The fundamental theorem in affine hypersurface theory

1988 ◽  
Vol 26 (2) ◽  
Author(s):  
Udo Simon
Keyword(s):  
Fundamental Theorem ◽  
Affine Hypersurface

On some definite integrals and a new method of reducing a function of spherical co-ordinates to a series of spherical harmonics

1903 ◽  
Vol 71 (467-476) ◽  
pp. 97-101 ◽  
Keyword(s):  
Spherical Harmonics ◽  
Fundamental Theorem ◽  
New Method ◽  
Definite Integrals

The expansion of a function f(θ) of an angle θ varying between 0 and π in terms of a series proceeding by the sines of the multiples of θ depends on the fundamental theorem, ∫ π 0 sin pθ sin qθ dθ = 0, where p and q are integer numbers not equal to each other.

scholarly journals Almost vanishing polynomials and an application to the Hough transform

2014 ◽  
Vol 13 (08) ◽  
pp. 1450057 ◽  
Author(s):  
Maria-Laura Torrente ◽  
Mauro C. Beltrametti
Keyword(s):  
Hough Transform ◽  
Affine Space ◽  
Bounded Region ◽  
Pattern Recognition Technique ◽  
Local Study ◽  
Affine Hypersurface ◽  
Automated Recognition ◽  
Real Affine ◽  
Necessary And Sufficient ◽  
Recognition Technique

We consider the problem of deciding whether or not an affine hypersurface of equation f = 0, where f = f(x1, …, xn) is a polynomial in ℝ[x1, …, xn], crosses a bounded region 𝒯 of the real affine space 𝔸n. We perform a local study of the problem, and provide both necessary and sufficient numerical conditions to answer the question. Our conditions are based on the evaluation of f at a point p ∈ 𝒯, and derive from the analysis of the differential geometric properties of the hypersurface z = f(x1, …, xn) at p. We discuss an application of our results in the context of the Hough transform, a pattern recognition technique for the automated recognition of curves in images.

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