TSQL - get closest coordinate on linestring to a point - sql-server

Question

Consider the overly simplistic example: POINT(0 0) and LINESTRING (1 -10, 1 10)
The closest point on the line to the POINT would be 1, 0.
How would one determine this in TSQL? My simple, not entirely accurate, approach was to make a linestring (POINT POINT) and extend out the X coord of one coords until the two linestrings intersected.
So:
linestring (0 0, 0.25 0) (no intersect)
linestring (0 0, 0.5 0) (no intersect)
linestring (0 0, 0.75 0) (no intersect)
linestring (0 0, 1 0) (intersection - so 1 0 is the point closest to POINT
This quasi worked, but doesn't seem to the most bestest/more performant way of accomplishing this.
For example, one inefficiency is that I move it one direction (positive increments), and if there was no match (after x attempts), then I would start over, but with negative increments.
To optimize, I tried moving in larger steps, then when intersected (probably went past the point), I backed off 1 increment and started from there with a smaller increment. I did this a couple of times - instead of going in tiny tiny increments so as not to overshoot by too much.
One acceptable assumption based on my processing that the POINT will be next to (left/right) of the LINESTRING.
Another acceptable assumption is that the LINESTRING will be fairly "perpendicular" to the POINT.

Answer 1

I think you can do this mathematically rather than with a brute-force iterative algorithm.
There is a post to get closest point to a line that describes the method.
I converted this method to SQL which returns the correct value (1,0). Your 'trivial' example is actually a bit of an edge case (vertical line with infinite slope) so it seems robust.
I also tested the source code with this example: https://www.desmos.com/calculator/iz07az84f5 and using the input for the line of (-1,2) (3,0) and a point at (2,2) got the correct answer (1.4, 0.8).
SQL code (also in SQL Fiddle at http://sqlfiddle.com/#!6/d87aa/15)
DECLARE #x int, #y int, #x1 int, #y1 int, #x2 int, #y2 int
DECLARE #atb2 float, #atp_dot_atb float
DECLARE #t float
--SELECT #x=0, #y=0
--SELECT #x1=1, #y1=10, #x2=1, #y2=-10
SELECT #x=2, #y=2
SELECT #x1=-1, #y1=2, #x2=3, #y2=0
SELECT #atb2 = SQUARE(#x2-#x1) + SQUARE(#y2-#y1) -- Basically finding the squared magnitude of a_to_b
SELECT #atp_dot_atb = (#x-#x1)*(#x2-#x1) + (#y-#y1)*(#y2-#y1) -- The dot product of a_to_p and a_to_b
SELECT #t = #atp_dot_atb / #atb2 -- The normalized "distance" from a to your closest point
SELECT #x1 + (#x2-#x1)*#t, #y1 + (#y2-#y1)*#t --Add the distance to A, moving towards B