-known fact.

To ensure that we can represent the tensor product representation by a triangular surface representation and vice versa the number of necessary coefficients increases. In particular, to represent a tensor product surface of bidegree (h, m), we need a triangular surface of degree n = h+m, whereas to represent a triangular surface of degree n we need a tensor product surface of bidegree (h, m) = (n, n), see Figure 3.4 for a visualization. When considering a surface which is not degenerated to a curve, both, h and m are greater than zero. By Figure 3.4 we can also easily see, that the tensor product representation possesses the base points (0, 1, 0) and (0, 0, 1) in the monomial representation, as there is only one nonzero coefficient for ti

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3.2. SURFACES 23 Figure 3.4: Coefficients needed to transform triangular (circles) and tensor product (crosses) surfaces

⨯ ⨯ ⨯ ⨯

⨯

Tensor product to triangular (h, m) = (3, 2), n = 5

Triangular to tensor product n = 3, (h, m) = (3, 3)

For the triangular Bernstein-B´ezier representation these base points transform to (1, 1, 0) and (1, 0, 1), respectively.

In the remainder of this thesis when considering surfaces we will focus on triangular surfaces with respect to the power basis. This is reasonable as we have seen, that in the triangular case their coefficients and the B´ezier control points coincide up to the scaling n_i, while tensor product surfaces cannot be dealt with by our approach since they possess base points.

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Chapter 4 Projective equivalences of curves

In this chapter we derive a method to determine all projective equivalences of two curves C and C0 in arbitrary dimension given by rational parameterizations. The de-termination of projective symmetries of one curve can be seen as a special case and similarly equivalences with respect to more special transformations such as affine ones, similarities and Euclidean ones are special cases. We show, how to discover them with some special attention on affine equivalences, in particular for polynomial curves, as this provides a major potential of simplification. The other two cases, i.e., similarities and Euclidean equivalences, are already thouroughly studied in literature (see e.g. Alcázar, 2014; Alcázar et al., 2014a,b, 2015a,b; Sánchez-Reyes, 2015) and we adress them as special cases of our approach.

The method is based on solving a polynomial system of equation that we derive from the polynomial parameterizations of the curve in homogeneous coordinates, using the fact that the reparameterization, correlating two proper parameterizations of the same curve, are linear rational (Lemma 4.17 in Sendra et al., 2008). By exploiting the special structure of this system and some basic linear algebra, we are able to reduce the number of unknowns and thus to simplify the system. We compare the method before and after this step of simplification both theoretically and by considering numerical examples.

This chapter is organized as follows. In Section 4.1 we discuss the representation of the input curves and their properties. Section 4.2 is the main part, in which we derive our method, i.e., the polynomial systems whose solutions determine all projective

This chapter is based on joint work with Bert J¨uttler published in Hauer and J¨uttler (2017)

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equivalences. First we consider a naive approach before we use some properties of the system to simplify it. In Section 4.3 we have a closer look at the special cases of affine equivalences of rational and polynomial curves and finally in the last section of this chapter we present some numerical examples, which confirm that the simplified system is superior to the naive approach.

4.1 Rational curves

We consider two parametric rational curves C and C0 ⊂ ¯Ed_{, where both curves are}

given by proper parameterizations

p : P1_{(R) → C ⊂ ¯}_Ed_, _{t 7→ p(t) = (p}

0(t0, t1) : p1(t0, t1) : · · · : pd(t0, t1)) ,

p0 : P1_{(R) → C}0 ⊂ ¯Ed, t 7→ p0(t) = (p0₀(t0, t1) : p01(t0, t1) : · · · : p0d(t0, t1))

with the parameter t = (t0, t1).

The domain of both parameterizations is the real projective line P1_(R).

Conse-quently, the homogeneous coordinates of both curves are homogeneous polynomials of degree n, pi(t) = n X j=0 cj,itn−j0 t j 1 and p 0 i(t) = n X j=0 c0_j,itn−j₀ tj₁, i = 0, . . . , d, with coefficient vectors

cj = (cj,0, cj,1, . . . , cj,d)T and c0j = (c 0 j,0, c 0 j,1, . . . , c 0 j,d) T . (4.1)

Note that every rational curve, which is not given by a proper parameterization, can be reparameterized to obtain a proper one. The proof for planar curves, which is given by Sendra et al. (2008), applies to any space dimension d.

Furthermore we assume that the parameterizations are in reduced form, i.e., gcd(p0(t), p1(t), . . . , pd(t)) = gcd(p00(t), p

0

1(t), . . . , p 0

d(t)) = 1 (4.2)

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4.2. DETECTING PROJECTIVE EQUIVALENCES 27 of a curve, hence the degrees of projectively equivalent curves have to be equal. In particular, this implies that both curves possess the same degree

max(deg_t_i(p0(t)), degti(p1(t)), . . . , degti(pd(t))) = n, max(deg_t i(p 0 0(t)), degti(p 0 1(t)), . . . , degti(p 0 d(t))) = n, (4.3) with respect to ti, i = 0, 1.

We will assume that neither of the two curves is contained in a hyperplane. Conse-quently, the matrices (cij) resp. (c0ij) formed by the coefficient vectors have rank d + 1.

Clearly, this is only possible if the degree satisfies n ≥ d.

4.2 Detecting projective equivalences

We present two methods to analyze whether two curves C, C0 ⊂ ¯Ed are projectively equivalent and to find all equivalences. This includes the construction of the associated projective transformations. We assume that the degrees satisfy n > d since any two curves of degree d, which are not contained in hyperplanes, are related by infinitely many projective transformations.

4.2.1 The direct method

We start with a simple technical lemma.

Lemma 7. Two rational parameterizations p(t) and p0(t) in reduced form are equivalent, i.e. p(t) ' p0(t) holds for all t ∈ P1_{(R), if and only if there exists a non-zero constant}

µ such that

cj = µc0j, j = 0, . . . , n.

Proof. The equivalence of the two curves implies that there exists a rational function µ(t) = µ1(t) µ0(t) = p 0 0(t) p0(t) = p 0 1(t) p1(t) = · · · = p 0 d(t) pd(t)

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Conse-quently, the two rational curves satisfy

µ0(t)p(t) = µ1(t)p0(t).

This function is indeed a constant since µ0| gcd(p00, p 0 1, . . . , p 0 d) | {z } =1 and µ1| gcd(p0, p1, . . . , pd) | {z } =1 .

Recall that any two proper parameterizations of a rational curve are related by a linear rational reparameterization (Sendra et al., 2008), which is simply a regular projective transformation of the real projective line

r(t) = α00 α01 α10 α11 ! | {z } =α t = α00t0+ α01t1 α10t0+ α11t1 !

described by a regular matrix α. A similar statement, that avails itself of the language from CAGD, was given in Berry and Patterson (1997) who show for rational Bézier curves the uniqueness of the Bézier control points and that the weights can only be varied by a projective transformation of the parameter space. By Lemma 3 in Chapter 3 we see the close relation between the monomial coefficients and the Bézier control points. Note here, that Berry and Patterson (1997) fix the starting point and the end point of a curve segment and therefore they have less degrees of freedom. In our case the coefficients can also vary by some specific linear combinations as we see in the following Lemma.

We investigate the transformation of the coefficients which is caused by a linear rational reparameterization.

Lemma 8. The reparameterized curve

(p ◦ r)(t) = ˆp(t) = n X j=0 ˆ cjtn−j0 t j 1

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4.2. DETECTING PROJECTIVE EQUIVALENCES 29 has the coefficients

ˆ cj(α) = n X i=0 ci j X `=0 n − i ` i j − ` αn−i−`₀₀ α`₀₁αi−j+`₁₀ αj−`₁₁ (4.4) for j = 0, . . . , n.

Proof. A short computation gives

(p ◦ r)(t) = n X i=0 ci(α00t0 + α01t1)n−i(α10t0 + α11t1)i = n X i=0 ci n−i X `=0 n − i ` αn−i−`₀₀ tn−i−`₀ α`₀₁t`₁ ! _i X m=0 i m α₁₀i−mti−m₀ α₁₁mtm₁ ! = n X i=0 ci n−i X `=0 i X m=0 n − i ` i m αn−i−`₀₀ α`₀₁αi−m₁₀ αm₁₁tn−m−`₀ tm+`₁ = n X i=0 ci n X j=0 X `+m=j n − i ` i m αn−i−`₀₀ α`₀₁αi−m₁₀ αm₁₁tn−j₀ tj₁ = n X j=0 tn−j₀ tj₁ n X i=0 ci X `+m=j n − i ` i m αn−i−`₀₀ α`₀₁αi−m₁₀ αm₁₁ Comparing the coefficients confirms (4.4).

Remark 9. For singular matrices α, the projective mapping r transforms the entire line into a single point, hence the coefficients of the reparameterized curve ˆp = p ◦ r are all linearly dependent.

We identify projective equivalences by analyzing whether the coefficients are related by a projective transformation.

Proposition 10. Let C and C0 be rational curves of degree n > d with proper param-eterizations p(t) and p0(t) satisfying (4.2) and (4.3). The two curves are projectively equivalent if and only if there exist a regular projective transformation matrix M and a projective transformation α of the real line, such that the coefficients of both curves satisfy

M c0_j = ˆcj(α), j = 0, . . . , n, (4.5)

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Proof. On the one hand, the conditions (4.5) imply that the two curves are projectively equivalent. On the other hand, we consider two projectively equivalent curves C0 and C. There exists a projective transformation f with the matrix M such that

f (C0) = C.

We define z(t) = M p0(t). Consequently z(t) and p(t) are two proper parameterizations of the same curve C. According to Lemma 4.17 of Sendra et al. (2008) there is a linear rational reparameterization r(t) – and hence an associated projective transformation α – such that

z(t) ' (p ◦ r)(t). Thus we obtain that

n X j=0 M c0_jtn−j₀ tj₁ = M p0(t) = z(t) ' (p ◦ r)(t) = ˆp(t) = n X j=0 ˆ cj(α)t n−j 0 t j 1

where we use – from left to right – the representation of p0, the definition of z(t), Lemma 4.17 of Sendra et al. (2008), the definition of ˆp and Lemma 8. We complete the proof by comparing the leftmost and rightmost terms and noting that Lemma 7 implies (4.5). Note that the constant µ of the homogeneous coordinates can be put into M .

Recall that we assume that neither of the two curves is contained in a hyperplane. Consequently, the regularity of M can be guaranteed by the regularity of α, cf. Remark 9. Thus, in addition to (4.5) we have

det α 6= 0.

In order to avoid the inequality constraint we replace it with the equation

(det α)u = 1 (4.6)

which involves the additional variable u. Moreover, as the representation of the pro-jective transformation α is only determined up to a constant factor we can choose a normalization of this transformation. We choose the first nonzero coefficient in the first

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4.2. DETECTING PROJECTIVE EQUIVALENCES 31 row of α to be equal to one and hence obtain two different cases

I) α00 = 1,

II) α00 = 0 and α01= 1,

(4.7)

which we have to consider.

Remark 11. When solving our polynomial system by computing the Gr¨obner basis, the computational costs for the first case I) considerably exceeds the costs for II).

Summing up, we arrive at a simple method for testing whether two curves are pro-jectively equivalent:

Corollary 12. Under the assumptions of the previous proposition, the two rational curves C and C0 are projectively equivalent if and only if there exist transformation ma-trices M and α and a constant u such that the equations (4.5) and (4.6) and one of the normalization conditions (4.7) are satisfied.

This corollary leads to two systems of (d + 1)(n + 1) + 2 polynomial equations for (d + 1)2_{+ 5 unknowns in M (containing (d + 1)}2 _{unknowns), α (4 unknowns) and u. The}

equations from formula (4.5) are linear in M but of degree n with respect the elements of α. Solving this system will be called the

Direct method for detecting Projectively equivalent curves. (DP)

4.2.2 Reducing the number of unknowns

We observe that the system given by equation (4.5) has a special structure, i.e., it is linear in the unknowns describing M and the right hand side are homogeneous polynomials of degree n in α. We use this knowledge to reduce the number of unknowns before solving the system with the help of standard computer algebra systems.

Proposition 13. Let C and C0 be as in Proposition 10. The coefficient matrices

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have d + 1 linearly independent column (coefficient) vectors and the kernel of the second matrix (c0_ij) is spanned by the basis vectors

bk= (bk_j)j=0,...,n, k = 1, . . . , n − d.

The two curves are projectively equivalent if and only if there is a regular transformation α of the projective line, such that the equations

n

X

j=0

ˆ

cij(α)bkj = 0, i = 0, . . . , d, k = 1, . . . , n − d (4.9)

are satisfied, where the coefficients ˆcij(α) are given in (4.4).

Proof. Clearly, the coefficient matrix has rank d+1, hence its kernel has dimension n−d. This confirms the existence of the linearly independent columns and kernel basis vectors. We show that the equations (4.9) together with the regularity of α are equivalent to the condition stated in Proposition 10. First we confirm that the latter condition implies equation (4.9) and the regularity. On the one hand, equation (4.5) ensures that the kernel of the coefficient matrix (cij) in (4.8) is contained in the kernel of the coefficient matrix

(ˆcij(α))i=0,...,d,j=0,...,n (4.10)

of the reparameterized curve, thereby proving equation (4.9). The regularity of α follows directly from the regularity of M and equation (4.5), see Remark 9.

Second we prove the other implication. Equation (4.9) guarantees that the kernel of the matrix (4.10) contains the kernel of (cij), given in (4.8). This implies that the space

spanned by the row vectors of the matrix in (4.10) is contained in the space spanned by row vectors of (cij), since these spaces are the orthogonal complements of the kernels.

This proves the existence of the matrix M in (4.5). Due to the regularity of α and the assumption on the linearly independent coefficient vectors we have that both

(ˆcij(α))i=0,...,d,j=0,...,n and (c0ij)i=0,...,d,j=0,...,n

have rank d + 1 which proves the regularity of M .

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4.2. DETECTING PROJECTIVE EQUIVALENCES 33 of α and we choose the same normalization condition (4.7) as in the direct method. Hence, Proposition 13 leads to two systems of (n − d)(d + 1) + 2 polynomial equations for only five unknowns, u and the elements in α. The maximum degree of the system is n. Solving this system will be called the

Reduced method for detecting Projectively equivalent curves. (RP)

4.2.3 Computation of projective equivalences

For both methods DP and RP, we first compute the Gr¨obner bases of the systems formed by the regularity condition (4.6), the normalization (4.7) together with the main system (4.5) and (4.9), respectively.

The direct method returns both the 2 × 2 matrices α that specify the reparameteriza-tions r and the associated projective transformareparameteriza-tions. In contrast to this, the method RP computes the 2 × 2 matrices α only. Once the reparameterizations have been found, the corresponding projective transformations M are obtained simply by solving the linear systems of equations

M c0_j(`) = ˆc_j(`)(α), ` = 0, . . . , d, (4.11) for the (d + 1)2 _{unknown elements of M , where c}0

j(`), for ` = 0, . . . , d, are linear in-dependent column coefficient vectors of the coefficient matrix (c0_ij)i=0,...,d,j=0,...,n. The

computational effort is negligible compared to the overall computation time.

The specific type of the equivalence can be found by investigating the properties of the transformation matrix M . More precisely, it is an affine equivalence if the elements satisfy

m0i= 0, for i = 1, . . . , d.

It is a similarity (or even a congruence transformation) if additionally the condition ATA = λI with A =mij

m00

i,j=1,...,d

is fulfilled, where I is the d × d identity matrix (and the factor even satisfies λ = 1 for congruence transformations).

When applied to pairs (C, C) of identical curves, each of the two methods allows us to identify all projective symmetries. Once again, this includes all affine or Euclidean

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sym-Table 4.1: Characteristics of the non-linear polynomial systems for detecting projective equivalences of rational degree n curves in d-dimensional space.

# of unknowns degree # equations

DP (d + 1)2_{+ 5} _n _{n(d + 1) + d + 2}

RP 5 n n(d + 1) − d2_{− d + 2}

metries, which are found by analyzing the properties of the corresponding transformation Matrix M , analogously to the discussion above.

Self-similarities of rational curves are always symmetries (see e.g. Theorem 6 in Alc´azar et al., 2015a), while non-rational curves (such as the logarithmic spiral) may possess more general self-similarities. Finally we note that the image of a curve with symmetries under a general affine (resp. projective) transformation has affine (resp. projective) symmetries, which are not symmetries.

Finally we compare the characteristics of the non-linear polynomial systems for the two different methods in Table 4.1. This is the most time consuming part of our algo-rithm. For the direct method DP, the unknowns include the elements of the matrix M , since it uses an all-at-once approach. For the reduced method RP both, the number of unknowns and the number of equations, are decreased by (d + 1)2_.

4.3 Special cases - affine equivalences and

polyno-mial input curves

We take a closer look at two special cases.

4.3.1 Affine equivalences

As mentioned in the end of the previous section, we may identify affine equivalences by investigating the transformation matrix M . However, if we are interested exclusively in this special case it might be advantageous to take this into account beforehand, for instance, if one is interested in the Gr¨obner Basis of this system per se. We will see in Section 4.4 that for DP this modification also improves the speed of computation, whereas the reduced methods RA and RP show a similar behaviour.

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4.3. AFFINE EQUIVALENCES AND POLYNOMIAL INPUT CURVES 35 Concerning the direct method for affine equivalences, Corollary 12 leads to a system of (d + 1)(n + 1) + 1 polynomial equations for d(d + 1) + 6 unknowns in M (containing d(d + 1) + 1 unknowns), α (4 unknowns) and u. The equations are linear in M but of degree n with respect the elements of α and u. Solving this system will be called the

Direct method for detecting Affinely equivalent curves. (DA) A similar modification can be derived for the reduced method RP as well. Instead of decreasing the number of unknowns, this increases the number of equations. More precisely, we obtain additional equations involving only the first homogeneous coordinate. These equations are of degree n:

Corollary 14. Let C and C0 be two rational curves as in Proposition 13. The two curves C and C0 are affinely equivalent if and only if there exist a regular projective transformation α, defining a reparameterization, and a constant ω such that the equations ωc0_0k = ˆc0k(α), k = 0, . . . , n, (4.12) n X j=0 ˆ cij(α)bkj = 0, i = 0, . . . , d, k = 1, . . . , n − d, (4.13) are satisfied.

Proof. On the one hand, if the two curves are affinely equivalent, then the equations are obviously satisfied with ω = m0,0, see Equation (2.1) and Proposition 10.

On the other hand, consider two curves satisfying the equations (4.12) and (4.13). First we note that due to the regularity of α and the rank d + 1 of the matrix

(ˆcij(α))i=0,...,d,j=0,...,n

not all of the right hand sides in (4.12) can be equal to zero which implies ω 6= 0. The two curves are projectively equivalent according to Proposition 13, hence there exists a projective transformation M with

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We will show that M represents an affine transformation. Any point x0 has a unique representation

x0 =

d

X

`=0

ξ`c0_j(`)

with certain real coefficients ξ` and some choice of d + 1 coefficient vectors c0_j(0), . . . , c

0 j(d) in general position. Its image under the projective transformation then satisfies

x = M x0 =

d

X

j=0

ξ`ˆcj(`)(α),

due to the linearity of the transformation. We now use these two representations and the additional Equation (4.12) to derive the relation

x0 = ωx00

between the 0-th coordinates. Consequently, the projective transformation M is indeed even an affine transformation since it maps any point x0 at infinity (where x0₀ = 0) to another point at infinity.

Consequently, in addition to equation (4.9) we add n+1 polynomial equations (4.12), which have degree n with respect to the four scalar unknowns in α and which are linear with respect to an additional variable ω. Solving this system will be called the

Reduced method for detecting Affinely equivalent curves. (RA) Remark 15. Both methods DA and RA can also be applied even if the degree and the space dimension are equal, i.e., n = d. When considering affine equivalences the equality of degree and dimension does not necessarily lead to an infinite family of solutions.

The methods DA and RA only give solutions such that m0i = 0 for i > 0 in the

transformation matrix M . In the case of the reduced method, M can again be computed by solving the linear system of equations

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4.3. AFFINE EQUIVALENCES AND POLYNOMIAL INPUT CURVES 37 Table 4.2: Characteristics of the non-linear polynomial systems for detecting affine equivalences of rational degree n curves in d-dimensional space.

# of unknowns maximum degree # equations

DA d(d + 1) + 6 n n(d + 1) + d + 2

RA 6 n n(d + 2) − d2_{− d + 3}

for the d(d + 1) + 1 unknown elements in M , where c0_j(`) are linear independent coefficient vectors. The computational effort is negligible compared to the overall computation time. Once more, the specific type of the equivalence can be identified by investigating the properties of the transformation matrix M . When applied to pairs (C, C) of identical curves, each of the two methods allows us to detect all affine symmetries, which are again Euclidean symmetries if ATA = I.

Finally we again compare the characteristics of the non-linear polynomial systems for the two different methods in Table 4.2, similar to the previous one.

4.3.2 Affine equivalences of polynomial curves

In the case of two polynomial input curves C and C0 the problem at hand simplifies as the reparameterization r(t) is no longer a linear rational function but becomes a linear transformation only.

Corollary 16. In the situation of Proposition 10, the projective transformation α defines a linear parameter transformation (i.e. α01 = 0) and the matrix M satisfies (2.1) if C

and C0 are two affinely equivalent polynomially parameterized curves.

Proof. Firstly, it is obvious that the matrix M has the structure (2.1), since M is an affine transformation. Secondly, the coefficients of a polynomial curve C satisfy c0j = 0

for j > 0, hence (4.4) gives the relation

ˆ c0j(α) = c00 n j αn−j₀₀ αj₀₁.

Together with (2.1), the equations (4.5) then imply α01= 0 since the coefficients of the

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Consequently it suffices to consider only linear reparameterizations, where α00= 1

and α01= 0, when detecting affine equivalences of polynomial curves. This observation

has several consequences:

• The formulas from Lemma 8 for representing the coefficients of a reparameterized curve simplify to ˆ cj(α) = n X i=0 ci i j αi−j₁₀ αj₁₁ (4.14)

• When applying RA, the equations (4.12) are automatically satisfied for k > 0 and the remaining one determines the value of ω. Hence we only consider (4.6), (4.9) as in RP, but with (4.14) instead of (4.4) and a different normalization condition

α00 = 1 and α01= 0

instead of (4.7). Thus, it suffices to solve only one system instead of two which consists of n(d + 1) − d2− d non-linear equations in the three unknowns u, α10 and

α11.

This observation cannot be extended to the case of projective equivalences. In fact, two projectively equivalent polynomial curves may be related by a linear rational repa-rameterization, which is not a linear parameter transformation, as we show by a simple example.

Example 17. We consider two cubic polynomial curves

p(t) =    t3 0 3t2₀t1+ 3t0t21− 3t31 t3₁    and p 0 (t) =    t3 0 −3t3 0+ 3t20t1+ 3t0t21 t3₁   

which are projectively equivalent as they are related for instance by

M p0(t) = p(αt) with α = 0 1 1 0 ! and M =    0 0 1 0 1 0 1 0 0   .

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4.4. EXAMPLES 39 the form ˆ M p(t) = p(αt) with α = 0 1 1 0 ! and M =ˆ    0 0 1 −3 1 3 1 0 0   .

Two of the five remaining projective symmetries are affine symmetries (including the

identity). _♦

4.4 Examples

The main computational costs of the methods are caused by solving the algebraic systems presented in Sections 4.2 and 4.3. Several numerical and symbolic methods for solving algebraic systems exist. The generation of efficient solvers for polynomial systems is an interesting and wide field of research. However, a detailed discussion of these methods is beyond the scope of this thesis. Instead we rely on existing methods, which have been implemented in well-established computer algebra systems. More precisely, we use Mathematica R _{Version 11 and Maple}TM _{2017 (Maplesoft, 2011 – 2017) for our}

numerical examples.

In the remainder of this section we first consider projective equivalences in the next subsection. We create the polynomial system of RP for a cubic planar rational curve before we show our results for some more complex curves in 2D and 3D. Second, we ad-dress the computation of affine equivalences in Section 4.4.2 and show the computational results for this case. Finally we investigate polynomial curves.

4.4.1 Projective equivalences

First we consider our methods for finding all projective equivalences, i.e. the methods DP and RP.

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Folium of Descartes - a cubic example We investigate the two rational curves

p(t) =    t3₀+ t3₁ 3t2 0t1 3t0t21    and p 0 (t) =    −208t3 0+ 108t20t1− 18t0t21+ t31 −200t3 0+ 144t20t1− 30t0t21+ 2t31 144t3 0− 60t20t1 + 6t0t21    (4.15)

for projective equivalences. The first one is the Folium of Descartes, since choosing t0 = 1

and t1 = t transforms p(t) to the usual parameterization. The second curve was derived

by applying a reparameterization and an affine transformation.

We omit the details of the direct method and show the equations generated by RP. The coefficient-matrices take the form

(cij) =    1 0 0 1 0 3 0 0 0 0 3 0    (c 0 ij) =    −208 108 −18 1 −200 144 −30 2 144 −60 6 0   

and the first one leads to

(ˆcij(α)) =    α3 00+ α310 3α200α01+ 3α210α11 3α00α201+ 3α10α211 α301+ α113 3α2 00α10 6α00α01α10+ 3α200α11 3α201α10+ 6α00α01α11 3α012 α11 3α00α210 3α01α210+ 6α00α10α11 6α01α10α11+ 3α00α211 3α01α211   .

The kernel of the second one consists of one vector b1 =

1 6 36 208

and hence we have the following system

0 = α3₀₀+ 18α2₀₀α01+ 108α00α201+ 208α301+ α310+ 18α210α11+ 108α10α211+ 208α311

0 = 3α2₀₀α10+ 36α00α01α10+ 108α201α10+ 18α200α11+ 216α00α01α11+ 624α201α11

0 = 3α00α102 + 18α01α210+ 36α00α10α11+ 216α01α10α11+ 108α00α211+ 624α01α211

1 = 9u(α3₀₀+ α3₁₀) ±1 = u

Equivalence detection of rational curves and surfaces / submitted by Michael Hauer