Define function that goes from one point to another parametrically

Question

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:

enter image description here

@Kuba That gives the complete line. I want the line to be defined from one point to another, not outside. — Arcotick, Apr 22 '14 at 10:10
@Arcotick You just got to restrict t to between 0 and 1. (You can use Piecewise for this.) — C. E., Apr 22 '14 at 10:14
I'm sorry I don't know what do you want. Please specify arguments/domain and expected result/codomain. — Kuba, Apr 22 '14 at 10:15
@Pickett But I want the function itself to be undefined when 1<t<0 — Arcotick, Apr 22 '14 at 10:18
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
Why can't you just use Line in your plot? You can insert Graphics into a plot with Prolog or 'Epilog`. — m_goldberg, Apr 22 '14 at 10:22
@m_goldberg Because I want to get the intersection point of that line and another function. And I need an equation for that. — Arcotick, Apr 22 '14 at 10:25
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

score 1 · Accepted Answer · edited Apr 13 '17 at 12:55

line[t_, p1_, p2_] := p1 + t (p2 - p1)

Solve[Join[
        {line[t1, {0, 0}, {1, 1}] == line[t2, {0, 1}, {1.2, 0}]},
        {0 <= t1 <= 1, 0 <= t2 <= 1}],
      {t1, t2}]

{{t1 -> 0.545455, t2 -> 0.454545}}

{line[t1, {0, 0}, {1, 1}], line[t2, {0, 1}, {1.2, 0}]} /. %

{{{0.545455, 0.545455}, 
  {0.545455, 0.545455}}}

Implementation of Balaban's Line intersection algorithm in Mathematica is also very closely related.

Define function that goes from one point to another parametrically

1 Answers1