Triangulation in three dimensions: Difference between revisions

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Revision as of 18:23, 22 May 2011

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== Development ==

[[Image: triangulation_illust_02.gif|700px600px|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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By the [[law of cosines]],

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(BD)^2 = (AB)^2+(AD)^2-2(AB)(AD)\cos(\angle{BAD})

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(CD)^2 = (AC)^2+(AD)^2-2(AC)(AD)\cos(\angle{CAD})

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(CD)^2 = (BC)^2+(BD)^2-2(BC)(BD)\cos(\angle{CBD})

</math><br /><br />

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[[image: facesABD_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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