Thursday, September 29, 2005

Math: Distance between two lines

เยี่ยมมาก คิดอยู่ตั้งนานว่าจะ proof ทฤษฎีพื้นฐานนี่ได้ยังไง ในที่สุดก็เจอ

Objective: Given two straight lines in 3D:
  • line 1:    r = a + t*b
  • line 2:    r = p  +s*q

    Show that the distance between those two lines are |(a - p).u|
    where u is a unit vector given by b x q / |b x q|.
The distance is the length of line segment that perpendicular to both line 1 and 2. Thus if we can find the length of that line segment we are done. So there are two subproblems now:
  1. Finding direction of the line segment
  2. Finding the length of the line segment that bridge line 1 and 2
To find direction of the line segment cross product can be used because output from cross product is a vector perpendicular to both input vectors. In other word, b x q gives out the direction we want.

To find the length, scalar projection can be applied. If we know a vector that bridges the two lines (in this case it is a - p), we can find the distance between those two lines by calculating scalar projection on the perpendicular vector. Remember that the distance is actually |a - p| cos theta where theta is an angle between (a - p) and the perpendicular vector.

The scalar projection is |a - p|cos theta = | (a - p) . (b x q) | / | b x q| = | (a - p) . u|.


1 comment:

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