RE: [Flightgear-devel] Simple Question for the Geometricians

[BERNDT, J. S. (J. (JSC-E. (LM) <jon...@js...>]

> I know that we have a few math/geometry hacks lurking on the list, so
> here's what I hope will be a simple problem:
> 
>   Given a line segment in 3D space defined by the points (x1, y1, z1)
>   and (x2, y2, z2), find a point (x3, y3, z3) on the segment where
>   only the z3 value is known.  It's safe to assume that z3 falls
>   between z1 and z2, and that z1 != z2.
> 
> All answers will be appreciated, but the appreciation for algorithmic
> answers will be more sincere than the appreciation for formulaic
> answers.

It's sort of a variation on similar triangles. The percentage of distance
that z3 falls along the whole distance z1z2 is the same as how far x3 will
fall along the distance x1x2:

x3 = [ (z3-z1)/(z2-z1)) * (x2-x1) ] + x1

-or-

factor = (z3-z1)/(z2-z1)

x3 = factor * (x2 - x1) + x1
y3 = factor * (y2 - y1) + y1
z3 = factor * (z2 - z1) + z1

Jon