Understanding the scope of the printed work Volume 1 This page is an extract from “Grassmann Algebra Volume 1: Foundations - Exploring extended vector algebra with Mathematica” by John Browne. First Edition 2012.

4  Geometric Interpretations  

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4.1  Introduction

157

4.2  Geometrically Interpreted 1-Elements

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Vectors 158 Points 160 Declaring a basis for a bound vector space 163 Composing vectors and points 164 Example: Calculation of the centre of mass 165

4.3  Geometrically Interpreted 2-Elements

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Simple geometrically interpreted 2-elements 166 Bivectors 167 Bound vectors 169 Composing bivectors and bound vectors 171 The sum of two parallel bound vectors 172 The sum of two non-parallel bound vectors 173 Sums of bound vectors 175 Example: Reducing a sum of bound vectors 176

4.4  Geometrically Interpreted m-Elements

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Types of geometrically interpreted m-elements 177 The m-vector 178 The bound m-vector 179 Bound simple m-vectors expressed by points 179 Bound simple bivectors 180 Composing m-vectors and bound m-vectors 182

4.5  Geometrically Interpreted Spaces

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Vector and point spaces 183 Coordinate spaces 184 Geometric dependence 184 Geometric duality 184

4.6  m-Planes

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m-planes defined by points or bound m-vectors 186 m-planes defined by m-vectors 186 m-planes as exterior quotients 186 Computing exterior quotients 187 The m-vector of a bound m-vector 188

4.7  Line Coordinates

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Lines in a plane 190 Lines in a 3-plane 193 Lines in a 4-plane 196 Lines in an m-plane 196 Exploration: Line simplicity 197

4.8  Plane Coordinates

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Planes in a 3-plane 200 Planes in a 4-plane 204 Planes in an m-plane 204 The coordinates of geometric entities 204

4.9  Calculation of Intersections

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The intersection of two lines in a plane 205 The intersection of a line and a plane in a 3-plane 206 The intersection of two planes in a 3-plane 207 Example: The osculating plane to a curve 208

4.10  Decomposition into Components

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The shadow 209 Decomposition in a 2-space 210 Decomposition in a 3-space 213 Decomposition in a 4-space 215 Decomposition of a point or vector in an n-space 216

4.11  Projective Space

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The relationship between Projective and Grassmann geometries 216 The intersection of two lines in a plane 217 The line at infinity in a plane 218 Projective 3-space 219 Homogeneous coordinates 219 Duality 221 Desargues' theorem 222 Pappus' theorem 223 Projective n-space 225

4.12  Projection

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Central projection 226 A general projection formula 228 Computing the image of a central projection 229 Examples in coordinate form 231 General projection 233 Parallel projection 234 The generality of the projection formulae 237

4.13  Regions of Space

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Regions of space 237 Regions of a plane 238 Regions of a line 239 Planar regions defined by two lines 240 Planar regions defined by three lines 242 Creating a pentagonal region 244 Creating a 5-star region 247 Creating a 5-star pyramid 251 Summary 253

4.14  Geometric Constructions

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Geometric expressions 254 Geometric equations for lines and planes 255 The geometric equation of a conic section in the plane 256 The geometric equation as a prescription to construct 257 The algebraic equation of a conic section in the plane 258 An alternative geometric equation of a conic section in the plane 260 Conic sections through five points 262 Dual constructions 265 Constructing conics in space 266 A geometric equation for a cubic in the plane 269 Pascal's Theorem 271 Hexagons in a conic 275 Pascal points 278 Pascal lines 281

4.15  Summary

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5  The Complement  

289 

5.1  Introduction

289

5.2  Axioms for the Complement

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The grade of a complement 291 The linearity of the complement operation 291 The complement axiom 291 The complement of a complement axiom 292 The complement of unity 294

5.3  Defining the Complement

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The symmetry relation for regressive products 294 The complement of an m-element 295 The defining identity for the complement of an m-element 296 Defining the complement of a basis element 296 Defining the complement of the basis n-element 297 Complements of basis elements in matrix form 298 Complements of cobasis elements in matrix form 298 Adjoints 300

5.4  Metrics

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Complements of basis m-elements 300 The metric on a linear space of m-elements 301 Metrics in a 3-space 302 Complements of basis m-elements in matrix form 304 Composing metrics 305 The determinant of an m-metric 306

5.5  Cometrics

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The relation between a metric and its cometric in 3-space 307 The general relation between a metric and its cometric 309 Computing a Q matrix 310 Tabulating Q matrices 312 Properties of Q matrices 313 Testing the formulae 315

5.6  The Euclidean Complement

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Tabulating Euclidean complements of basis elements 315 Formulae for the Euclidean complement of basis elements 317 Products leading to a scalar or n-element 318

5.7  Exploring Complements

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Alternative forms for complements 319 Orthogonality 320 Visualizing the complement axiom 322 The regressive product in terms of complements 322 The complement of a simple element is simple 323 Glimpses of the inner product 323 Idempotent complements 324

5.8  Working with Metrics

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Working with metrics 325 The default metric 325 Declaring a metric 326 Declaring a general metric 326 Calculating induced metrics 327 Creating palettes of induced metrics 328 The determinant of the metric tensor 330

5.9  Calculating Complements

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Entering complements 330 Creating palettes of complements of basis elements 331 Converting complements of basis elements 333 Simplifying expressions involving complements 337 Converting expressions involving complements to specified forms 338 Converting regressive products of basis elements in a metric space 339

5.10  Complements in a vector space

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The Euclidean complement in a vector 2-space 340 The non-Euclidean complement in a vector 2-space 341 The Euclidean complement in a vector 3-space 343 The non-Euclidean complement in a vector 3-space 345

5.11  Complements in a bound space

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Metrics in a bound space 347 The hybrid metric 348 The orthogonality of the origin to m-vectors 348 Unit elements in a bound vector space 349 Complement equivalences in a bound vector 2-space 349 Complement equivalences in a bound vector 3-space 352 Complement equivalences in a bound vector n-space 354 The complement of the complement of an m-vector in a bound space 355 Calculating with vector space complements 356

5.12  Complements of bound elements

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The Euclidean complement of a point in the plane 357 The Euclidean complement of a point in a bound 3-space 359 The complement of a point in a bound n-space 360 The complement of a bound element 361 The complement of the complement of a bound element 362 Entities which are of the same type as their complements 363 Euclidean complements of bound elements 364 The regressive product of point complements 366

5.13  Reciprocal Bases

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Reciprocal bases 367 The complement of a basis element 368 The complement of a cobasis element 370 Products of basis elements 370

5.14  Summary

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