Question: I'm trying to extrapolate from a line segment to find a point on the line but a 3rd of the way 'back', i.e. trying to find point `new`, given points `A` and `B` below:

![enter image description here](https://i.stack.imgur.com/FTMi3.png)

Given a line, I can interpolate it to get a position at any particular percentage along it:

```
=# select st_line_interpolate_point(
   st_makeline('0101000020E6100000300DC347C49418C03EE8D9ACFAA44A40', 
               '0101000020E6100000FB743C66A03218C0CDCCCCCCCC7C4A40'), 
   0.333);
0101000020E6100000ED45B41D537718C069C6A2E9EC984A40

```

I tried entering a negative number to find a point along the line in the opposite direction, but that fails as the interpolation argument has to be in the range [0, 1]

I thought about first scaling the line, but that doesn't use the centre of the line as the origin, so is useless for my purposes.

Answer: 
Another way I have solved a similar problem previously is to break it down into the following steps.

```
-- get the points A and B given a line L
A := ST_STARTPOINT(L);
B := ST_ENDPOINT(L);

-- get the bearing from point B --> A
azimuth := ST_AZIMUTH(B,A);

-- get the length of the line A --> B
length := ST_DISTANCE(A,B);
newlength := length + (length * (1/3));   -- increase the line length by 1/3

-- create a new point 1/3 as far away from A as B is from A
newpoint := ST_TRANSLATE(A, sin(azimuth) * newlength, cos(azimuth) * newlength);

```

EDIT: fixed the assignment of newlength so that it is 1 1/3 the length, rather than 1/3 and switched A & B around to match diagram.
Have solved it with:

```
F = 1.3333
st_affine(A, F, 0, 
             0, F, 
            (F-1)*-st_x(st_line_interpolate_point(st_makeline(A, B), 0.5)), 
            (F-1)*-st_y(st_line_interpolate_point(st_makeline(A, B), 0.5))
          )

```

Explanation:

(2-d) Scale the start point by a factor of 1.3333, taking the midpoint of the line segment as the origin for scaling.

Get out the graph paper!

<http://en.wikipedia.org/wiki/Affine_transformation>