Midpoint Formula for any Fraction

Date: 01/26/99 at 00:04:23
From: T.J.
Subject: Rectangular coordinate system

I need to come up with a formula similar to the midpoint formula. I 
need this formula to find "any fractional part" of the line from P1 
to P2.

This is the midpoint formula:

        X1 + X2             Y1 + Y2 
   Xm = -------  and  Ym =  -------
           2                   2

The formula I came up with is:
                
        X1 + X2            Y1 + Y2
   Xa = -------  and  Ya = -------
           N                  N

where N represents any fraction of the line and Xa is the point on the 
line. Will this formula actually work? 
   
This is where I am stuck. 
Thanks!

Date: 01/26/99 at 12:05:52
From: Doctor Peterson
Subject: Re: Rectangular coordinate system

Hi, T.J. Let's start out in one dimension to make it easier. We have 
two points A and B, and want to find a point C that is F = m/n of the 
way from one to the other. For instance, F might be 2/3. Here's what 
we have:

    0  A                     C      B
    +--+---------------------+------+----
       |<-------------------------->|
       |             B-A     |      |
       |<------------------->|      |
       |       F*(B-A)       |      |

The coordinate of point C is

    A + F*(B-A) = A + F*B - F*A = (1-F)*A + F*B)

So the point you are looking for in your two-dimensional problem is

    Xf = (1-F)*X1 + F*X2
    Yf = (1-F)*Y1 + F*Y2

If you have ever used weighted averages, you'll see that that is just 
what this formula is. The midpoint is a simple average (mean), while 
your fractional point is the weighted average of the endpoints.
                            
- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/