The past decade has witnessed sustained interest in elucidating the basic theory of Pythagorean–hodograph (PH) curves, developing construction algorithms, formulating generalizations, and investigating applications. The rapid pace of this activity, encompassing diverse lines of inquiry, makes it desirable at this point to take a broad perspective of these recent developments, and assess their relationships. In the present article, we aim to address this need by categorizing recent results into a number of broad themes—extensions and specializations of the basic polynomial PH curves; rational orthonormal frames along spatial PH curves; construction and analysis algorithms for PH curves; surface design based on PH curves; and the use of PH curves in practical applications.

New Developments in Theory, Algorithms, and Applications for Pythagorean–Hodograph Curves

https://cris.unibo.it/bitstream/11585/614146/5/PHB_planar.pdf

Planar Pythagorean-Hodograph B-Spline curves

We introduce a new class of Pythagorean-Hodograph (PH) space curves - called Algebraic-Trigonometric Pythagorean-Hodograph (ATPH) space curves - that are defined over a six-dimensional space mixing algebraic and trigonometric polynomials. After providing a general definition for this new class of curves, their quaternion representation is introduced and the fundamental properties are discussed. Then, as previously done with their quintic polynomial counterpart, a constructive approach to solve the first-order Hermite interpolation problem in ℝ3 is provided. Comparisons with the polynomial case are illustrated to point out the greater flexibility of ATPH curves with respect to polynomial PH curves.

/pdf/algebraic-trigonometric-pythagorean-hodograph-space-curves-e5w3s93ypl.pdf

Algebraic-Trigonometric Pythagorean-Hodograph space curves

In this paper, we construct $$C^2$$
 Algebraic-Trigonometric Pythagorean Hodograph (ATPH) splines by solving a non-linear system of equations in complex variables. We compare these splines, which depend on several shape parameters, with their polynomial PH counterpart as well as with the well-known $$C^2$$
 cubic B-splines. We finally present criteria for choosing the free shape parameters based on the minimization of certain fairness functionals.

Design of $$C^2$$ C 2 algebraic-trigonometric pythagorean hodograph splines with shape parameters

Singular cases of planar and spatial C1 Hermite interpolation problems based on quintic Pythagorean-hodograph curves

In this article we define a new class of Pythagorean-Hodograph curves built-upon a six-dimensional mixed algebraic-trigonometric space, we show their fundamental properties and compare them with their well-known quintic polynomial counterpart. A complex representation for these curves is introduced and constructive approaches are provided to solve different application problems, such as interpolating C 1 Hermite data and constructing spirals as G 2 transition elements between a line segment and a circle, as well as between a pair of external circles.

/pdf/algebraic-trigonometric-pythagorean-hodograph-curves-and-3tdxwgv1r3.pdf

Algebraic-Trigonometric Pythagorean-Hodograph curves and their use for Hermite interpolation

This paper deals with Pythagorean hodograph curves of the spaces of algebraic trigonometric functions and trigonometric polynomials, $\text {span}\left \{1,t,\left \{{\cos \limits } \left ({kt}\right ), {\sin \limits } \left ({kt}\right )\right \}_{k=1}^{m}\right \} $ and $ \text {span}\left \{1,\left \{{\cos \limits } \left ({kt}\right ), {\sin \limits } \left ({kt}\right )\right \}_{k=1}^{m}\right \}, $ respectively. First, we propose a general characterization of planar PH curves in these spaces. Next, we consider the particular cases m = 1 and m = 2. For each of them, we give the general form of the control polygon of the PH curves, the implicit relations defining these curves, and their geometrical interpretations. Some examples and particular cases complete this study.

Characterization and extensive study of cubic and quintic algebraic trigonometric planar PH curves

Animation 3D: le problème de mouvement de camera

RESUME. L’article presente un systeme qui permet la simulation 3D suivie de son execution sur un plateau d’un mouvement de camera realiste pour la stop motion. Les travaux de recherche portent sur deux points : un etat de l’art des outils actuels pour produire des mouvements de camera en animation stop motion ; les moyens de reproduire sur un plateau de stop motion une animation faite en 3D. Le systeme, en cours de developpement, permettra a terme a la chaine des acteurs impliques dans une animation stop motion de previsualiser et de produire des mouvements de camera realistes en 3D. Ces mouvements, parametrables interactivement grâce a une interface haptique, reproductibles a l’identique et modifiables au cours de la prise de vue, donnent a la mise en scene une liberte accrue. MOTS-CLES : animation, stop motion, mouvement de camera, contraintes, simulation realiste.

Animation 3D: mouvements de caméra réalistes pour la stop motion

This paper deals with the construction of the Algebraic Trigonometric Pythagorean Hodograph (ATPH) cubic-like Hermite interpolant. A characterization of solutions according to the tangents at both ends and a global free shape parameter α is performed. Since this degree of freedom can be used for adjustments, we study how the curve evolves with respect to α. Several examples illustrating the construction process and a simple fitting method to determine the unique ATPH curve passing through a given point are proposed. 

/pdf/hermite-interpolation-by-planar-cubic-like-atph-3dr7xksy.pdf

Laura Saini

Papers

Algebraic-Trigonometric Pythagorean-Hodograph curves and their use for Hermite interpolation

Characterization and extensive study of cubic and quintic algebraic trigonometric planar PH curves

Animation 3D: le problème de mouvement de camera

Animation 3D: mouvements de caméra réalistes pour la stop motion

Hermite interpolation by planar cubic-like ATPH