Re: Cirlce in 3D?

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• Subject: [mg132729] Re: Cirlce in 3D?
• From: Murray Eisenberg <murray at math.umass.edu>
• Date: Wed, 14 May 2014 05:25:38 -0400 (EDT)
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Is the Grassmann Calculus notebook to which you refer currently included with your Grassmann Algebra package? If not, how can one obtain it?

On May 12, 2014, at 10:25 PM, djmpark <djmpark at comcast.net> wrote:

> Using John Browne's Grassmann algebra application and with the three points
> at Cartesian coordinates {ax, ay, az}, {bx, by, bz}, and {cx, cy, cz} I
> calculated the center of the circle at:
>
> {x -> -(4 (az (by - cy) + bz cy - by cz +
>         ay (-bz +
>            cz)) (-(ax^2 + ay^2 + az^2 - bx^2 - by^2 - bz^2) (-by cx +
>             ay (-bx + cx) + ax (by - cy) + bx cy) -
>         2 (az - bz) (az by cx - ay bz cx - az bx cy + ax bz cy +
>            ay bx cz - ax by cz)) - (2 (ax^2 + ay^2 + az^2 - bx^2 -
>            by^2 - bz^2) (bz - cz) -
>         2 (az - bz) (bx^2 + by^2 + bz^2 - cx^2 - cy^2 -
>            cz^2)) (-2 (ay - by) (ay (bx - cx) + by cx - bx cy +
>            ax (-by + cy)) -
>         2 (az - bz) (az (bx - cx) + bz cx - bx cz +
>            ax (-bz + cz))))/(-4 (az (bx - cx) + bz cx - bx cz +
>         ax (-bz + cz)) (-2 (ay - by) (ay (bx - cx) + by cx - bx cy +
>            ax (-by + cy)) -
>         2 (az - bz) (az (bx - cx) + bz cx - bx cz +
>            ax (-bz + cz))) +
>      4 (az (by - cy) + bz cy - by cz +
>         ay (-bz + cz)) (2 (ax - bx) (-by cx + ay (-bx + cx) +
>            ax (by - cy) + bx cy) +
>         2 (az - bz) (az (by - cy) + bz cy - by cz + ay (-bz + cz)))),
>  y -> (-2 (ax^2 + ay^2 + az^2 - bx^2 - by^2 - bz^2) (bz - cz) +
>     2 (az - bz) (bx^2 + by^2 + bz^2 - cx^2 - cy^2 -
>        cz^2) + (4 (az (bx - cx) + bz cx - bx cz +
>          ax (-bz + cz)) (4 (az (by - cy) + bz cy - by cz +
>             ay (-bz +
>                cz)) (-(ax^2 + ay^2 + az^2 - bx^2 - by^2 -
>                 bz^2) (-by cx + ay (-bx + cx) + ax (by - cy) +
>                bx cy) -
>             2 (az - bz) (az by cx - ay bz cx - az bx cy + ax bz cy +
>                ay bx cz - ax by cz)) - (2 (ax^2 + ay^2 + az^2 -
>                bx^2 - by^2 - bz^2) (bz - cz) -
>             2 (az - bz) (bx^2 + by^2 + bz^2 - cx^2 - cy^2 -
>                cz^2)) (-2 (ay - by) (ay (bx - cx) + by cx - bx cy +
>                ax (-by + cy)) -
>             2 (az - bz) (az (bx - cx) + bz cx - bx cz +
>                ax (-bz + cz)))))/(-4 (az (bx - cx) + bz cx - bx cz +
>           ax (-bz + cz)) (-2 (ay - by) (ay (bx - cx) + by cx -
>              bx cy + ax (-by + cy)) -
>           2 (az - bz) (az (bx - cx) + bz cx - bx cz +
>              ax (-bz + cz))) +
>        4 (az (by - cy) + bz cy - by cz +
>           ay (-bz + cz)) (2 (ax - bx) (-by cx + ay (-bx + cx) +
>              ax (by - cy) + bx cy) +
>           2 (az - bz) (az (by - cy) + bz cy - by cz +
>              ay (-bz + cz)))))/(4 (az (by - cy) + bz cy - by cz +
>       ay (-bz + cz))),
> z -> (-az^2 by + by cx^2 + az^2 cy - bx^2 cy - by^2 cy - bz^2 cy +
>      by cy^2 + ax^2 (-by + cy) + ay^2 (-by + cy) + by cz^2 +
>      ay (bx^2 + by^2 + bz^2 - cx^2 - cy^2 - cz^2))/(2 (-bz cy +
>        az (-by + cy) + ay (bz - cz) + by cz)) + ((-by cx +
>        ay (-bx + cx) + ax (by - cy) +
>        bx cy) (4 (az (by - cy) + bz cy - by cz +
>           ay (-bz +
>              cz)) (-(ax^2 + ay^2 + az^2 - bx^2 - by^2 -
>               bz^2) (-by cx + ay (-bx + cx) + ax (by - cy) +
>              bx cy) -
>           2 (az - bz) (az by cx - ay bz cx - az bx cy + ax bz cy +
>              ay bx cz - ax by cz)) - (2 (ax^2 + ay^2 + az^2 - bx^2 -
>              by^2 - bz^2) (bz - cz) -
>           2 (az - bz) (bx^2 + by^2 + bz^2 - cx^2 - cy^2 -
>              cz^2)) (-2 (ay - by) (ay (bx - cx) + by cx - bx cy +
>              ax (-by + cy)) -
>           2 (az - bz) (az (bx - cx) + bz cx - bx cz +
>              ax (-bz + cz)))))/((az (by - cy) + bz cy - by cz +
>        ay (-bz + cz)) (-4 (az (bx - cx) + bz cx - bx cz +
>           ax (-bz + cz)) (-2 (ay - by) (ay (bx - cx) + by cx -
>              bx cy + ax (-by + cy)) -
>
>           2 (az - bz) (az (bx - cx) + bz cx - bx cz +
>              ax (-bz + cz))) +
>        4 (az (by - cy) + bz cy - by cz +
>           ay (-bz + cz)) (2 (ax - bx) (-by cx + ay (-bx + cx) +
>              ax (by - cy) + bx cy) +
>           2 (az - bz) (az (by - cy) + bz cy - by cz +
>              ay (-bz + cz)))))}
>
> I have a notebook on it but it uses the GrassmannCalculus and Presentations
> Applications. Presentations has Circle3D and Disk3D primitives. The notebook
> also calculates and displays random cases.
>
>
> David Park
> djmpark at comcast.net
> http://home.comcast.net/~djmpark/index.html
>
>
>
> From: Ste[hen Gray [mailto:stevebg at roadrunner.com]
>
> I'm looking for a neat formula to find the center of a circle in 3D through
> 3 points. I also need a good way to display it, preferably thickened so I
> can show several and see whether they are linked, etc. To my surprise I did
> not find anything on the Wolfram sites about these problems. (I have
> Mathematica 7, if that matters.)
>
>

Murray Eisenberg                                murray at math.umass.edu
Mathematics & Statistics Dept.
Lederle Graduate Research Tower      phone 240 246-7240 (H)
University of Massachusetts
710 North Pleasant Street
Amherst, MA 01003-9305

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