Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
Operator Duality
Complement reflection with respect to a point dual to a plane
In[315]:=
<<Wolfram`GeometricAlgebra`ProjectiveGeometry`
In[316]:=
g=PGAPlane[{1,0,0},1/4]
Out[316]=
e
423
1
4
e
321
In[317]:=
g
Out[317]=
Point[{4,0,0}]
In[662]:=
reflection=PGAPoint[AntiGeometricProduct[g,Point[{x,y,0}],]]〚1,;;2〛
g
Out[662]=
--x,y
1
2
In[663]:=
f[{x_,y_}]:=Sin[2Pix]DensityPlot[f[reflection],{x,-4,4},{y,-4,4},ColorFunction"BlueGreenYellow",PlotPoints100,FrameTrue,AxesTrue,Epilog{Thick,Red,InfiniteLine[{{-1/4,0},{-1/4,4}}]},AxesLabel{Style["x",Italic,18],Style["y",Italic,18]}]
Out[664]=
In[655]:=
antiReflection=PGAPoint
[,Point[{x,y,0}],]1,;;2
 | GeometricProduct |
g
g
Out[655]=
,-
x
-1+
x
2
y
-1+
x
2
In[658]:=
VectorDensityPlot[antiReflection,{x,-2,6},{y,-4,4},FrameTrue,AxesTrue,Epilog{Thick,PointSize[Large],Red,InfiniteLine[{{0,0},{0,4}}],Point[{4,0}],White,InfiniteLine[{{2,0},{2,1}}]},AxesLabel{Style["x",Italic,18],Style["y",Italic,18]}]
Out[658]=
In[665]:=
ComplexPlot-I,{z,-2-4I,6+4I}
Re[z]
-1+
Re[z]
2
Im[z]
-1+
Re[z]
2
Out[665]=
Complement rotation:
In[638]:=
R=PGARotor[InfiniteLine[{{1/2,0,0},{1/2,0,1}}],phi]
Out[638]=
Sin+-Sin+Cos
phi
2
e
43
1
2
phi
2
e
31
phi
2
In[652]:=
stream=Simplify[PGAPoint@AntiGeometricProduct[R,Point[{x,y,0}],]]〚1,;;2〛
R
Out[652]=
+-+xCos[phi]-ySin[phi],yCos[phi]-+xSin[phi]
1
2
1
2
Sin[phi]
2
In[653]:=
StreamPlot[D[stream,phi]/.phi0,{x,-4,4},{y,-4,4}]
Out[653]=
In[656]:=
antiStream=SimplifyPGAPoint@
,Point[{x,y,0}],1,;;2
 | GeometricProduct |
R
R
Out[656]=
,
2xCos[phi]-2ySin[phi]
2+x-xCos[phi]+ySin[phi]
2(yCos[phi]+xSin[phi])
2+x-xCos[phi]+ySin[phi]
In[657]:=
StreamPlot[D[antiStream,phi]/.phi0,{x,-4,4},{y,-4,4}]
Out[657]=
""