Given 6 lengths l1..l6 in R,

assuming that the Tetrahedron has a face lies on the x-y plane;

and there is an edge of the Tetrahedron lies on the x-axis.

Find the 4 3D coordinates v1..v4 that define the Tetrahedron.

Thank you

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- September 10th 2009, 01:27 AMwilliamlai3a6 lengths define the 3D coordinate of a Tetrahedron
Given 6 lengths l1..l6 in R,

assuming that the Tetrahedron has a face lies on the x-y plane;

and there is an edge of the Tetrahedron lies on the x-axis.

Find the 4 3D coordinates v1..v4 that define the Tetrahedron.

Thank you - September 10th 2009, 12:17 PMOpalg
We can take and . Let , with being the lengths of the edges . Then

Multiply out the bracket in (2), and substitute (1) into (2), to get

From (3) you can find p, and then from (1) you can find q. So that determines the coordinates of .

The procedure to find the location of is similar, except that this time we have to work in three dimensions rather than two. Let , with being the lengths of the edges . Then

Multiply out the brackets in (5) and (6), and substitute (4) into those equations to replace by . That will leave you with two linear equations for a and b, which you can solve. Finally, substitute these values for a and b into (4) to find c.