In this section we have introduced multivectors and covered their basic mathematical products such as Ù and ¿, and standard operations such as Clifford conjugation §#, inverse -1, projection ¯ and meet Ç in some depth. We have seen how blades provide an elegant system for representing and manipulating the intersections of k-spheres and k-planes, while mixed grade multivectors can embody transformations such as rotations, shears and translations. All for arbitary dimension N and for both Euclidean and nonEuclidean geometries such as Minkowski and spherical.
    We have described a coordinate-based C++ implementation of a sparsity-exploiting multivector class for large N and seen how multivector operations can be programmed.

    It is ultimately most natural to regard multivectors as generalised numbers   ameniable to various geometrical interpretations. The addition of differing grade blades so objectionable to those disinclined to "add different things"  may under a different geometrical interpretaion correspond to the widely accepted addition of 1-vectors.
    Multivectors were "discovered" by William Clifford in 1870s following Grassman but languished in mathematical obscurity until developed and championed by David Hestenes in 1960s, recieving serious attention in 1980s. The contractive product ¿ and associated projection ¯ of Lounesto and Dorst; the General Homegenised embedding of Li, Hestenes, and Rockwood ; and the Bouma delta product D were important recent developments in the 1990s. The field remains surprisingly young given its historical age and likely importance.

    Now we turn to the multivector generalisation of vector calculus.
    Next : Multivector Calculus

Sierpinski (<1K) References/Source Material for Multivector Methods

    David Hestenes, Hongbo Li, Alyn Rockwood. "New Algebraic Tools for Classical Geometry"

    David Hestenes, Garret Sobczyk "Clifford Algebra to Geometric Calculus" D. Reidel Publishing 1984,1992 [Amazon US UK]

    Hongbo Li, David Hestenes, Alyn Rockwood. "Generalised Homogeneous Coordinates for Computational Geometry"

    David Hestenes, Hongbo Li, Alyn Rockwood. "Spherical Conformal Geometry with Geometric Algebra"

    Leo Dorst. "Honing Geometric Algebra for Computer Sciences"

    Leo Dorst. "The Inner Products of Geometric Algebra"

    Patrick Reany. "The Square Root of a Vector"

    Pertti Lounesto "Counterexamples for Validation and Discovering of New Theorems" [ "Geometric Algebra with Applications in Science and Engineering" ed. Corrchano & Sobczyk [ISBN 0-8176-4199-8] [Amazon US UK]

    Pertti Lounesto "Clifford Algebras and Spinors [2nd Ed]" Cambridge University Press 2001 [Amazon US UK]

    Tim Bouma,Leo Dorst,H G J Pijls "Geometric Algebra for SubSpace Operations" Acta Applicandae Mathematicae ,73 . Kluwa Academic Publications pp285-300. 2002

    Stephen Mann,Leo Dorst.Daniel Fontinje "The Making of GABLE: A Geometric Algebra Learning Environment in Matlab" [ "Geometric Algebra with Applications in Science and Engineering" ed. Corrchano & Sobczyk [ISBN 0-8176-4199-8] [Amazon US UK]

    Daniel Fontinje,Tim Bouma,Leo Dorst. "Gaigen: a Geometric Algbera Implementation Generator"

    Daniel Fontinje "Efficient Numerical Implimentation of High Dimensional Clifford Algebras [DRAFT]"

    Anthony Lasenby "Recent Applications of Conformal Geometric Algebra"   2005

    Leo Dorst. Daniel Fontijne, Steve Mann "Geometric Algebra for Computer Science" Morgan Kaufmann 2007 [Amazon US UK]

    Leo Dorst. Robert Valkenburg "Square Root and Logarithm of Rotors in 3D Conformal Geometric Algebra Using Polar Decomposition" [ "Guide to Geometric Algebra in Practice"Geometric Algebra for Computer Science" ed Dorst & Lasenby Springer 2011 [ISBN 978--=85729-810-2] [Amazon US UK]

    Nina Amenta, Sunghee Choi, Ravi Krishna, Kolluri "The Power Crust, Unions of Balls, and the Medial Axis Theorem"

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