53 research outputs found
Minkowski Actions of Quaternion Sets and their Applications
Applications of Laguerre geometry to computer aided geometric design are presented, realized through Minkowski actions. Basic Laguerre geometry is first discussed. Then quaternions, set multiplication, and the use of quaternion sets to define Minkowski actions are described and used to achieve results used in geometric design
Anisotropic Wavefronts and Laguerre Geometry
Motivated by the study of wave fronts in anisotropic media, we propose an
incidence geometry of anisotropic spheres in a Finsler-Minkowski space. An
anisotropic version of the Laguerre functional is considered. In some
circumstances, this functional can be used to determine that two wavefronts
observed at distinct times in a homogeneous, anisotropic medium, do not
originate from the same source
Algorithms for curve design and accurate computations with totally positive matrices
Esta tesis doctoral se enmarca dentro de la teoría de la Positividad Total. Las matrices totalmente positivas han aparecido en aplicaciones de campos tan diversos como la Teoría de la Aproximación, la Biología, la Economía, la Combinatoria, la Estadística, las Ecuaciones Diferenciales, la Mecánica, el Diseño Geométrico Asistido por Ordenador o el Álgebra Numérica Lineal. En esta tesis nos centraremos en dos de los campos que están relacionados con matrices totalmente positivas.This doctoral thesis is framed within the theory of Total Positivity. Totally positive matrices have appeared in applications from fields as diverse as Approximation Theory, Biology, Economics, Combinatorics, Statistics, Differential Equations, Mechanics, Computer Aided Geometric Design or Linear Numerical Algebra. In this thesis, we will focus on two of the fields that are related to totally positive matrices.<br /
Ruled Laguerre minimal surfaces
A Laguerre minimal surface is an immersed surface in the Euclidean space
being an extremal of the functional \int (H^2/K - 1) dA. In the present paper,
we prove that the only ruled Laguerre minimal surfaces are up to isometry the
surfaces R(u,v) = (Au, Bu, Cu + D cos 2u) + v (sin u, cos u, 0), where A, B, C,
D are fixed real numbers. To achieve invariance under Laguerre transformations,
we also derive all Laguerre minimal surfaces that are enveloped by a family of
cones. The methodology is based on the isotropic model of Laguerre geometry. In
this model a Laguerre minimal surface enveloped by a family of cones
corresponds to a graph of a biharmonic function carrying a family of isotropic
circles. We classify such functions by showing that the top view of the family
of circles is a pencil.Comment: 28 pages, 9 figures. Minor correction: missed assumption (*) added to
Propositions 1-2 and Theorem 2, missed case (nested circles having nonempty
envelope) added in the proof of Pencil Theorem 4, missed proof that the arcs
cut off by the envelope are disjoint added in the proof of Lemma
On organizing principles of Discrete Differential Geometry. Geometry of spheres
Discrete differential geometry aims to develop discrete equivalents of the
geometric notions and methods of classical differential geometry. In this
survey we discuss the following two fundamental Discretization Principles: the
transformation group principle (smooth geometric objects and their
discretizations are invariant with respect to the same transformation group)
and the consistency principle (discretizations of smooth parametrized
geometries can be extended to multidimensional consistent nets). The main
concrete geometric problem discussed in this survey is a discretization of
curvature line parametrized surfaces in Lie geometry. We find a discretization
of curvature line parametrization which unifies the circular and conical nets
by systematically applying the Discretization Principles.Comment: 57 pages, 18 figures; In the second version the terminology is
slightly changed and umbilic points are discusse
Characterizing envelopes of moving rotational cones and applications in CNC machining
Motivated by applications in CNC machining, we provide a characterization of surfaces which are enveloped by a one-parametric family of congruent rotational cones. As limit cases, we also address ruled surfaces and their offsets. The characterizations are higher order nonlinear PDEs generalizing the ones by Gauss and Monge for developable surfaces and ruled surfaces, respectively. The derivation includes results on local approximations of a surface by cones of revolution, which are expressed by contact order in the space of planes. To this purpose, the isotropic model of Laguerre geometry is used as there rotational cones correspond to curves (isotropic circles) and higher order contact is computed with respect to the image of the input surface in the isotropic model. Therefore, one studies curve-surface contact that is conceptually simpler than the surface-surface case. We show that, in a generic case, there exist at most six positions of a fixed rotational cone that have third order contact with the input surface. These results are themselves of interest in geometric computing, for example in cutter selection and positioning for flank CNC machining.RYC-2017-2264
Implicitization of curves and (hyper)surfaces using predicted support
We reduce implicitization of rational planar parametric curves and (hyper)surfaces to linear algebra, by interpolating the coefficients of the implicit equation.
For predicting the implicit support, we focus on methods that exploit input and output structure in the sense of sparse (or toric) elimination theory, namely by computing the Newton polytope of the implicit polynomial, via sparse resultant theory.
Our algorithm works even in the presence of base points but, in this case, the implicit equation shall be obtained as a factor of the produced polynomial.
We implement our methods on Maple, and some on Matlab as well, and study their numerical stability and efficiency on several classes of curves and surfaces.
We apply our approach to approximate implicitization,
and quantify the accuracy of the approximate output,
which turns out to be satisfactory on all tested examples; we also relate our measures to Hausdorff distance.
In building a square or rectangular matrix, an important issue is (over)sampling the given curve or surface: we conclude that unitary complexes offer the best tradeoff between speed and accuracy when numerical methods are employed, namely SVD, whereas for exact kernel computation random integers is the method of choice.
We compare our prototype to existing software and find that it is rather competitive
Rational rolling ball blending of natural quadrics
We construct a blending surface of two natural quadrics using rational variable rolling ball approach, i.e. as a canal surface with a rational spine curve and a rational radius. All general positions of the given quadric surfaces are considered. The proposed construction is Laguerre invariant. In particular, the blending surface has rational offset of the same degree.
Natūralių kvadrikių jungimas racionalaus apriedančio rutuliuko metodu
Santrauka
Natūralios kvadrikos (sferos, apskritiminiai cilindrai ir kūgiai) dažnai naudojamos geometriniame modeliavime. Šiame darbe siūlomas naujas dvieju natūraliu kvadrikiu glodaus jungimo metodas, naudojant kintamo racionalaus spindulio apriedančio rutuliuko metoda, t.y. jungiamasis paviršius ‐ tai kanalinis paviršius, kuris turi racionalia ašine kreive ir racionalu spinduli. Metodas tinka visiems dvieju kvadrikiu bendru poziciju atvejams. Konstrukcija yra invariantiška Laguerre geometrijos atžvilgiu: pavyzdžiui, jungiamasis paviršius turi to paties laipsnio racionalu ofseta.
First Published Online: 14 Oct 201
Spinor representation in isotropic 3-space via Laguerre geometry
We give a detailed description of the geometry of isotropic space, in
parallel to those of Euclidean space within the realm of Laguerre geometry.
After developing basic surface theory in isotropic space, we define spin
transformations, directly leading to the spinor representation of conformal
surfaces in isotropic space. As an application, we obtain the Weierstrass-type
representation for zero mean curvature surfaces, and the Kenmotsu-type
representation for constant mean curvature surfaces, allowing us to construct
many explicit examples.Comment: 30 pages, 9 figure
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