Book Volume 1- Table of Contents: page 2 of 3

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

## 157

### 4.1 Introduction

### 157

### 4.2 Geometrically Interpreted 1-Elements

### 158

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

### 166

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

### 177

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

### 183

Vector and point spaces
183
Coordinate spaces
184
Geometric dependence
184
Geometric duality
184
### 4.6 m-Planes

### 185

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

### 190

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

### 200

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

### 205

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

### 209

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

### 216

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

### 226

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

### 237

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

### 254

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

### 287

## 5 The Complement

## 289

### 5.1 Introduction

### 289

### 5.2 Axioms for the Complement

### 291

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

### 294

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

### 300

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

### 307

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

### 315

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

### 319

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

### 325

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

### 330

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

### 340

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

### 347

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

### 357

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

### 367

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

### 371

© John Browne 2012.