17 Facts About Geometric algebra

In mathematics, a geometric algebra is an extension of elementary algebra to work with geometrical objects such as vectors.

Geometric algebra is built out of two fundamental operations, addition and the geometric product.

Geometric algebra has been advocated, most notably by David Hestenes and Chris Doran, as the preferred mathematical framework for physics.

Essential product in the algebra is called the geometric product, and the product in the contained exterior algebra is called the exterior product.

The above definition of the geometric algebra is abstract, so we summarize the properties of the geometric product by the following set of axioms.

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Since every element of the Geometric algebra can be expressed as the sum of products of this form, this defines the exterior product for every pair of elements of the Geometric algebra.

Every multivector of the geometric algebra can be expressed as a linear combination of the canonical basis elements.

Elements of the geometric algebra that are scalar multiples of are grade- blades and are called scalars.

Many of the elements of the Geometric algebra are not graded by this scheme since they are sums of elements of differing grade.

Geometric algebra represents subspaces of as blades, and so they coexist in the same algebra with vectors from.

From Vectors to Geometric Algebra covers basic analytic geometry and gives an introduction to stereographic projection.

Projective geometric algebra, known as the homogeneous model, provides a complete algebra containing representations of all Euclidean isometries and the linear subspaces on which they operate.

The motor Geometric algebra is the correct generalization of dual quaternions to the full set of objects that appear in PGA.

Every product in the algebra has a matching "antiproduct" that performs the same operation on the duals of its operands.

All objects in projective geometric algebra are homogeneous, including magnitudes that would simply be scalars in nonprojective settings.

Geometric algebra calculus extends the formalism to include differentiation and integration including differential geometry and differential forms.

In mathematics, Emil Artin's Geometric Algebra discusses the algebra associated with each of a number of geometries, including affine geometry, projective geometry, symplectic geometry, and orthogonal geometry.