TY - JOUR
T1 - Shape analysis of planar PH curves with the Gauss–Legendre control polygons
AU - Moon, Hwan Pyo
AU - Kim, Soo Hyun
AU - Kwon, Song Hwa
N1 - Publisher Copyright:
© 2020 Elsevier B.V.
PY - 2020/10
Y1 - 2020/10
N2 - Kim and Moon (2017) have recently proposed rectifying control polygons as an alternative to Bézier control polygons and a way of controlling planar PH curves by the rectifying control polygons. While a Bézier control polygon determines a unique polynomial curve, a rectifying control polygon gives a multitude of PH curves. This multiplicity of PH curves naturally raises the selection problem of the “best” PH curves, which is the main topic of this paper. To resolve the problem, we first classify PH curves of degree 2n+1 into 2n subclasses by defining the types of PH curves, and propose the absolute hodograph winding number as a topological index to characterize the topological behavior of PH curves in shape. We present a lower bound of the topological index of a PH curve which is given solely by its type, and prove the uniqueness of the best PH curve by exploiting it. The existence theorems are also proved for cubic and quintic PH curves. Finally, we propose a selection rule of the best PH curve only based on its type.
AB - Kim and Moon (2017) have recently proposed rectifying control polygons as an alternative to Bézier control polygons and a way of controlling planar PH curves by the rectifying control polygons. While a Bézier control polygon determines a unique polynomial curve, a rectifying control polygon gives a multitude of PH curves. This multiplicity of PH curves naturally raises the selection problem of the “best” PH curves, which is the main topic of this paper. To resolve the problem, we first classify PH curves of degree 2n+1 into 2n subclasses by defining the types of PH curves, and propose the absolute hodograph winding number as a topological index to characterize the topological behavior of PH curves in shape. We present a lower bound of the topological index of a PH curve which is given solely by its type, and prove the uniqueness of the best PH curve by exploiting it. The existence theorems are also proved for cubic and quintic PH curves. Finally, we propose a selection rule of the best PH curve only based on its type.
KW - Gauss–Legendre control polygon
KW - Gauss–Legendre quadrature
KW - Hodograph winding number
KW - Pythagorean hodograph curve
KW - Rectifying control polygon
UR - http://www.scopus.com/inward/record.url?scp=85089554438&partnerID=8YFLogxK
U2 - 10.1016/j.cagd.2020.101915
DO - 10.1016/j.cagd.2020.101915
M3 - Article
AN - SCOPUS:85089554438
SN - 0167-8396
VL - 82
JO - Computer Aided Geometric Design
JF - Computer Aided Geometric Design
M1 - 101915
ER -