Using linear regression to find the intersection of two 2-d

lines.

------------------------------------------------------------------

Linear regression can be used to find the line through two

2-d points as follows:

1. Clear the stats.

2. Do y1 ENTER x1 sigma+.

3. Do y2 ENTER x2 sigma+.

Then (on the 33s), do L.R.; the line is y = m*x + b, with m

and b displayed.

To get the value into the x-reg, get the value displayed

and press ENTER.

Since, projectively, two lines determine a point just as

well as two points

determine a line, we should be able to use linear

regression to determine

the intersection of two lines.

The problem is to determine a representation of lines such

that we can easily

enter the representations of two lines and get their

intersection.

L.R. is given (xi, yi) (i = 1..2) and returns (m, b).

We would like to enter (mi, bi) (i=1..2) and get (x, y).

We can't quite do this, but if we write b = -x*m + y, we

see that b corresponds

to y and -m corresponds to x.

That is,

y = m*x + b

b = x*(-m) + y

So, if we enter (-mi, bi) for each line as the (x, y) input

for sigma+,

we should get x for m and y for b from L.R.

Example: Find the intersection of y=6x-1 and y=3x+2.

The (m, b) values are (6, -1), (3, 2), so we enter (-6, -1)

and (-3, 2)

(i.e., CLEAR sums(4) -1 ENTER -6 sigma+ 2 ENTER -3 sigma+).

The result is m=1, b=5, so the intersection is (1, 5)

----------------------------------

Martin Cohen

5/27/04