Intersecting LinesDate: 05/08/97 at 18:41:18 From: Alain CUYPERS Subject: Intersection of two lines in space Hi Dr. Math, Could you please help me with the following problem? I have to calculate the coordinates of the intersection point of two lines in space (if there is an intersection). The two given lines are both determinated by two points, so I don't have their equations. Line A contains the points (x1,y1,z1) and (x2,y2,z2) Line B contains the points (x3,y3,z3) and (x4,y4,z4) How do I solve this problem? Thanks, Alain Cuypers Date: 05/09/97 at 09:40:43 From: Doctor Jerry Subject: Re: Intersection of two line in space Hi Alain, Line A can be described by the three parametric equations: x = x1+t*(x2-x1) y = y1+t*(y2-y1) z = z1+t*(z2-z1) Line B can be described by the three parametric equations: x = x3+s*(x4-x3) y = y3+s*(y4-y3) z = z3+s*(z4-z3) To determine whether these two lines intersect, solve simultaneously the equations: x1+t*(x2-x1) = x3+s*(x4-x3) y1+t*(y2-y1) = y3+s*(y4-y3) Are there values of t and s that satisfy these two equations? If there are, then the lines intersect. If so, check to see if the two z-values are also equal. If not, then the lines do not intersect. Note: if the same parameter, for example, t, is used to describe both lines, then you may run into trouble. The equations that result from this set-up model another, related question. -Doctor Jerry, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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