Get[FileNameJoin[{NotebookDirectory[],"Packages","ConformalAlgebra3D.wl"}]]

p:=px e[1,5]+py e[2,5]+pz e[3,5]+pw e[4,5]

q:=qx e[1,5]+qy e[2,5]+qz e[3,5]+qw e[4,5]

l:=lvx e[4,1,5]+lvy e[4,2,5]+lvz e[4,3,5]+lmx e[2,3,5]+lmy e[3,1,5]+lmz e[1,2,5]

k:=kvx e[4,1,5]+kvy e[4,2,5]+kvz e[4,3,5]+kmx e[2,3,5]+kmy e[3,1,5]+kmz e[1,2,5]

g:=gx e[4,2,3,5]+gy e[4,3,1,5]+gz e[4,1,2,5]+gw e[3,2,1,5]

h:=hx e[4,2,3,5]+hy e[4,3,1,5]+hz e[4,1,2,5]+hw e[3,2,1,5]

a:=ax e[1]+ay e[2]+az e[3]+aw e[4]+au e[5]

b:=bx e[1]+by e[2]+bz e[3]+bw e[4]+bu e[5]

d:=dvx e[4,1]+dvy e[4,2]+dvz e[4,3]+dmx e[2,3]+dmy e[3,1]+dmz e[1,2]+dpx e[1,5]+dpy e[2,5]+dpz e[3,5]+dpw e[4,5]

f:=fvx e[4,1]+fvy e[4,2]+fvz e[4,3]+fmx e[2,3]+fmy e[3,1]+fmz e[1,2]+fpx e[1,5]+fpy e[2,5]+fpz e[3,5]+fpw e[4,5]

c:=cgx e[4,2,3]+cgy e[4,3,1]+cgz e[4,1,2]+cgw e[3,2,1]+cvx e[4,1,5]+cvy e[4,2,5]+cvz e[4,3,5]+cmx e[2,3,5]+cmy e[3,1,5]+cmz e[1,2,5]

o:=ogx e[4,2,3]+ogy e[4,3,1]+ogz e[4,1,2]+ogw e[3,2,1]+ovx e[4,1,5]+ovy e[4,2,5]+ovz e[4,3,5]+omx e[2,3,5]+omy e[3,1,5]+omz e[1,2,5]

s:=su e[1,2,3,4]+sx e[4,2,3,5]+sy e[4,3,1,5]+sz e[4,1,2,5]+sw e[3,2,1,5]

t:=tu e[1,2,3,4]+tx e[4,2,3,5]+ty e[4,3,1,5]+tz e[4,1,2,5]+tw e[3,2,1,5]

Join and Meet

Join of two round points

a∧b//BasisForm

Manipulate[With[{p1=RoundPoint[0,0,0,1,1.5],p2=RoundPoint[x,0,0,1,r]},
Show[DrawRoundPoint[p1],DrawRoundPoint[p2],DrawDipole[p1∧p2],PlotRange->{{-2,7},{-4,4},{-4,4}},Boxed->False,ViewAngle->20°]],
{{x,4},0.1,5},{{r,1},0,3}]

Join of flat point and round point

p∧a//BasisForm

Join of dipole and round point

d∧a//BasisForm

Manipulate[With[{p1=RoundPoint[0,0,0,1,1],p2=RoundPoint[4,0,0,1,3],p3=RoundPoint[2,0,z,1,-r]},
Show[DrawRoundPoint[p1],DrawRoundPoint[p2],DrawRoundPoint[p3],DrawDipole[p1∧p2],DrawDipole[p2∧p3],DrawDipole[p3∧p1],DrawCircle[p1∧p2∧p3],PlotRange->{{-2,8},{-4,4},{-2,8}},Boxed->False,ViewAngle->20°]],
{{z,4},0.1,5},{{r,1},0,3}]

Join of dipole and flat point

d∧p//BasisForm

Join of line and round point

l∧a//BasisForm

Join of circle and round point

c∧a//BasisForm

Manipulate[With[{p1=RoundPoint[0,0,z,1,r],cir=RoundCircle[4,0,0,0,0,0,1]},
Show[DrawRoundPoint[p1],DrawCircle[cir],DrawSphere[p1∧cir],PlotRange->{{-4,4},{-4,4},{-4,6}},Boxed->False,ViewAngle->20°]],
{{z,4},0.1,5},{{r,1},0,3}]

Join of two dipoles

d∧f//BasisForm

Manipulate[With[{d1=RoundDipole[2,0,0,0,0,1,0],d2=RoundDipole[r,x,0,0,0,0,1]},
Show[DrawDipole[d1],DrawDipole[d2],DrawSphere[d1∧d2],PlotRange->{{-2,8},{-4,4},{-4,4}},Boxed->False,ViewAngle->20°]],
{{x,4},0.1,5},{{r,1},0,4}]

Meet of two planes

g∨h//BasisForm

Meet of plane and line

g∨l//BasisForm

Meet of two spheres

s∨t//BasisForm

Manipulate[Module[{r1=4,r2=1,s1,s2},s1=RoundSphere[r1,0,0,0];s2=RoundSphere[r2,x,0,0];Show[DrawSphere[s1],DrawSphere[s2],DrawCircle[s1∨s2],PlotRange->{{-4,4},{-4,4},{-4,4}},Boxed->False,ViewAngle->20°]],{{x,2},-4,4}]

Manipulate[Module[{r1=4,r2=-2,s1,s2},s1=RoundSphere[r1,-2.5,0,0];s2=RoundSphere[r2,x,0,0];Show[DrawSphere[s1],DrawSphere[s2],DrawCircle[s1∨s2],DrawSphere[Container[s1∨s2]],PlotRange->{{-4,4},{-4,4},{-4,4}},Boxed->False,ViewAngle->20°]],{{x,2},-4,4}]

Manipulate[Module[{r1=-4,r2=-1,s1,s2},s1=RoundSphere[r1,0,0,0];s2=RoundSphere[r2,x,0,0];Show[DrawSphere[s1],DrawSphere[s2],DrawCircle[s1∨s2],PlotRange->{{-4,4},{-4,4},{-4,4}},Boxed->False,ViewAngle->20°]],{{x,2},-4,4}]

Meet of sphere and plane

s∨g//BasisForm

Manipulate[Module[{r1=4,s1},s1=RoundSphere[r1,0,0,z];Show[DrawSphere[s1],DrawPlane[e[4,1,2,5]],DrawCircle[s1∨e[4,1,2,5]],PlotRange->{{-4,4},{-4,4},{-4,4}},Boxed->False,ViewAngle->20°]],{{z,1},-4,4}]

Meet of sphere and circle

s∨c//BasisForm

Manipulate[With[{s1=RoundSphere[9,0,0,0],c1=RoundCircle[r,x,1,0,0,Cos[φ],Sin[φ]]},
Show[DrawSphere[s1],DrawCircle[c1],DrawDipole[c1∨s1],PlotRange->{{-9,9},{-4,4},{-4,4}},Boxed->False,Axes->False,ViewAngle->20°]],
{{x,4},-7,7},{{r,1},0,9},{φ,0,Pi}]

Meet of sphere and line

s∨l//BasisForm

Meet of sphere and dipole

s∨d//BasisForm

Manipulate[With[{dip=RoundDipole[r,0,0.5,z,0,0,1],sph=RoundSphere[2,0,0,1]},
Show[DrawSphere[sph],DrawDipole[dip],DrawRoundPoint[dip∨sph],PlotRange->{{-3,3},{-3,3},{-2,5}},Boxed->False,ViewAngle->20°]],
{{z,2},0.1,5},{{r,1},0,3}]

Meet of sphere and flat point

s∨p//BasisForm

Manipulate[With[{dip=FlatPoint[0,0,z],sph=RoundSphere[r,0,0,1]},
Show[DrawSphere[sph],DrawFlatPoint[dip],DrawRoundPoint[dip∨sph],PlotRange->{{-3,3},{-3,3},{-2,5}},Boxed->False,ViewAngle->20°]],
{{z,2},0.1,5},{{r,1},0,4}]

Meet of circle and plane

g∨c//BasisForm

Meet of two circles

c∨o//BasisForm

Manipulate[With[{c1=RoundCircle[9,0,0,0,0,0,1],c2=RoundCircle[r,x,0,0,0,1,0]},
Show[DrawCircle[c1],DrawCircle[c2],DrawPlane[Carrier[c1]],DrawPlane[Carrier[c2]],DrawRoundPoint[c1∨c2],PlotRange->{{-9,9},{-6,6},{-6,6}},Boxed->False,Axes->False,ViewAngle->20°]],
{{x,4},-7,7},{{r,1},0,9}]

Animate[With[{c1=RoundCircle[9,0,0,0,0,0,1],c2=RoundCircle[4,x,0,1,0,1,0]},
Show[DrawCircle[c1],DrawCircle[c2],DrawRoundPoint[c1∨c2],PlotRange->{{-10,10},{-10,10},{-10,10}},Boxed->False,Axes->False,ViewCenter->{0.5,0.5,0.5},ViewPoint->{0,-2,1},ViewAngle->20°]],
{x,-8.9,9,18/(20*30)},AnimationRate->20]

Meet of circle and line

c∨l//BasisForm

Manipulate[With[{c1=RoundCircle[r,0,0,0,0,0,1],l1=FlatLine[0,0,1,0,-x,0]},
Show[DrawCircle[c1],DrawLine[l1],DrawRoundPoint[c1∨l1],PlotRange->{{-7,7},{-4,4},{-4,4}},Boxed->False,Axes->False,ViewAngle->20°]],
{{x,2},-7,7},{{r,8},0,16}]

Meet of two lines

l∨k//BasisForm

Meet of plane and dipole

g∨d//BasisForm

Manipulate[With[{dip=RoundDipole[r,0,0,z,0,0,1],pln=FlatPlane[0,0,1,0]},
Show[DrawPlane[pln],DrawDipole[dip],DrawRoundPoint[dip∨pln],PlotRange->{{-5,5},{-5,5},{-5,5}},Boxed->False,ViewAngle->20°]],
{{z,2},-2,5},{{r,1},0,9}]

Expansion

Expand round point onto sphere

a∧Antidual[s]//BasisForm

Manipulate[With[{pnt=RoundPoint[0,0,0,1,1],sph=RoundSphere[r,x,0,0]},
Show[DrawSphere[sph],DrawRoundPoint[pnt],DrawDipole[pnt∧Antidual[sph]],PlotRange->{{-2,8},{-3,3},{-3,3}},Boxed->False,ViewAngle->20°]],
{{x,4},-2,5},{{r,1},0,6}]

Expand round point onto plane

a∧Antidual[g]//BasisForm

Manipulate[With[{pnt=RoundPoint[0,0,0,1,r],sph=FlatPlane[-1,0,0,x]},
Show[DrawPlane[sph],DrawRoundPoint[pnt],DrawDipole[pnt∧Antidual[sph]],PlotRange->{{-2,8},{-3,3},{-3,3}},Boxed->False,ViewAngle->20°]],
{{x,4},-2,5},{{r,1},0,6}]

Expand dipole onto sphere

d∧Antidual[s]//BasisForm

Manipulate[With[{dip=RoundDipole[1,0,0,0,0,0,1],sph=RoundSphere[r,x,0,0]},
Show[DrawSphere[sph],DrawDipole[dip],DrawCircle[dip∧Antidual[sph]],PlotRange->{{-2,8},{-3,3},{-3,3}},Boxed->False,ViewAngle->20°]],
{{x,4},-2,5},{{r,1},0,6}]

Expand dipole onto plane

d∧Antidual[g]//GCollect

Manipulate[With[{dip=RoundDipole[r,0,0,0,0,0,1],sph=FlatPlane[-1,0,0,x]},
Show[DrawPlane[sph],DrawDipole[dip],DrawCircle[dip∧Antidual[sph]],PlotRange->{{-2,8},{-3,3},{-3,3}},Boxed->False,ViewAngle->20°]],
{{x,4},-2,5},{{r,1},0,6}]

Expand flat point onto sphere

p∧Antidual[s]//BasisForm

Manipulate[With[{dip=e[4,5],sph=RoundSphere[r,x,0,0]},
Show[DrawSphere[sph],DrawFlatPoint[dip],DrawLine[dip∧Antidual[sph]],PlotRange->{{-2,8},{-3,3},{-3,3}},Boxed->False,ViewAngle->20°]],
{{x,4},-2,5},{{r,1},0,6}]

Expand flat point onto plane

p∧Antidual[g]//BasisForm

Expand circle onto sphere

c∧Antidual[s]//BasisForm

Manipulate[With[{cir=RoundCircle[1,0,0,0,1,0,0],sph=RoundSphere[r,x,0,0]},
Show[DrawSphere[sph],DrawCircle[cir],DrawSphere[cir∧Antidual[sph]],PlotRange->{{-2,6},{-3,3},{-3,3}},Boxed->False,ViewAngle->20°]],
{{x,4},-2,5},{{r,1},0,4}]

Expand circle onto plane

c∧Antidual[g]//BasisForm

Manipulate[With[{cir=RoundCircle[r,0,0,1/2,Sqrt[3]/2,0,-1/2],sph=FlatPlane[-1,0,0,x]},
Show[DrawPlane[sph],DrawCircle[cir],DrawSphere[cir∧Antidual[sph]],PlotRange->{{-2,6},{-3,3},{-3,3}},Boxed->False,ViewAngle->20°]],
{{x,1.8},-2,5},{{r,1},0,4}]

Expand line onto sphere

l∧Antidual[s]//BasisForm

Manipulate[With[{cir=FlatLine[0,0,1,0,0,0],sph=RoundSphere[r,x,0,0]},
Show[DrawSphere[sph],DrawLine[cir],DrawPlane[cir∧Antidual[sph]],PlotRange->{{-1,5},{-3,3},{-3,3}},Boxed->False,ViewAngle->20°]],
{{x,4},-2,5},{{r,1},0,4}]

Expand line onto plane

l∧Antidual[g]//BasisForm

Expand round point onto circle

a∧Antidual[c]//BasisForm

Manipulate[With[{pnt=RoundPoint[0,0,2,1,1],cir=RoundCircle[r,x,0,0,0,0,1]},
Show[DrawCircle[cir],DrawRoundPoint[pnt],DrawCircle[pnt∧Antidual[cir]],PlotRange->{{-2,8},{-3,3},{-3,3}},Boxed->False,Axes->False,ViewAngle->20°]],
{{x,4},-2,5},{{r,1},0,6}]

Expand round point onto line

a∧Antidual[l]//BasisForm

Manipulate[With[{pnt=RoundPoint[0,0,1,1,r],cir=FlatLine[0,1,0,0,0,x]},
Show[DrawLine[cir],DrawRoundPoint[pnt],DrawCircle[pnt∧Antidual[cir]],PlotRange->{{-2,8},{-3,3},{-3,3}},Boxed->False,Axes->False,ViewAngle->20°]],
{{x,2.2},-2,5},{{r,1},0,6}]

Expand flat point onto circle

p∧Antidual[c]//BasisForm

Manipulate[With[{pnt=FlatPoint[-1,0,-1],cir=RoundCircle[r,x,0,0,0,0,1]},
Show[DrawCircle[cir],DrawFlatPoint[pnt],DrawPlane[pnt∧Antidual[cir]],PlotRange->{{-2,8},{-3,3},{-3,3}},Boxed->False,Axes->False,ViewAngle->20°]],
{{x,4},-2,5},{{r,1},0,6}]

Expand flat point onto line

p∧Antidual[l]//BasisForm

Expand dipole onto circle

d∧Antidual[c]//BasisForm

Manipulate[With[{dip=RoundDipole[1,0,0,0,0,0,1],cir=RoundCircle[r,x,0,0,0,1,0]},
Show[DrawDipole[dip],DrawCircle[cir],DrawSphere[dip∧Antidual[cir]],PlotRange->{{-2,8},{-3,3},{-3,3}},Boxed->False,Axes->False,ViewAngle->20°]],
{{x,4},-2,5},{{r,1},0,6}]

Expand dipole onto line

d∧Antidual[l]//BasisForm

Manipulate[With[{dip=RoundDipole[1,0,0,0,0,0,1],cir=(Cos[φ]e[2]+Sin[φ]e[3])∧(x e[1,5]+e[4,5])},
Show[DrawDipole[dip],DrawLine[cir],DrawSphere[dip∧Antidual[cir]],PlotRange->{{-2,8},{-3,3},{-3,3}},Boxed->False,Axes->False,ViewAngle->20°]],
{{x,2.5},-2,5},{{φ,Pi/2},-Pi,Pi}]

Expand round point onto dipole

a∧Antidual[d]//BasisForm

Manipulate[With[{pnt=RoundPoint[0,0,0,1,0.5],dip=RoundDipole[r,x,0,2,0,0,1]},
Show[DrawDipole[dip],DrawRoundPoint[pnt],DrawSphere[pnt∧Antidual[dip]],PlotRange->{{-2,8},{-3,3},{-3,3}},Boxed->False,Axes->False,ViewAngle->20°]],
{{x,4},-2,5},{{r,1},0,6}]

Expand round point onto flat point

a∧Antidual[p]//BasisForm

Manipulate[With[{pnt=RoundPoint[0,0,0,1,r],dip=FlatPoint[x,0,0]},
Show[DrawFlatPoint[dip],DrawRoundPoint[pnt],DrawSphere[pnt∧Antidual[dip]],PlotRange->{{-2,8},{-3,3},{-3,3}},Boxed->False,Axes->False,ViewAngle->20°]],
{{x,2.5},-2,5},{{r,1},0,6}]

License

MIT License

Copyright (c) 2025 Eric Lengyel

Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files (the “Software”), to deal
in the Software without restriction, including without limitation the rights
to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
copies of the Software, and to permit persons to whom the Software is
furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all
copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED “AS IS”, WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
SOFTWARE.