Eliminating parameters for computing an intersection

dear Reinhard,

the package by Mario Riotorto will be your friend.
Follow this link for more elaborated examples of this package:

http://www.telefonica.net/web2/biomates/maxima/gpdraw

As far as I remember Mario has given a fantastic example of intersecting
cylinders.

hth
Wolfgang


-----Urspr?ngliche Nachricht-----
Von: Reinhard Jansohn <jansohn at betontest.de>
An: maxima at math.utexas.edu <maxima at math.utexas.edu>
Datum: Mittwoch, 3. Dezember 2008 16:10
Betreff: [Maxima] Eliminating parameters for computing an intersection


|Hello Maxima friends!
|
|I computed the intersection of two cylinders by hand as shown below but I
want to compute similar intesections with Maxima.
|
|Please note that there is an article "The Geometry of Intersecting Tubes
Applied to Controlling a Robotic Welding Torch" from by John M. Stockie on
the web http://www.math.sfu.ca/~stockie/weld/ showing how this is done in
Maple. Unfortunately I could not do that in Maxima.
|
|cylinder 1:
|x1 = v1
|y1 = R1 * sin(u1)
|z1 = R1 * cos(u1)
|
|cylinder 2:
|x2 = R2 * cos(u2)
|y2 = R2 * sin(u2)
|z2 = v2
|
|The intersection maybe computed simply by setting x1=x2, y1=y2 and z1=z2
which gives us
|R2 * cos(u2) = v1
|R2 * sin(u2) = R1 * sin(u1)
|v2           = R1 * cos(u1)
|
|Now we three equations with four parameters u1, v1, u2 and v2.
|To get the intersection curve we eliminate three parameters. If we do that
we end up in
|x = R2 * cos(u2)
|y = R2 * sin(u2)
|z = sqrt(R1^2 - R2^2 * sin^2(u2))
|
|My question is how can this be done in Maxima.
|
|Does anyone know how to get this done in Maxima.
|
|Thanks for help.
|Kindest regrads
|Reinhard
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