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Math Help - Cylinder in 3d space

  1. #1
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    Cylinder in 3d space

    Given 2 points P1= (x1, y1, z1) and P2= (x2, y2, z2), how can we describe a cylinder around the line formed?

    I was thinking of finding a parallel line segment and rotating it at distance r around the line segment defined by P1 and P2, but i'm not sure how to define these.

    I don't want to find just the surface area, per se, but the actual points that the cylinder would go through.

    thanks. if that's not clear enough, please let me know.
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  2. #2
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    Hi,

    this is the right way to go.

    What you need are two unit-vectors perpendicular to the line segment and to each other.

    If the line-segment-vector is (a1,a2,a3), then you can usually use (0,a3,-a2) as the first one, and (a1,a2,a3)x(0,a3,-a2) as the second one (there are rare cases when this doesnīt work, figure ot yourself, please).

    Now normalize those vectors and name them e1 and e2.

    Now you cet a circle around P1 as c(t) = P1 + r*cos(t)*e1 + r*sin(t)*e2, t running from 0 to 2pi.

    Thus your parametrization of the whole cylinder is

    cyl(t,u) = P1 + u*(P2-P1) + + r*cos(t)*e1 + r*sin(t)*e2, t running from 0 to 2pi, u running from 0 to 1.

    I can make an Archimedes Geo3D construction to illustrate this, if you donīt get me.

    Regards,

    Andreas
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