translate coordinates relative to line segment

Question

I need to write a function (in Javascript) that accepts the endpoints of a line segment and an additional point and returns coordinates for the point relative to the start point. So, based on this picture:

function perpendicular_coords(x1,y1, x2,y2, xp,yp)

returns [xp',yp'] where xp' is the distance from (xp,yp) along a line perpendicular to the line segment, and yp' is the distance from (x1,y1) to the point where that perpendicular line intersects the line segment.

What I've tried so far:

function rotateRad(cx, cy, x, y, radians) {
    var cos = Math.cos(radians),
        sin = Math.sin(radians),
        nx = (cos * (x - cx)) + (sin * (y - cy)) + cx,
        ny = (cos * (y - cy)) - (sin * (x - cx)) + cy;
    return [nx, ny];
}

[xp', yp'] = rotateRad(x1,y1, xp, yp, Math.atan2(y2-y1,x2-x1));

I didn't write the function; got it from https://stackoverflow.com/a/17411276/1368860

Answer 1

I took a different approach, by combining two functions:

1. To get yp' I find the intersection point using the answer from here: Perpendicular on a line from a given point
2. To get xp' I calculate the distance between (xp, yp) and the line, using the equation from here: https://en.wikipedia.org/wiki/Distance_from_a_point_to_a_line

Not the most elegant solution, but it seems to work.

Resulting code https://jsfiddle.net/qke0m4mb/

function perpendicular_coords(x1, y1, x2, y2, xp, yp) {
  var dx = x2 - x1,
      dy = y2 - y1;

  //find intersection point    
  var k = (dy * (xp-x1) - dx * (yp-y1)) / (dy*dy + dx*dx);
  var x4 = xp - k * dy;
  var y4 = yp + k * dx;

  var ypt = Math.sqrt((y4-y1)*(y4-y1)+(x4-x1)*(x4-x1));
  var xpt = distance(x1, y1, x2, y2, xp, yp);
  return [xpt, ypt];
}

// Distance of point from line
function distance(x1, y1, x2, y2, xp, yp) {
  var dx = x2 - x1;
  var dy = y2 - y1;

  return Math.abs(dy*xp - dx*yp + x2*y1 - y2*x1) / Math.sqrt(dy*dy + dx*dx);
}