Very basic question on a line and a point

Hi

I am looking at https://processing.org/discourse/beta/num_1276644884.html.

Shame on me, but I miss to understand the geometry (and/or algebra) which justifies the line of code
float mX = (-x1+x)*ca + (-y1+y)*sa;.
How is it that that works fine?
Thanks

1 Like

reposting the snippet as a reference:

/**
 * Returns a point on the line (x1,y1) -> (x2,y2) 
 * that is closest to the point (x,y)
 * 
 * The result is a PVector. 
 * result.x and result.y are points on the line. 
 * The result.z variable contains the distance from (x,y) to the line, 
 * just in case you need it :) 
 */
PVector getDistance( float x1, float y1, float x2, float y2, float x, float y ){
  PVector result = new PVector(); 
  
  float dx = x2 - x1; 
  float dy = y2 - y1; 
  float d = sqrt( dx*dx + dy*dy ); 
  float ca = dx/d; // cosine
  float sa = dy/d; // sine 
  
  float mX = (-x1+x)*ca + (-y1+y)*sa; 
  
  if( mX <= 0 ){
    result.x = x1; 
    result.y = y1; 
  }
  else if( mX >= d ){
    result.x = x2; 
    result.y = y2; 
  }
  else{
    result.x = x1 + mX*ca; 
    result.y = y1 + mX*sa; 
  }
  
  dx = x - result.x; 
  dy = y - result.y; 
  result.z = sqrt( dx*dx + dy*dy ); 
  
  return result;   
}

let’s say we use the figure from wikipedia
image

this code (I flipped inside the parenthesis)

  float mX = (x-x1)*ca + (y-y1)*sa; 

is a dot product of (x-x1, y-y1) and (ca, sa). The former is a vector from (x1, y1) to (x, y), corresponding to a in the figure. The latter is a vector from (x1, y1) to (x2, y2) but normalized (as divided by d): b in the figure. The dot product can be thought as a projection, and since (ca, sa) is normalized and is a unit vector, the result mX is the length of (x-x1, y-y1) or a projected on b (in the figure, mX corresponds to the length of a1).

4 Likes

Thanks for the reply. Yes, I understand I miss some basic linear algebra, I’ll try to check out a book.

Still, I miss to see how it is that you may define result.x = x1 + (-x1+x)*ca + (-y1+y)*sa)*ca. Where does it come from?

I appreciate your help

again (-x1+x)*ca + (-y1+y)*sa is the dot product explained above. Say the dot product is a, then x = x1 + a * ca is basically moving from the point (x, y) along the line… but I guess you should look for some material instead of someone explaining this here… I studied math in Japan so I don’t know any good material in English but if anyone suggestions that would be great.

Yes, you are correct. Need some spare time.