Triangulation in three dimensions: Difference between revisions

Revision as of 15:56, 24 May 2011

Revision as of 01:55, 5 June 2011

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'''Triangulation in three dimensions''' is a method of finding the ___location of a point in [[three dimensional|three dimensions]] based on other known coordinates and distances, it is commonly used in [[surveying]] and [[astronomy]].

[[Triangulation]] is also used in 2 dimensions to find the ___location of a point on a [[Plane_Plane (geometry)|plane]], this is commonly used in [[navigation]] to plot positions on a map.

===One method to triangulate a ___location in 3D===

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== Development ==

[[Image: triangulation_illust_02triangulation illust 02.gif|600px|thumb|Figure 1. The apex and its mirror reflection about the plane of ABC precipitate D and D'.]]

Three sticks of known lengths ''AD, BD, CD'' are anchored in the ground at known coordinates '''A, B, C'''. This development calculates the coordinates of the apex where the other ends of the three sticks will meet. These coordinates are given by the vector '''D'''. In the mirror case, '''D'''' is sub-apex where the three sticks would meet below the plane of '''A, B, C''' as well.

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[[image: facesABD_ACD_BCD_2facesABD ACD BCD 2.gif|700px|thumb|Figure 2. The normals are dropped on the sides from the apex and their intersections with AB, AC and BC are determined.]]

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The projection<ref>Borisenko, A. I. and Tarapov, I. E., (1968) "Vector and Tensor Analysis", General Publishing Company, p. 6. ISBN 0-486-63833-2</ref> of '''AD''' onto '''AB''' and '''AC''', and the projection of '''BD''' onto '''BC''' results in,

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[[image: Intersect_normals_ABC_3Intersect normals ABC 3.gif|700px|thumb|Figure 3. The red normals intersect at a common point.]]<br />

The three [[unit normal]]s to '''AB''', '''AC''' and '''BC''' in the [[Plane (geometry)|plane]] of ABC are:

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== Example ==

<!-- Deleted image removed: [[Image: 3 spheres2abcplane.gif|750px|thumb|Figure 4. The data is input into the equations to obtain the solution. The shaded area is the plane of ABC. A, B and C are the centers of each of the three spheres.]] -->

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[[Image:3_spheres2ABCcenters3 spheres2ABCcenters.gif|750px|thumb|Figure 5. Showing points D and D' as the result of 3 intersecting spheres at centers A, B, C from data given in Figure 4. The line adjoining D and D' imbedded in the interiors of all three spheres is the axis of symmetry of the three spheres.]]

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