[PDF] 1 Distance between a point and a line 2 Distance - KIT



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1 Distance between a point and a line 2 Distance - KIT

4 DISTANCE BETWEEN TWO PLANES

1 Distance between a point and a line

Letabe a point and`be a linevt+p, wherevis a unit direction vector. Then dist(a;`) =dist(ap;vt) =japj2(va)2:

Another formula: Ifpandqare points on`, then

dist(a;`) =j(ap)(aq)jjpqj:

2 Distance between a point and a plane

Letabe a point andPbe a plane with normal equationnx=, wherenis a unit normal vector. Then dist(a;P) =janj:

Another formula: Ifb2P, then

dist(a;P) =j(ab)nj:

3 Distance between two lines in R

3

Given two line`1and`2:

Check whether the lines intersect by setting their parametric equations equal. If they intersect, the distance is zero. If they do not intersect and parallel (these can be observed by comparing the direction vectors), late any point on one line and calculate the distance to another line. If the lines do not intersect and are nor parallel, they belong to two parallel planes with normal vectorn. This vector is orthogonal to each of the direction vectors of the lines. Find the equation of such a planePthrough`1, pick an arbitrary pointA2`2, nd the distance betweenAandP.

4 Distance between two planes

Check whether the planes intersect by considering their normal vectors. If normal vectors are nor parallel or if the planes coincide, the planes intersect, the distance is zero.

If they do not intersect, take a point in one plane and nd a distance to another plane.Prof. Maria Axenovichhttp://www.math.kit.edu/iag6/edu/graphtheo2013w/

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