Another generalization we achieved, was the extension to base-point free, properly parameterized rational surfaces. We showed, that two base-point free birational param-eterizations of one surface are related by a projective transformation of the parameter domain, which we identified with the projective plane. In the generic case the assump-tions on the parameterization (base-point free and birational) are fulfilled. We used this result to propose a method for finding projective equivalences of rationally parameterized surfaces.
The proposed method creates a polynomial system of equations in the variables that specify the reparameterization (four for curves and nine for surfaces) and one additional variable that guarantees the regularity of the transformation. We applied this method to several examples and solved the resulting systems using the Gr¨obner basis implemen-tation of MapleTM 2017.
The computational experiments showed a good behaviour. For various special curves and surfaces we found all possible equivalences within a fraction of a second and even for randomly generated examples we were able to solve the system in a reasonable time. For example, we were able to determine the projective equivalences of randomly generated rationally parameterized surfaces up to a degree of 6 and the affine equivalences of polynomial surfaces up to a degree of 14 within two minutes.
6.2. OUTLOOK 85 concentrated on base-point free triangular surfaces, where we identified the parameter domain with the projective plane P2(R). As clarified in Section 3.2.2 this theory is not applicable to tensor product surfaces, as they alway possess base points for the projective plane as parameter domain. We believe that a similar theory can also be worked out for the tensor product case by identifying the parameter domain with P1(R)×P1(R). In that case, not every tensor product surfaces possesses base points.
A different approach could be to try to weaken the assumptions about birationality and base points at all. This may result in more complex reparameterizations then those, which can be represented by a projective transformation of the parameter domain. How-ever, if all the possible reparameterizations can be described by a closed form similar to Equation (5.4), then the detection of equivalences would be straightforward.
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List of Figures
2.1 Dependencies between the different types of equivalence relations . . . . 10
3.1 Reparameterization relating power basis and Bernstein-B´ezier representa-tion . . . 16
3.2 Reparameterization relating power basis and Bernstein-B´ezier representa-tion of curves . . . 17
3.3 Reparameterization relating power basis and Bernstein-B´ezier representa-tion of surfaces . . . 21
3.4 Coefficients needed to transform triangular (circles) and tensor product (crosses) surfaces . . . 23
4.1 Examples 1-10 of projectively and affinely equivalent curves . . . 44
5.1 Steiner surface . . . 53
5.2 The general cases of quadratic rational surfaces . . . 70
5.3 Remaining base-point-free quadratic rational surfaces . . . 71
5.4 Cubics . . . 72
5.5 Quadrics (oval quadrik, ring quadrik, cone) . . . 73
5.6 The general cases of quadratic polynomial surfaces (left 3-3-1c, right 3-3-2b) 76 5.7 Rational degree 3 surfaces: projectively transformed Enneper . . . 80
5.8 Rational degree 3 surfaces with 6 symmetries . . . 80
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List of Tables
2.1 Projective equivalence of manifolds and its special cases . . . 10 4.1 Characteristics of the non-linear polynomial systems for detecting
projec-tive equivalences of rational degreen curves in d-dimensional space. . . . 34 4.2 Characteristics of the non-linear polynomial systems for detecting affine
equivalences of rational degreen curves in d-dimensional space. . . 37 4.3 Parameterizations of the curves considered in Sections 4.4.1 and 4.4.2 . . 42 4.4 Computation time in Maple for projective symmetries and equivalences
of planar curves (time in seconds) . . . 45 4.5 Parameterizations of the curves considered in Section 4.4.1 . . . 45 4.6 Computation time in Maple for projective symmetries and equivalences
of space curves (time in seconds) . . . 46 4.7 Computation time for projective equivalences with random curves (in
sec-onds) . . . 47 4.8 Computation time in Maple for affine equivalences of planar curves (time
in seconds) . . . 48 5.1 Specifications of the polynomial systems . . . 67 5.2 Projective classification of 100 randomly generated surfaces with
coeffi-cients|c(k,i)| ≤ 100 . . . 74 5.3 Number of projective symmetries (including the identity) . . . 75 5.4 Loglog-plot of the computation time (in sec.) of Gr¨obner basis for
projec-tive equivalences of quadratic rational surfaces with random values. . . . 75 5.5 Number of affine classes of quadratic parameterizable surfaces . . . 77 5.6 Number of the affine symmetries (including the identity) . . . 78
95
5.7 Affine classification of 100 polynomial surfaces with random coefficients
|ck,i| ≤100 . . . 78 5.8 Computation time (in sec.) of the Gr¨obner basis for affine equivalences of
quadratic polynomial surfaces with random values. . . 79 5.9 Computation time (in sec.) of Gr¨obner basis for projective symmetries . 81 5.10 Computation time (in sec.) of Gr¨obner basis for affine symmetries and
equivalences of polynomial surfaces. . . 82
Curriculum Vitae
Personal Details
Name Dipl.Ing. Michael Hauer Date of birth February 2, 1989
Place of birth Gmunden, Austria Nationality Austrian
Education
2014–present Doctorate Degree Program of Engineering Sciences at Johannes Kepler University in Linz
2012–2013 Master’s Degree Program of Industrial Mathematics at Johannes Kepler University in Linz
2008–2012 Bachelor’s Degree Program of Technical Mathematics at Johannes Kepler University in Linz
1999–2007 Secondary school BG/BRG Gmunden 1995–1999 Primary school in Gschwandt
Working positions and special activities
2014–present University assistant at the Institute of Applied Geometry, JKU Linz 2011 Exchange semester (Erasmus) at the university Joseph Fourier in
Grenoble, France
2009 Vacation work at MIBA HTC in Vorchdorf, Austria
2008 Internship for 5 months at MIBA Steeltec in Vrable, Slovakia
2007 Military service
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