Concept# Hyperboloid

Summary

In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by deforming it by means of directional scalings, or more generally, of an affine transformation.
A hyperboloid is a quadric surface, that is, a surface defined as the zero set of a polynomial of degree two in three variables. Among quadric surfaces, a hyperboloid is characterized by not being a cone or a cylinder, having a center of symmetry, and intersecting many planes into hyperbolas. A hyperboloid has three pairwise perpendicular axes of symmetry, and three pairwise perpendicular planes of symmetry.
Given a hyperboloid, one can choose a Cartesian coordinate system such that the hyperboloid is defined by one of the following equations:
: {x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2}= 1,
or
: {x^2 \over a^2} + {y^2 \over

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Iva Bogdanova Vandergheynst, Pierre Vandergheynst

We review the known construction of the continuous wavelet transform (CWT) on the two-sphere. Next we describe the construction of a CWT on the upper sheet of a two- sheeted hyperboloid, emphasizing the similarities between the two cases. Finally we give some indications on the CWT on a paraboloid and we introduce a unified approach to the CWT on conic sections.

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In this paper we build a Continuous Wavelet Transform (CWT) on the upper sheet of the 2-hyperboloid $H_+^2$. First, we define a class of suitable dilations on the hyperboloid through conic projection. Then, incorporating hyperbolic motions belonging to $SO_0(1,2)$, we define a family of hyperbolic wavelets. The continuous wavelet transform $W_f(a,x)$ is obtained by convolution of the scaled wavelets with the signal. The wavelet transform is proved to be invertible whenever wavelets satisfy a particular admissibility condition, which turns out to be a zero-mean condition. We then provide some basic examples and discuss the limit at null curvature.

2007Iva Bogdanova Vandergheynst, Pierre Vandergheynst

We review the coherent state or group-theoretical construction of the continuous wavelet transform (CWT) on the two-sphere. Next we describe the construction of a CWT on the upper sheet of a two-sheeted hyperboloid, emphasizing the similarities between the two cases.. Finally we give some indications on the CWT on a paraboloid and we introduce a unified approach to the CWT on conic sections.

2008