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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 51F: Metric Geometry

Introduction

History

Applications and related fields

Subfields

• 51F05: Absolute planes
• 51F10: Absolute spaces
• 51F15: Reflection groups, reflection geometries, See also 20H10, 20H15; for Coxeter groups, See 20F55
• 51F20: Congruence and orthogonality, See also 20H05
• 51F25: Orthogonal and unitary groups, See also 20H05
• 51F99: None of the above but in this section

Parent field: 51: Geometry

Browse all (old) classifications for this area at the AMS.

Textbooks, reference works, and tutorials

Software and tables

Other web sites with this focus

Selected topics at this site

A common question is this: how can you determine positions if you know the distances to three other points? A couple of relevant threads:

• Given 3 known points in 3-space, and the distances from each known point to an unknown point, how to determine the position of the unknown point?
• Fortran code to determine location by triangulation. [Author hasn't given permission to post his name.]
• Determine present location from distances to three fixed points.
• Using the Cholesky factorization of a matrix to find an isometric embedding of a finite set of points.

Here's an interesting mix of practical and theoretical considerations, of algebra and geometry. A USENET poster asked if one could determine positions within a triangle knowing the angles of sighting to the vertices of the triangle, and how one could compute this.

• The original posting and a (pretty predictable) response. In this discussion the problem was purely geometrical.
• An explanation of the practical origins of the triangulation problem.
• A follow-up post giving a ruler-and-compass solution to the triangulation problem at hand.
• Calculating position from three angles to fixed locations

Other files for 51F:

• Representing a rotation in R^3 using rotations around only two axes
• Extracting the axis of rotation from a 3x3 orthogonal matrix.
• Proving the analogue of the Pythagorean theorem in higher dimensions.
• Determine a Euclidean motion from shadows of 5 moved points
• Finding the Euclidean isometry best matching n pre- and post-point pairs
• When can this freely-moving linkage be constructed?
• Determine locations of points given all bipartite distances
• Points in the plane with only two distances between them