TY - JOUR
T1 - G1 Hermite interpolation method for spatial PH curves with PH planar projections
AU - Song, Yoonae
AU - Kim, Soo Hyun
AU - Moon, Hwan Pyo
N1 - Publisher Copyright:
© 2022 Elsevier B.V.
PY - 2022/8
Y1 - 2022/8
N2 - The research subject of this paper is the spatial Pythagorean hodograph (PH) curves whose projections to the horizontal plane are planar PH curves. Because of this geometric configuration, we name them PH curves over PH curves, or PHoPH curve. We investigate the algebraic structure of PHoPH curves and show that their hodographs are obtained by applying two squaring maps successively to quaternion generator polynomials. The simplest nontrivial PHoPH curves generated from linear quaternion generators are quintic curves, which have adequate degrees of freedom to solve the G1 Hermite interpolation problem. From the algebraic structure, we can derive a system of nonlinear equation for G1 interpolation, which is addressable by numerical methods. We also suggest the choice of initial values for the numerical method. The solvability is not guaranteed for arbitrary G1 data in general, however, we show the feasibility of the system for the G1 data taken from a small segment of reference curves without inflection points using extensive Monte-Carlo simulation. We also present a few illustrative examples of PHoPH spline curves that approximate the given reference curves.
AB - The research subject of this paper is the spatial Pythagorean hodograph (PH) curves whose projections to the horizontal plane are planar PH curves. Because of this geometric configuration, we name them PH curves over PH curves, or PHoPH curve. We investigate the algebraic structure of PHoPH curves and show that their hodographs are obtained by applying two squaring maps successively to quaternion generator polynomials. The simplest nontrivial PHoPH curves generated from linear quaternion generators are quintic curves, which have adequate degrees of freedom to solve the G1 Hermite interpolation problem. From the algebraic structure, we can derive a system of nonlinear equation for G1 interpolation, which is addressable by numerical methods. We also suggest the choice of initial values for the numerical method. The solvability is not guaranteed for arbitrary G1 data in general, however, we show the feasibility of the system for the G1 data taken from a small segment of reference curves without inflection points using extensive Monte-Carlo simulation. We also present a few illustrative examples of PHoPH spline curves that approximate the given reference curves.
KW - G Hermite interpolation
KW - Monte-Carlo simulation
KW - PH curve
KW - PHoPH curve
KW - Quaternion representation
UR - http://www.scopus.com/inward/record.url?scp=85134412249&partnerID=8YFLogxK
U2 - 10.1016/j.cagd.2022.102132
DO - 10.1016/j.cagd.2022.102132
M3 - Article
AN - SCOPUS:85134412249
SN - 0167-8396
VL - 97
JO - Computer Aided Geometric Design
JF - Computer Aided Geometric Design
M1 - 102132
ER -