Introduction to Differential Geometry of Space Curves and Surfaces by Sochi Taha

Author:Sochi, Taha [Sochi, Taha]
Language: eng
Format: azw3, epub, mobi
Publisher: Createspace
Published: 2017-07-03T16:00:00+00:00

3.10  Sphere Mapping

Sphere mapping or Gauss mapping is a correlation between the points of a surface and the unit sphere where each point on the surface is projected onto its unit normal as a point on the unit sphere which is centered at the origin of coordinates. This sort of mapping for surfaces is similar to the spherical indicatrix mapping (see § 5.5↓) for space curves. In technical terms, let S be a surface embedded in an ℝ3 space and S1 represents the origin-centered unit sphere in this space, then Gauss mapping is given by: (302) {N:S → S1,  N(P) = P̌}

where the point P(x, y, z) on the trace of S is mapped by N onto the point P̌(x̌, y̌, ž) on the trace of the unit sphere with x, y, z being the coordinates of P and x̌, y̌, ž being the coordinates of the origin-based position vector of the normal vector to the surface, n, at P. To have a single-valued sphere mapping, the functional relation representing the surface S should be one-to-one.

The image D̃ on the unit sphere of a Gauss mapping of a patch D on a surface S is called the spherical image of D. The limit of the ratio of the area of a region Z̃ on the spherical image to the area of the corresponding region Z on the surface S in the neighborhood of a given point P on S equals the absolute value of the Gaussian curvature |K| at P as Z shrinks to the point P, that is: (303)

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