Stereographic projection

Summary

In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the pole or center of projection), onto a plane (the projection plane) perpendicular to the diameter through the point. It is a smooth, bijective function from the entire sphere except the center of projection to the entire plane. It maps circles on the sphere to circles or lines on the plane, and is conformal, meaning that it preserves angles at which curves meet and thus locally approximately preserves shapes. It is neither isometric (distance preserving) nor equiareal (area preserving). The stereographic projection gives a way to represent a sphere by a plane. The metric induced by the inverse stereographic projection from the plane to the sphere defines a geodesic distance between points in the plane equal to the spherical distance between the spherical points they represent. A two-dimensional coordinate system on the stereographic plane is an alternativ

Official source

https://en.wikipedia.org/wiki/Stereographic_projection

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CAVE systems are nowadays one of the best Virtual Reality (VR) immersive devices available for rendering virtual environments. Unfortunately, such kind of hardware is extremely expensive, complex and cum-bersome, thus limited in its spread. Several cheaper solutions already exist, but they implement usually only a subset of features of a professional CAVE (like a reduced number of wall-displays, no stereographic at all or with a low-quality support, low brightness, etc.). In this paper we describe how we have built a low cost CAVE with four screens (three walls and a floor), stereographic rendering and user tracking by only using hardware commonly available on the market and free software, we show the different solutions and work-around we implemented to solve the problems we encountered and we conclude with a fair evaluation of our system by using two applications we developed with it.

2007

We are developing a Virtual Reality based training system for micromanipulation in collaboration with the National University Hospital in Singapore. While conventional approaches judge a virtual environment by its resemblance to the real environment, we use simple environments with only selected features of the real task, and develop fast algorithms to investigate the learning of dexterity primitives using various multi-sensory cues. For the needle maneuvering primitive, this paper introduces a method using stereographic projection to compute the distance between the curved needle and a curved tube, necessary to investigate multi-sensory cues systematically. This analytical algorithm is shown to be faster by orders of magnitude than numerical ones. The computation time barely increases with increasing precision, a critical condition to simulate the microworld.

2004

Wavelet Transform Analysis of Truncated Fringe Patterns in 3-D Surface Profilometry

Sai Siva Gorthi

Wavelet transform analysis of projected fringe pattern for phase recovery in 3-D shape measurement of objects is investigated. The present communication specifically outlines and evaluates the errors that creep in to the reconstructed profiles when fringe images do not satisfy periodicity. Three specific cases that give raise to non-periodicity of fringe image are simulated and leakage effects caused by each one of them are analyzed with continuous complex Morlet wavelet transform. Same images are analyzed with FFT method to make a comparison of the reconstructed profiles with both methods. Simulation results revealed a significant advantage of wavelet transform profilometry (WTP), that the distortions that arise due to leakage are confined to the locations of discontinuity and do not spread out over the entire projection as in the case of Fourier transform profilometry (FTP).

2005