# Project point to line segment

## Alec Jacobson

## November 08, 2010

Here some simple MATLAB code to project a point to a line segment, i.e. find the closest point on the line segment to a given point in space:
```
function [q] = project_point_to_line_segment(A,B,p)
% returns q the closest point to p on the line segment from A to B
% vector from A to B
AB = (B-A);
% squared distance from A to B
AB_squared = dot(AB,AB);
if(AB_squared == 0)
% A and B are the same point
q = A;
else
% vector from A to p
Ap = (p-A);
% from http://stackoverflow.com/questions/849211/
% Consider the line extending the segment, parameterized as A + t (B - A)
% We find projection of point p onto the line.
% It falls where t = [(p-A) . (B-A)] / |B-A|^2
t = dot(Ap,AB)/AB_squared;
if (t < 0.0)
% "Before" A on the line, just return A
q = A;
else if (t > 1.0)
% "After" B on the line, just return B
q = B;
else
% projection lines "inbetween" A and B on the line
q = A + t * AB;
end
end
end
```