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2011•Unpublished venueRequires access

Intrinsic Geometry of Intersection Curves of Surfaces

Sayed Abdel-Naeim Badr, Nassar H. Abdel-All, M. A. Soliman

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Abstract

This book is motivated by surface-surface intersection problems (SSI) in R³, self-intersection surfaces problem in R³ and three hypersurfaces intersection problems (SSSI) in R⁴. It presents algorithms for computing all the differential geometry properties of Frenet apparatus (t,n,b,κ,τ) of intersection curves of surfaces in R³ for transversal and tangential intersection. The surfaces may be implicit parameterization or parametric parameterization. It is also presents algorithms for computing all the differential geometry properties of intersection curves of Frenet apparatus (t,n,b₁,b₂,κ₁,κ₂,κ₃) of three hypersurfaces in R⁴ for transversal intersection. Some techniques for 4D visualization of objects in R⁴ have been given. Also some applications have been given and plotted.

About this research paper

What this paper is about

This book is motivated by surface-surface intersection problems (SSI) in R³, self-intersection surfaces problem in R³ and three hypersurfaces intersection problems (SSSI) in R⁴. It presents algorithms for computing all the differential geometry properties of Frenet apparatus (t,n,b,κ,τ) of intersection curves of surfaces in R³ for transversal and tangential intersection. The surfaces may be implicit parameterization or parametric parameterization. It is also presents algorithms for computing all the differential geometry properties of intersection curves of Frenet apparatus (t,n,b₁,b₂,κ₁,κ₂,κ₃) of three hypersurfaces in R⁴ for transversal intersection. Some techniques for 4D visualization of objects in R⁴ have been given. Also some applications have been given and plotted.

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Available abstract

This book is motivated by surface-surface intersection problems (SSI) in R³, self-intersection surfaces problem in R³ and three hypersurfaces intersection problems (SSSI) in R⁴. It presents algorithms for computing all the differential geometry properties of Frenet apparatus (t,n,b,κ,τ) of intersection curves of surfaces in R³ for transversal and tangential intersection. The surfaces may be implicit parameterization or parametric parameterization. It is also presents algorithms for computing all the differential geometry properties of intersection curves of Frenet apparatus (t,n,b₁,b₂,κ₁,κ₂,κ₃) of three hypersurfaces in R⁴ for transversal intersection. Some techniques for 4D visualization of objects in R⁴ have been given. Also some applications have been given and plotted.

Key concepts: Intersection (aeronautics), Frenet–Serret formulas, Parametric surface, Geometry, Transversal (combinatorics), Mathematics, Surface (topology), Differential geometry

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