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Concept# Spherical coordinate system

Summary

In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin; its polar angle measured from a fixed polar axis or zenith direction; and the azimuthal angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the fixed axis, measured from another fixed reference direction on that plane.
When radius is fixed, the two angular coordinates make a coordinate system on the sphere sometimes called spherical polar coordinates.
The radial distance is also called the radius or radial coordinate. The polar angle may be called colatitude, zenith angle, normal angle, or inclination angle.
The polar angle is often replaced by the elevation angle measured from the reference plane towards the positive Z axis; the depression angle is the negative of the elevation angle.
The use of symbols and the o

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Anaïs Marie Rassat, Jean-Luc Starck

Context. Future cosmological surveys will provide 3D large scale structure maps with large sky coverage, for which a 3D spherical Fourier-Bessel (SFB) analysis in spherical coordinates is natural. Wavelets are particularly well-suited to the analysis and denoising of cosmological data, but a spherical 3D isotropic wavelet transform does not currently exist to analyse spherical 3D data.

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In this paper, a complete system is presented which mimics a QWERTY keyboard on an arbitrary surface. The system consists of a pattern projector and a true-3D range camera for detecting the typing events. We exploit depth information acquired with the 3D range camera and detect the hand region using a pre-computed reference frame. The fingertips are found by analyzing the hands' contour and fitting the depth curve with different feature models. To detect a keystroke, we analyze the feature of the depth curve and map it back to a global coordinate system to find which key was pressed. These steps are fully automated and do not require human intervention. The system can be used in any application requiring zero form factor and minimized or no contact with a medium, as in a large number of cases in human-to-computer interaction, virtual reality, game control, 3D designs, etc.

2005