Define function that goes from one point to another parametrically

I have two points p1 and p2, and I want to get the parametric equation of a linear function that passes through those, but ranging from one point to another.

For example, if I have 2 points {0,0} and {1,1/2}, the equation would be {t,t/2}.

The goal of having a parametric equation is to have the function defined even when the slope is $\infty$.

I tried this (here the angle[] function is Atan2, is basically ArcTan but ranging from 0-2π):

Piecewise[
{
{{t, Last@p1 + Tan@angle[p1, p2] (t - First@p1)},
Min[First@p1, First@p2] <= t && t <= Max[First@p1, First@p2]},
{{First@p1, t}, First@p1 == First@p2}
},
Indeterminate
]

But when a point is on top of the other the function is a vertical line, ranging from $-\infty$ to $\infty$; I would only want the function to be defined from Last@p1 to Last@p2.

In other words, I want a Graphics' Line, but using parametric equations.

Something like this: • p1 + t (p2 - p1) ?
– Kuba
Apr 22 '14 at 10:07
• @Kuba That gives the complete line. I want the line to be defined from one point to another, not outside. Apr 22 '14 at 10:10
• @Arcotick You just got to restrict t to between 0 and 1. (You can use Piecewise for this.) Apr 22 '14 at 10:14
• line[t_] := p1 + t (p2 - p1) /; 0 <= t <= 1 but your question is still unclear. Please refer to my previous question. And focus on what you are writing.
– Kuba
Apr 22 '14 at 10:21
• You can still proceed with the function I've given, just Solve/NMinimize for t1, t2 with restrictions. Also, take a look at Line intersection algorithm
– Kuba
Apr 22 '14 at 10:35