IBM Journal of Research and Development
An algebraic approach to curves and surfaces on the sphere and on other quadrics
Selected papers of the international symposium on Free-form curves and free-form surfaces
Fundamentals of computer aided geometric design
Fundamentals of computer aided geometric design
The conformal map z→z2 of the hodograph plane
Computer Aided Geometric Design
Hermite interpolation by Pythagorean hodograph quintics
Mathematics of Computation
Geometric Hermite interpolation with Tschirnhausen cubics
Journal of Computational and Applied Mathematics
Applications of Laguerre geometry in CAGD
Computer Aided Geometric Design
Minkowski pythagorean hodographs
Computer Aided Geometric Design
Construction and shape analysis of PH Hermite interpolants
Computer Aided Geometric Design
Hermite interpolation by pythagorean hodograph curves of degree seven
Mathematics of Computation
Structural invariance of spatial Pythagorean hodographs
Computer Aided Geometric Design
C1 Hermite interpolation using MPH quartic
Computer Aided Geometric Design
Clifford algebra, Lorentzian geometry, and rational parametrization of canal surfaces
Computer Aided Geometric Design
G2 curve design with a pair of Pythagorean Hodograph quintic spiral segments
Computer Aided Geometric Design
Evolution-based least-squares fitting using Pythagorean hodograph spline curves
Computer Aided Geometric Design
Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable
Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable
G1 Hermite interpolation by Minkowski Pythagorean hodograph cubics
Computer Aided Geometric Design
Euclidean and Minkowski Pythagorean hodograph curves over planar cubics
Computer Aided Geometric Design
On rational Minkowski Pythagorean hodograph curves
Computer Aided Geometric Design
A unified Pythagorean hodograph approach to the medial axis transform and offset approximation
Journal of Computational and Applied Mathematics
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In our contribution we study cubic and quintic Pythagorean Hodograph (PH) curves in the Euclidean and Minkowski planes. We analyze their control polygons and give necessary and sufficient conditions for cubic and quintic curves to be PH. In the case of Euclidean cubics the conditions are known and we provide a new proof. For the case of Minkowski cubics we formulate and prove a new simple geometrical condition. We also give conditions for the control polygons of quintics in both types of planes. Moreover, we introduce the new notion of the preimage of a transformation, which is closely connected to the so-called preimage of a PH curve. We determine which transformations of the preimage curves produce similarities of PH curves in both Euclidean and Minkowski plane. Using these preimages of transformations we provide simple proofs of the known facts that up to similarities there exists only one Euclidean PH cubic (the so-called Tschirnhausen cubic) and two Minkowski PH cubics. Eventually, with the help of this novel approach we classify and describe the systems of Euclidean and Minkowski PH quintics.