Maths - choosing bases - 3D

Here is how I generated the tables for this page.

The tables were generated using this program.

The output of this program is shown below. To produce the results the program needs to have an XML input code. At the bottom of this page I have listed this input code.

Table for: g 3,0,0

a*bb.eb.e1b.e2b.e3b.e12b.e31b.e23b.e123
a.eee1e2e3e12e31e23e123
a.e1e1ee12-e31e2-e3e123e23
a.e2e2-e12ee23-e1e123e3e31
a.e3e3e31-e23ee123e1-e2e12
a.e12e12-e2e1e123-ee23-e31-e3
a.e31e31e3e123-e1-e23-ee12-e2
a.e23e23e123-e3e2e31-e12-e-e1
a.e123e123e23e31e12-e3-e2-e1-e

analysing commutivity: table does not commute: for example: e1*e2 != e2*e1

analysing associativity: table associates

Table for: g+ 3,0,0

a*bb.eb.e12b.e31b.e23
a.eee12e31e23
a.e12e12-ee23-e31
a.e31e31-e23-ee12
a.e23e23e31-e12-e

analysing commutivity: table does not commute: for example: e12*e31 != e31*e12

analysing associativity: table associates

Table for: g 2,1,0

a*bb.eb.e1b.e2b.e3b.e12b.e31b.e23b.e123
a.eee1e2e3e12e31e23e123
a.e1e1-ee12-e31-e2e3e123-e23
a.e2e2-e12ee23-e1e123e3e31
a.e3e3e31-e23ee123e1-e2e12
a.e12e12e2e1e123e-e23-e31e3
a.e31e31-e3e123-e1e23ee12e2
a.e23e23e123-e3e2e31-e12-e-e1
a.e123e123-e23e31e12e3e2-e1e

analysing commutivity: table does not commute: for example: e1*e2 != e2*e1

analysing associativity: table associates

Table for: g+ 2,1,0

a*bb.eb.e12b.e31b.e23
a.eee12e31e23
a.e12e12e-e23-e31
a.e31e31e23ee12
a.e23e23e31-e12-e

analysing commutivity: table does not commute: for example: e12*e31 != e31*e12

analysing associativity: table associates

Table for: g 1,2,0

a*bb.eb.e1b.e2b.e3b.e12b.e31b.e23b.e123
a.eee1e2e3e12e31e23e123
a.e1e1-ee12-e31-e2e3e123-e23
a.e2e2-e12-ee23e1e123-e3-e31
a.e3e3e31-e23ee123e1-e2e12
a.e12e12e2-e1e123-e-e23e31-e3
a.e31e31-e3e123-e1e23ee12e2
a.e23e23e123e3e2-e31-e12ee1
a.e123e123-e23-e31e12-e3e2e1-e

analysing commutivity: table does not commute: for example: e1*e2 != e2*e1

analysing associativity: table associates

Table for: g+ 1,2,0

a*bb.eb.e12b.e31b.e23
a.eee12e31e23
a.e12e12-e-e23e31
a.e31e31e23ee12
a.e23e23-e31-e12e

analysing commutivity: table does not commute: for example: e12*e31 != e31*e12

analysing associativity: table associates

Table for: g 0,3,0

a*bb.eb.e1b.e2b.e3b.e12b.e31b.e23b.e123
a.eee1e2e3e12e31e23e123
a.e1e1-ee12-e31-e2e3e123-e23
a.e2e2-e12-ee23e1e123-e3-e31
a.e3e3e31-e23-ee123-e1e2-e12
a.e12e12e2-e1e123-e-e23e31-e3
a.e31e31-e3e123e1e23-e-e12-e2
a.e23e23e123e3-e2-e31e12-e-e1
a.e123e123-e23-e31-e12-e3-e2-e1e

analysing commutivity: table does not commute: for example: e1*e2 != e2*e1

analysing associativity: table associates

Table for: g+ 0,3,0

a*bb.eb.e12b.e31b.e23
a.eee12e31e23
a.e12e12-e-e23e31
a.e31e31e23-e-e12
a.e23e23-e31e12-e

analysing commutivity: table does not commute: for example: e12*e31 != e31*e12

analysing associativity: table associates

Table for: g 2,0,1

a*bb.eb.e1b.e2b.e3b.e12b.e31b.e23b.e123
a.eee1e2e3e12e31e23e123
a.e1e1ee12-e31e2-e3e123e23
a.e2e2-e120e230e12300
a.e3e3e31-e23ee123e1-e2e12
a.e12e12-e20e1230e2300
a.e31e31e3e123-e1-e23-ee12-e2
a.e23e23e1230e20-e1200
a.e123e123e230e120-e200

analysing commutivity: table does not commute: for example: e1*e2 != e2*e1

analysing associativity: table associates

Table for: g+ 2,0,1

a*bb.eb.e12b.e31b.e23
a.eee12e31e23
a.e12e120e230
a.e31e31-e23-ee12
a.e23e230-e120

analysing commutivity: table does not commute: for example: e12*e31 != e31*e12

analysing associativity: table associates

Table for: g 1,1,1

a*bb.eb.e1b.e2b.e3b.e12b.e31b.e23b.e123
a.eee1e2e3e12e31e23e123
a.e1e1-ee12-e31-e2e3e123-e23
a.e2e2-e120e230e12300
a.e3e3e31-e23ee123e1-e2e12
a.e12e12e20e1230-e2300
a.e31e31-e3e123-e1e23ee12e2
a.e23e23e1230e20-e1200
a.e123e123-e230e120e200

analysing commutivity: table does not commute: for example: e1*e2 != e2*e1

analysing associativity: table associates

Table for: g+ 1,1,1

a*bb.eb.e12b.e31b.e23
a.eee12e31e23
a.e12e120-e230
a.e31e31e23ee12
a.e23e230-e120

analysing commutivity: table does not commute: for example: e12*e31 != e31*e12

analysing associativity: table associates

Table for: g 1,0,2

a*bb.eb.e1b.e2b.e3b.e12b.e31b.e23b.e123
a.eee1e2e3e12e31e23e123
a.e1e10e12-e3100e1230
a.e2e2-e120e230e12300
a.e3e3e31-e23ee123e1-e2e12
a.e12e1200e1230000
a.e31e310e123-e100e120
a.e23e23e1230e20-e1200
a.e123e12300e120000

analysing commutivity: table does not commute: for example: e1*e2 != e2*e1

analysing associativity: table associates

Table for: g+ 1,0,2

a*bb.eb.e12b.e31b.e23
a.eee12e31e23
a.e12e12000
a.e31e3100e12
a.e23e230-e120

analysing commutivity: table does not commute: for example: e31*e23 != e23*e31

analysing associativity: table associates

Table for: g 0,0,3

a*bb.eb.e1b.e2b.e3b.e12b.e31b.e23b.e123
a.eee1e2e3e12e31e23e123
a.e1e10e12-e3100e1230
a.e2e2-e120e230e12300
a.e3e3e31-e230e123000
a.e12e1200e1230000
a.e31e310e12300000
a.e23e23e123000000
a.e123e1230000000

analysing commutivity: table does not commute: for example: e1*e2 != e2*e1

analysing associativity: table associates

XML input code

To produce the results the program needs to have an XML input code listed here:

<classDef>
<outputTable type="product" format="html" name="g 3,0,0" analyse="on" enableLabels="on">
<mathTypeMulti name="a" type="3" sign="0" zero="0" subAlgebra="all"/>
</outputTable>
<outputTable type="product" format="html" name="g+ 3,0,0" analyse="on" enableLabels="on">
<mathTypeMulti name="a" type="3" sign="0" zero="0" subAlgebra="even"/>
</outputTable>
<outputTable type="product" format="html" name="g 2,1,0" analyse="on" enableLabels="on">
<mathTypeMulti name="b" type="3" sign="1" zero="0" subAlgebra="all"/>
</outputTable>
<outputTable type="product" format="html" name="g+ 2,1,0" analyse="on" enableLabels="on">
<mathTypeMulti name="b" type="3" sign="1" zero="0" subAlgebra="even"/>
</outputTable>
<outputTable type="product" format="html" name="g 1,2,0" analyse="on" enableLabels="on">
<mathTypeMulti name="b" type="3" sign="3" zero="0" subAlgebra="all"/>
</outputTable>
<outputTable type="product" format="html" name="g+ 1,2,0" analyse="on" enableLabels="on">
<mathTypeMulti name="b" type="3" sign="3" zero="0" subAlgebra="even"/>
</outputTable>
<outputTable type="product" format="html" name="g 0,3,0" analyse="on" enableLabels="on">
<mathTypeMulti name="b" type="3" sign="7" zero="0" subAlgebra="all"/>
</outputTable>
<outputTable type="product" format="html" name="g+ 0,3,0" analyse="on" enableLabels="on">
<mathTypeMulti name="b" type="3" sign="7" zero="0" subAlgebra="even"/>
</outputTable>
<outputTable type="product" format="html" name="g 2,0,1" analyse="on" enableLabels="on">
<mathTypeMulti name="b" type="3" sign="0" zero="2" subAlgebra="all"/>
</outputTable>
<outputTable type="product" format="html" name="g+ 2,0,1" analyse="on" enableLabels="on">
<mathTypeMulti name="b" type="3" sign="0" zero="2" subAlgebra="even"/>
</outputTable>
<outputTable type="product" format="html" name="g 1,1,1" analyse="on" enableLabels="on">
<mathTypeMulti name="b" type="3" sign="1" zero="2" subAlgebra="all"/>
</outputTable>
<outputTable type="product" format="html" name="g+ 1,1,1" analyse="on" enableLabels="on">
<mathTypeMulti name="b" type="3" sign="1" zero="2" subAlgebra="even"/>
</outputTable>
<outputTable type="product" format="html" name="g 1,0,2" analyse="on" enableLabels="on">
<mathTypeMulti name="b" type="3" sign="0" zero="3" subAlgebra="all"/>
</outputTable>
<outputTable type="product" format="html" name="g+ 1,0,2" analyse="on" enableLabels="on">
<mathTypeMulti name="b" type="3" sign="0" zero="3" subAlgebra="even"/>
</outputTable>
<outputTable type="product" format="html" name="g 0,0,3" analyse="on" enableLabels="on">
<mathTypeMulti name="b" type="3" sign="0" zero="7" subAlgebra="all"/>
</outputTable>
</classDef>


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flag flag flag flag flag flag Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics (Fundamental Theories of Physics). This book is intended for mathematicians and physicists rather than programmers, it is very theoretical. It covers the algebra and calculus of multivectors of any dimension and is not specific to 3D modelling.

 

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