# Distance from a point to a line in vector geometry - real world applications?

In vector geometry it is a standard example how to calculate the distance between a point and a line in space. However are there any interesting real world applications of this. I.e. where one wants to calculate the distance between a point and a line using vector geometry.

I don't want general applications of vector geometry or linear algebra but explicit interesting applications where the concrete problem of calculation (using vector geometry) the distance between a point and a line in space occurs.

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An obvious application of this is ray-tracing software and in general all 3D-graphic applications; they use vectors and linear algebra all the time to calculate angles of incidence and reflections for lighting, shading, etc.

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Could you give some more details where in this example the distance between a point and a line is needed? – student May 31 '11 at 18:47
I'm by no means well-versed in the subject but you would need the distance between a point and a line (or a point and a plane) if you were to calculate the amount of light the point gets from what's reflected from that line or plane (which is proportional to its distance, I think). – F M May 31 '11 at 18:53
why don't you guys vote each other up? – David Kohler Jun 1 '11 at 1:15
Thought I had already voted this up. – F M Jun 1 '11 at 3:17

To expand on fmartin's answer, point-line distance calculations are used for intersecting rays with spheres; if the distance from the center of the sphere to the line represented by the ray is greater than the sphere's radius, then there can be no intersection. (And when the distance is less than or equal to the radius, then the intersection points shake out with just a little extra math past the distance calculation: essentially, just apply the Pythagorean Theorem to the triangle(s) formed by the center of the sphere, the closest point on the line to that center, and the intersection point(s) of the line with the sphere.)

A more interesting conceptual example, though - not precisely an application of the specific formula you're looking for, but conceptually related - is the notion of Least-Squares approximation, which essentially fits a line through a cloud of points by determining the parameters of the line that minimize the sum of (squared) distances from all of the points to the line.

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fmartin is right on :) ..Any 3-D drawing/image you see on your computer screen, television screen, etc., is provided by the formulas used in Projective geometry. All the calculations used to make that image look realistic is thanks to Projective geometry! :)

Here is a link to a great article about Projective geometry:

http://www.geometer.org/mathcircles/projective.pdf

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