GrassmannCalculus`
Origin (★)  |
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Details and Options
Examples
(1)
Basic Examples
(1)
In[1]:=
<<GrassmannCalculus`
Set the GrassmannPlane coordinate system.
In[2]:=
 | SetActiveSpacePreferences |
 | PublicGrassmannAtlas |
The x axis can be specified by the exterior product of the Origin and the x direction, and then parametrized. These are the implicit and explicit representations of the x axis.
In[3]:=
★⋀
[%][1].{1,x}
e
x
 | ComposeSpan |
Out[3]=
★⋀
e
x
Out[3]=
★+x
e
x
In an expression like this the vector should not be considered as being at any specific location. It represents a transport of the rest of the expression, whatever that might be, in the x direction, which is where the term vector (Latin "carrier") arose.
e
x
A general point in the plane and a specific point in the plane:
In[4]:=
point1={1,x,y}.Basispoint2=
[a]
| ComposePoint |
Out[4]=
★+x+y
e
x
e
y
Out[4]=
★++
e
x
a
1
e
y
a
2
Using the Euclidean metric, the distance between the two points is:
In[5]:=
| Measure |
Out[5]=
2
(x-)
a
1
2
(y-)
a
2
For plotting, the coordinates of a point can be obtained by . The Origin is always taken to have coordinates .
ToListCoordinates
{0,0,…}
In[6]:=
{point1,point2,★+2+3}%//
e
x
e
y
| ToListCoordinates |
Out[6]=
{★+x+y,★++,★+2+3}
e
x
e
y
e
x
a
1
e
y
a
2
e
x
e
y
Out[6]=
{{x,y},{,},{2,3}}
a
1
a
2
Points can be added to obtain weighted points. factors out the weight and drops the weight, giving the location of the point.
ToWeightedPointForm
ToPointForm
In[7]:=
point1+point2+★+2+3//Simplify%//
%//
e
x
e
y
| ToWeightedPointForm |
| ToPointForm |
Out[7]=
3★+(2+x+)+(3+y+)
e
x
a
1
e
y
a
2
Out[7]=
3★+(2+x+)+(3+y+)
1
3
e
x
a
1
1
3
e
y
a
2
Out[7]=
★+(2+x+)+(3+y+)
1
3
e
x
a
1
1
3
e
y
a
2
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