Triangulation in three dimensions: Difference between revisions

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In [[vector analysis]], [[triangulation]] is a method of finding points in [[three dimensional]] spaces using distances, angles and vector functions such as [[magnitude]], [[dot product]] and [[cross product]]. Among its uses are in [[surveying]], [[navigation]], and [[astronomy]].

This article describes a method for determining the [[coordinates]] of the point where three lines meet, given the [[scalar]] lengths of the lines and the coordinates of their bases. If these three lines are the radii of three [[sphere]]s of known centers, this method can be used to calculate the intersection of the three spheres ''if they intersect''. In the event that the three spheres don't intersect, this method obtains the closest solution to the axis of symmetry between three spheres.

== Development ==

Suppose threeThree sticks of known lengthlengths ''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 the vector to the coordinates of the sub-apex where the three sticks would meet below the plane of '''A, B, C''' as well. This problem also calculates the axis of symmetry of three intersecting spheres. See the article [[Trilateration|Trilateration]] for the spherical case. Even if the three spheres don't intersect, this problem still calculates their axis of symmetry. This axis is the line drawn between '''D''' and '''D''''. To proceed, three [[sphere]]s of known centers '''A, B, C''' and known [[Radius|radii]] ''AD, BD, CD'' intersect at two points '''D''' and '''D''''. Similarly, three sticks of known lengths ''AD, BD, CD'' are planted in the ground at known [[coordinates]] '''A, B, C'''. The other ends meet at a calculated [[apex]]. Calculating '''D''' and '''D'''', 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,

Calculating '''D''' and '''D'''', 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: triangulation_illust_02.gif]]