Re: Cirlce in 3D?

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• Subject: [mg132731] Re: Cirlce in 3D?
• From: Francisco Javier GarcÃa CapitÃn <garciacapitan at gmail.com>
• Date: Thu, 15 May 2014 02:25:38 -0400 (EDT)
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```That was my approach:

BisectorPlane[{ptA_, ptB_}] :=
Factor[({x, y, z} - ptA)^2 - ({x, y, z} - ptB)^2] == 0

Circumcenter[{ptA_, ptB_, ptC_}] := {x, y, z} /. Solve[
Map[BisectorPlane, {{ptA, ptB}, {ptB, ptC}, {ptC, ptA}}], {x, y,
z}][[1]]

ShowGraphic[{ptA_, ptB_, ptC_}] :=
Module[{ptO, vecu, vecn, vecv, k, n},
n = 100;
ptO = Circumcenter[{ptA, ptB, ptC}];
vecn = Cross[ptB - ptA, ptC - ptA];
vecu = ptA - ptO;
vecv = Cross[vecu, vecn] Norm[vecu]/Norm[Cross[vecu, vecn]];
instr = {
PointSize[0.02],
Map[Point, {ptA, ptB, ptC}],
Red,
Line[Table[
ptO + Cos[2 k \[Pi]/n] vecu + Sin [2 k \[Pi]/n] vecv, {k, 0,
n}]],
Point[ptO]
};
Graphics3D[instr]
]

data = {{1, 2, 3}, {3, 4, 1}, {3, 2, 1}};

Circumcenter[data]
{2, 3, 2}

ShowGraphic[data]

2014-05-14 11:25 GMT+02:00 Murray Eisenberg <murray at math.umass.edu>:

> Is the Grassmann Calculus notebook to which you refer currently included
> with your Grassmann Algebra package? If not, how can one obtain it?
>
> [David: Please reply privately as well as to this group. Thanks]
>
> 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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