Manipulating multivectors
In this tutorial we cover GALua's overload of the indexing operator ("[]") for multivector values.
We have already seen that if the value used as an index with this operator is a multivector
value, then we just get the inner product. If, however, you provide an integer index,
the result is a multivector that is the part of the indexed multivector being homogeneous
of the grade that is the intergral index. An example may make this clearer.
local vecA = e1 + e2 local vecB = e2 + e3 local rotor = vecA * vecB print( "rotor = " .. tostring( rotor ) ) print( "rotor[0] = " .. tostring( rotor[0] ) ) print( "rotor[1] = " .. tostring( rotor[1] ) ) print( "rotor[2] = " .. tostring( rotor[2] ) )
Running this Lua code, we get the following output.
rotor = 1.00 + e1^e2 + e1^e3 + e2^e3 rotor[0] = 1.00 rotor[1] = 0 rotor[2] = e1^e2 + e1^e3 + e2^e3
We can also use the indexing operator ("[]") to assign certain grade
parts of a given multivector.
rotor[0] = e2 .. e1 rotor[2] = e2 ^ e1
Be aware that if you try to assign a multivector homogeneous of the wrong grade, or try to assign a multivector that is not homogeneous of any grade, you will get a fatal error.
rotor[2] = vecB -- Error! rotor[0] = vecA * vecB -- Error!
Being able to manipulate multivectors this way, it also becomes important to realize that a multivector variable is nothing more than a reference to a multivector value. Consider the following Lua code.
local multivecA = 5 + e1 + e1^e2 local multivecB = multivecA multivecB[1] = e2 print( "multivecA = " .. tostring( multivecA ) )
This script will print "5 + e2 + e1^e2", not "5 + e1 + e1^e2"
as one might expect. What the programmer may have intended to do is the following.
local multivecA = 5 + e1 + e1^e2 local multivecB = multivecA:copy() multivecB[1] = e2 print( "multivecA = " .. tostring( multivecA ) )
By utilizing the :copy() meta-method of the multivector data-type, we're
able to return a new copy of the value referenced by the multivecA variable to
be referenced by the multivecB variable. This new code now prints
"5 + e1 + e1^e2" as originally intended.