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# Maths -- 3d lines Revisited

infinisa
Hello AaronHarrison

 Quote: How, mathematically, can you describe a line in 3-space. z = mx + c = ny + d ?? Does this work? I need to know for some 3d programming: once I have the equation intersections will be easy...

You almost got it right, but the formula is not general enough (just as y=mx + c doesn't cover the case x=0).

The general formula for a line in 3 dimensions is:

(x - a)/p = (y - b)/q = (z - c)/r
[= real parameter t - see below]

In this formula, (a, b, c) are the coordinates of a point the line passes through;
(p, q, r) are the coordinates of a direction vector along the line.

In fact it is easy to see that this is equivalent to the vector equation r = a + td,
where
r is the vector (x, y, z),
a is the vector (a, b, c),
t is a real parameter (can take any real value),
d is the vector (p, q, r).

The vector equation is valid in any number of dimensions, and its cartesian version can easily be generalised to any number of dimensions - including 2.

BTW, I can't understand why Bikerman locked your topic, as your question is about 3D, and his link was to the 2D case.

Hope this helps!
Bikerman
I thought the 3d case was considered in the other thread which is why I redirected to it.
Did I make a boo-boo?