Geometric Algebra

Note on incremental publishing:

This volume is published as it takes shape according to the Leanpub philosophy. The original Italian version (https://leanpub.com/Algebra-Geometrica) guides the development, followed at a short distance by the translations personally curated in English and French (https://leanpub.com/Algebre-geometrique).

Also available in the multilingual bundle [Geometric Algebra Multilingual Bundle](https://leanpub.com/b/geometric-algebra-multilingual-edition-it-en-fr)

To guarantee the maximum expository and mathematical rigor, the publication cadence is not rigid: the indicative objective is the release of a new chapter (or of the relative translated block) roughly every two months. By purchasing now, you will receive for free all the subsequent chapters and errata up to the complete work. You can follow the progress or report typos through the book page on Leanpub.

Audience to which the text is addressed:

All those (mathematicians, engineers, physicists or computer scientists, for example) interested in a vector and multivector calculus independent of Cartesian coordinates, to be used in practical applications (for example, algorithms for sensor fusion), and not entirely satisfied by the current approaches to geometric algebras (both metric and projective) or curious to explore different ones.

Brief description of the content and its objective (Geometry as a "White Box"):

In the traditional analytical approach, space is treated as a pre-existing container and geometric problems are translated into matrices of numerical coordinates. This way of proceeding often behaves like a "black box": the algorithm processes numbers, but the intermediate steps lose the visual and conceptual link with the physical reality of space. Inspired by the original intuition of Hermann Grassmann and the program of Leibniz's characteristica geometrica, this volume proposes a paradigm shift: the algebraic symbols directly denote the geometric objects and the portions of space (for example, points and arrows). In this invariant calculus ("white box"), the introduction of Cartesian coordinates is postponed, operating preliminarily with polynomials of points and vectors capable of preserving in every step their geometric and kinematic meaning.

What Chapter 1 contains: First steps

The first chapter (57 pages with formal treatment, proofs and applicative examples) lays the axiomatic foundations of the poly-point algebra:

Points, weights and affine spaces: The point as an algebraic and geometric generator; distinction between free vectors (linear combinations with null weight) and points of the affine space (combinations with unitary weight).

Polynomials of points and vectors: The associative and distributive product by juxtaposition. The axiom of commutativity: two points or two vectors commute if and only if they are linearly dependent.

Overcoming of Viète's Lex Homogeneorum: The use of non-homogeneous polynomials (multi-points and multi-vectors) understood not as inert forms, but as unified transformation operators.

2-points and bound vectors (cursors): Explicit representation of the line of action, invariance by sliding (principle of transmissibility), pole of reduction and natural decomposition of forces and moments without external artificial constructs.

2-vectors and cyclic bi-points: Equivalence between free bi-vectors and closed polygonal chains of cursors with null translational resultant. Critical dimensional comparison with Gibbs' cross product in ℝ³.

Algebra of the ratio and Cayley's Bracket: Calculation of the ratio between collinear cursors through barycentric transition matrices and invariant determinants. Emergence of the couple (pure moment) in the case of cursors on distinct lines.

3-points and geometric Cramer's Rule: Interpretation of the 3-point as an oriented area and intrinsic proof of Cramer's rule seen as a simple balance of areas of parallelograms subjected to sliding by Cavalieri's principle.

Geometric and physical abstraction: Application of barycentric coordinates to the color space (RGB additive synthesis) and geometric and physical meaning of negative barycentric coordinates as compensation/desaturation light.

Invariant formula of intersection between coplanar lines: Determination of the point of incidence without a priori Cartesian coordinates. A posteriori choice of the basis and visual diagnosis of numerical ill-conditioning.

Prerequisites

The text requires solely two starting skills:

1. Algebraic prerequisites: Knowing how to manipulate polynomials of letters (associative property, distributive property, factorizations).
2. Geometric prerequisites: Intuitive capacity to visualize points, arrows, lines and planes in space.