Finding the Equation of a 3D Line from Two Points

[tandoorichicken]

How do you find the equation of a line between two points in three dimensional space? I sort of forgot. =\

 

[0rthodontist]

c(t) = (P-Q)t+P

 

[d_leet]

c(t) = (P-Q)t+P

Shouldn't that be c(t) = (P-Q)t+Q

because at t = 0 you should get Q, and at t = 1 you should get P which isn't what happens in your equation.

 

[0rthodontist]

It really doesn't matter how you parametrize it so long as it is the same line. c(t) = .3455(P-Q)t+(P+Q)/2 is equally valid.

 

[HallsofIvy]

Shouldn't that be c(t) = (P-Q)t+Q

because at t = 0 you should get Q, and at t = 1 you should get P which isn't what happens in your equation.

Orthodontist's form, c= (P- Q)t+ P, gives P when t=0 and Q when t= -1. That perfectly valid.

By the way, Tandoorichicken, since a line is one-dimensional, in a three dimensional space you need more than one equation. Orthodontist gave a "vector" equation where P and Q are the position vectors of two points and t is a parameter. Writing that in component form gives three parametric equations. Given the single variable t, you can calculate x, y, and z.