An algebraic approach to curves and surfaces on the sphere and on other quadrics

Dietz, Roland; Hoschek, Josef; Jüttler, Bert

doi:10.1016/0167-8396(93)90037-4

Cited by 134 publications

(84 citation statements)

References 12 publications

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“…The implicit representations of α and α show the relations between the degrees of F and G = α(F ). Let F be given by (3). Inserting (10) into F yields…”

Section: Propositionmentioning

confidence: 99%

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Offsets, conchoids and pedal surfaces

2014

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Abstract. We discuss three geometric constructions and their relations, namely the offset, the conchoid and the pedal construction. The offset surface F d of a given surface F is the set of points at fixed normal distance d of F . The conchoid surface G d of a given surface G is obtained by increasing the radius function by d with respect to a given reference point O. There is a nice relation between offsets and conchoids: The pedal surfaces of a family of offset surfaces are a family of conchoid surfaces. Since this relation is birational, a family of rational offset surfaces corresponds to a family of rational conchoid surfaces and vice versa.We present theoretical principles of this mapping and apply it to ruled surfaces and quadrics. Since these surfaces have rational offsets and conchoids, their pedal and inverse pedal surfaces are new classes of rational conchoid surfaces and rational offset surfaces. Mathematics Subject Classification (2010). 51M,51N,53A.

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“…The implicit representations of α and α show the relations between the degrees of F and G = α(F ). Let F be given by (3). Inserting (10) into F yields…”

Section: Propositionmentioning

confidence: 99%

“…Combining universal rational parametrizations of S 2 with the respective maps ϕ and γ results in respective universal parameterizations of offset and conchoid surfaces. Following [3] one chooses four arbitrary polynomials…”

Section: The Relation Between Offsets and Conchoidsmentioning

confidence: 99%

Offsets, conchoids and pedal surfaces

2014

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“…We will need P := {[w : x : y : 0] ∈ E 3 with w, x, y ∈ R} which corresponds to the plane z = 0 in cartesian coordinates. The creation γ i uses the next projections from [3]. First, the hyperbolic projection ϑ transforms a point [e 0 : e 1 : e 2 : e 3 ] to a point in P given by ϑ(e) := e 2 0 + e 2 3 : e 0 e 1 − e 2 e 3 : e 1 e 3 + e 0 e 2 : 0 .…”

Section: Methodsmentioning

confidence: 99%

“…Hence, we want to contribute in the development of algorithms for treatments of mesh-free molecular geometries. Each atom in a molecule is represented as an imaginary sphere whose radius corresponds to the Van der Walls radius [2,3]. Let us denote by B(ω, ρ) the ball of center ω and radius ρ.…”

Section: Introductionmentioning

confidence: 99%

Molecular Surface Decomposition Using Graphical Modeling

Randrianarivony

Brunnett

Informatik Aktuell

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Abstract. We aim at decomposing molecular surfaces which are in the form of Van der Waals model into a structure adapted for mesh-free modeling. We focus on the generation of spherical patches by using rational Bézier surfaces and homogeneous coordinates. We can achieve exact global continuity by using generalized stereographic projection. We support the theoretical descriptions by practical decompositions of a few molecular surfaces.

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“…These polynomials can be found by parameterizing the corresponding quadrics. This can be done, e.g., by stereographic projection, or using the method of generalized stereographic projection [8], or its equivalent formulation via Pythagorean hodograph representation maps [5]. 2.…”

Section: Constructionmentioning

confidence: 99%

MOS Surfaces: Medial Surface Transforms with Rational Domain Boundaries

Kosinka

Jüttler

Mathematics of Surfaces XII

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Abstract. We consider rational surface patches s (u, v) in the fourdimensional Minkowski space IR 3,1 , which describe parts of the medial surface (or medial axis) transform of spatial domains. The corresponding segments of the domain boundary are then obtained as the envelopes of the associated two-parameter family of spheres. If the Plücker coordinates of the line at infinity of the (two-dimensional) tangent plane of s satisfy a sum-of-squares condition, then the two envelope surfaces are shown to be rational surfaces. We characterize these Plücker coordinates and analyze the case, where the medial surface transform is contained in a hyperplane of the four-dimensional Minkowski space.

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An algebraic approach to curves and surfaces on the sphere and on other quadrics

Cited by 134 publications

References 12 publications

Offsets, conchoids and pedal surfaces

Offsets, conchoids and pedal surfaces

Molecular Surface Decomposition Using Graphical Modeling

MOS Surfaces: Medial Surface Transforms with Rational Domain Boundaries

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