Construction of spline curves on smooth manifolds by action of Lie groups

AbstractPolynomials for blending parametric curves in Lie groups are defined. Properties of these polynomials are proved. Blending parametric curves in Lie groups with these polynomials is considered. Then application of the proposed technique to construction of spline curves on smooth manifolds is presented. As an example, construction of spherical spline curves using the proposed approach is depicted.

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A Tool-Path Interpolation Algorithm Based on Unfolding-Bending Parametric Curves

Key Engineering Materials ◽

10.4028/www.scientific.net/kem.407-408.220 ◽

2009 ◽

Vol 407-408 ◽

pp. 220-224

Author(s):

Li Cheng Fan ◽

Li Ning Sun ◽

Zhi Jiang Du

Keyword(s):

Tool Path ◽

Discrete Data ◽

Sculptured Surface ◽

Interpolation Algorithm ◽

Parametric Curves ◽

Chord Error ◽

B Spline ◽

Spline Curves ◽

Cutting Vibration ◽

Simulation Results

To grind sculptured surface by swing motion, it’s needed to interpolate the discrete data nodes on the arc-driving surface. To ensure that all the sample points locate on the arc surface, a novel unfolding parameter curve is proposed. Firstly, all the discrete data are stretched onto a plane, inside which the cubic B-spline curves are interpolated. After that, some points are sampled from the spline curves and bended to the arc-driving surface according to the swing radius. To eliminate cutting vibration and chord error exceeds limitation; radial speed is tuned by controlling the chord error. The simulation results are given to prove the proposed algorithm.

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Almost Hypercomplex Manifolds with Hermitian-Norden Metrics and 4-dimensional Indecomposable Real Lie Algebras Depending on One Parameter

10.20944/preprints202001.0267.v1 ◽

2020 ◽

Author(s):

Hristo Manev

Keyword(s):

Lie Groups ◽

Lie Algebras ◽

Smooth Manifolds ◽

Geometrical Characteristics ◽

Hypercomplex Structure ◽

Norden Metrics

We study almost hypercomplex structure with Hermitian-Norden metrics on 4-dimensional Lie groups considered as smooth manifolds. All the basic classes of a classification of 4-dimensional indecomposable real Lie algebras depending on one parameter are investigated. There are studied some geometrical characteristics of the respective almost hypercomplex manifolds with Hermitian-Norden metrics.

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