This is a notebook that was created by Eric Lengyel. I am re-publishing it here with attribution.
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Get["https://ftp.ericcbrown.com/pub/Geometric–Algebra/Packages/ConformalAlgebra3D.wl"]
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p:=pxe[1,5]+pye[2,5]+pze[3,5]+pwe[4,5]
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q:=qxe[1,5]+qye[2,5]+qze[3,5]+qwe[4,5]
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l:=lvxe[4,1,5]+lvye[4,2,5]+lvze[4,3,5]+lmxe[2,3,5]+lmye[3,1,5]+lmze[1,2,5]
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k:=kvxe[4,1,5]+kvye[4,2,5]+kvze[4,3,5]+kmxe[2,3,5]+kmye[3,1,5]+kmze[1,2,5]
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g:=gxe[4,2,3,5]+gye[4,3,1,5]+gze[4,1,2,5]+gwe[3,2,1,5]
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h:=hxe[4,2,3,5]+hye[4,3,1,5]+hze[4,1,2,5]+hwe[3,2,1,5]
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a:=axe[1]+aye[2]+aze[3]+awe[4]+aue[5]
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b:=bxe[1]+bye[2]+bze[3]+bwe[4]+bue[5]
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d:=dvxe[4,1]+dvye[4,2]+dvze[4,3]+dmxe[2,3]+dmye[3,1]+dmze[1,2]+dpxe[1,5]+dpye[2,5]+dpze[3,5]+dpwe[4,5]
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f:=fvxe[4,1]+fvye[4,2]+fvze[4,3]+fmxe[2,3]+fmye[3,1]+fmze[1,2]+fpxe[1,5]+fpye[2,5]+fpze[3,5]+fpwe[4,5]
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c:=cgxe[4,2,3]+cgye[4,3,1]+cgze[4,1,2]+cgwe[3,2,1]+cvxe[4,1,5]+cvye[4,2,5]+cvze[4,3,5]+cmxe[2,3,5]+cmye[3,1,5]+cmze[1,2,5]
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o:=ogxe[4,2,3]+ogye[4,3,1]+ogze[4,1,2]+ogwe[3,2,1]+ovxe[4,1,5]+ovye[4,2,5]+ovze[4,3,5]+omxe[2,3,5]+omye[3,1,5]+omze[1,2,5]
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s:=sue[1,2,3,4]+sxe[4,2,3,5]+sye[4,3,1,5]+sze[4,1,2,5]+swe[3,2,1,5]
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t:=tue[1,2,3,4]+txe[4,2,3,5]+tye[4,3,1,5]+tze[4,1,2,5]+twe[3,2,1,5]
Join and Meet
Join and Meet
Join of two round points
Join of two round points
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a∧b//BasisForm
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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},SaveDefinitions–>True]
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Join of flat point and round point
Join of flat point and round point
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p∧a//BasisForm
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Join of dipole and round point
Join of dipole and round point
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d∧a//BasisForm
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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},SaveDefinitions–>True]
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Join of dipole and flat point
Join of dipole and flat point
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d∧p//BasisForm
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Join of line and round point
Join of line and round point
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l∧a//BasisForm
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Join of circle and round point
Join of circle and round point
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c∧a//BasisForm
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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},SaveDefinitions–>True]
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Join of two dipoles
Join of two dipoles
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d∧f//BasisForm
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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},SaveDefinitions–>True]
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Meet of two planes
Meet of two planes
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g∨h//BasisForm
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Meet of plane and line
Meet of plane and line
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g∨l//BasisForm
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Meet of two spheres
Meet of two spheres
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s∨t//BasisForm
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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},SaveDefinitions–>True]
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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},SaveDefinitions–>True]
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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},SaveDefinitions–>True]
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Meet of sphere and plane
Meet of sphere and plane
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s∨g//BasisForm
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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},SaveDefinitions–>True]
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Meet of sphere and circle
Meet of sphere and circle
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s∨c//BasisForm
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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},SaveDefinitions–>True]
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Meet of sphere and line
Meet of sphere and line
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s∨l//BasisForm
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lmzsu+(lmysz–lmzsy+lvxsw)+lmxsu+(–lmxsz+lmzsx+lvysw)+lmysu+(lmxsy–lmysx+lvzsw)+lvxsu+lvysu+lvzsu+(–lvxsx–lvysy–lvzsz)
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Meet of sphere and dipole
Meet of sphere and dipole
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s∨d//BasisForm
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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},SaveDefinitions–>True]
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Meet of sphere and flat point
Meet of sphere and flat point
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s∨p//BasisForm
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pxsu+pysu+pzsu+pwsu+(–pwsw–pxsx–pysy–pzsz)
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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},SaveDefinitions–>True]
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Meet of circle and plane
Meet of circle and plane
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g∨c//BasisForm
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Meet of two circles
Meet of two circles
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c∨o//BasisForm
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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},SaveDefinitions–>True]
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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,SaveDefinitions–>True]
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Meet of circle and line
Meet of circle and line
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c∨l//BasisForm
Out[]//TraditionalForm=
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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},SaveDefinitions–>True]
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Meet of two lines
Meet of two lines
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l∨k//BasisForm
Out[]//TraditionalForm=
e
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Meet of plane and dipole
Meet of plane and dipole
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g∨d//BasisForm
Out[]//TraditionalForm=
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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},SaveDefinitions–>True]
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Expansion
Expansion
Expand round point onto sphere
Expand round point onto sphere
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a∧Antidual[s]//BasisForm
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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},SaveDefinitions–>True]
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Expand round point onto plane
Expand round point onto plane
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a∧Antidual[g]//BasisForm
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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},SaveDefinitions–>True]
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Expand dipole onto sphere
Expand dipole onto sphere
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d∧Antidual[s]//BasisForm
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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},SaveDefinitions–>True]
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Expand dipole onto plane
Expand dipole onto plane
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d∧Antidual[g]//GCollect
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GCollect[–dmzgwe[1,2,5]+dpygxe[1,2,5]–dpxgye[1,2,5]–dmxgwe[2,3,5]+dpzgye[2,3,5]–dpygze[2,3,5]–dmygwe[3,1,5]–dpzgxe[3,1,5]+dpxgze[3,1,5]–dmxgxe[3,2,1]–dmygye[3,2,1]–dmzgze[3,2,1]–dvygxe[4,1,2]+dvxgye[4,1,2]–dvxgwe[4,1,5]–dpwgxe[4,1,5]–dvzgye[4,2,3]+dvygze[4,2,3]–dvygwe[4,2,5]–dpwgye[4,2,5]+dvzgxe[4,3,1]–dvxgze[4,3,1]–dvzgwe[4,3,5]–dpwgze[4,3,5]]
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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},SaveDefinitions–>True]
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Expand flat point onto sphere
Expand flat point onto sphere
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p∧Antidual[s]//BasisForm
Out[]//TraditionalForm=
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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},SaveDefinitions–>True]
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Expand flat point onto plane
Expand flat point onto plane
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p∧Antidual[g]//BasisForm
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Expand circle onto sphere
Expand circle onto sphere
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c∧Antidual[s]//BasisForm
Out[]//TraditionalForm=
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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},SaveDefinitions–>True]
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Expand circle onto plane
Expand circle onto plane
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c∧Antidual[g]//BasisForm
Out[]//TraditionalForm=
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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},SaveDefinitions–>True]
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Expand line onto sphere
Expand line onto sphere
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l∧Antidual[s]//BasisForm
Out[]//TraditionalForm=
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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},SaveDefinitions–>True]
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Expand line onto plane
Expand line onto plane
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l∧Antidual[g]//BasisForm
Out[]//TraditionalForm=
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Expand round point onto circle
Expand round point onto circle
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a∧Antidual[c]//BasisForm
Out[]//TraditionalForm=
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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},SaveDefinitions–>True]
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Expand round point onto line
Expand round point onto line
In[]:=
a∧Antidual[l]//BasisForm
Out[]//TraditionalForm=
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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},SaveDefinitions–>True]
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Expand flat point onto circle
Expand flat point onto circle
In[]:=
p∧Antidual[c]//BasisForm
Out[]//TraditionalForm=
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4315
In[]:=
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},SaveDefinitions–>True]
Out[]=
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Expand flat point onto line
Expand flat point onto line
In[]:=
p∧Antidual[l]//BasisForm
Out[]//TraditionalForm=
e
3215
e
4125
e
4235
e
4315
Expand dipole onto circle
Expand dipole onto circle
In[]:=
d∧Antidual[c]//BasisForm
Out[]//TraditionalForm=
e
1234
e
3215
e
4125
e
4235
e
4315
In[]:=
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},SaveDefinitions–>True]
Out[]=
| | ||||||||||
| |||||||||||
Expand dipole onto line
Expand dipole onto line
In[]:=
d∧Antidual[l]//BasisForm
Out[]//TraditionalForm=
e
1234
e
3215
e
4125
e
4235
e
4315
In[]:=
Manipulate[With[{dip=RoundDipole[1,0,0,0,0,0,1],cir=(Cos[φ]e[2]+Sin[φ]e[3])∧(xe[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},SaveDefinitions–>True]
Out[]=
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| |||||||||||
Expand round point onto dipole
Expand round point onto dipole
In[]:=
a∧Antidual[d]//BasisForm
Out[]//TraditionalForm=
e
1234
e
3215
e
4125
e
4235
e
4315
In[]:=
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},SaveDefinitions–>True]
Out[]=
| | ||||||||||
| |||||||||||
Expand round point onto flat point
Expand round point onto flat point
In[]:=
a∧Antidual[p]//BasisForm
Out[]//TraditionalForm=
–awpw+(aupw–axpx–aypy–azpz)+awpz+awpx+awpy
e
1234
e
3215
e
4125
e
4235
e
4315
In[]:=
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},SaveDefinitions–>True]
Out[]=
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| |||||||||||
License
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.
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.
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