Dianne Hansford, Gerald Farin

Practical Linear Algebra

A Geometry Toolbox

• Broschiertes Buch

Produktbeschreibung

Linear algebra is growing in importance. The Fourth Edition of this popular text introduces linear algebra in a comprehensive, geometric, and algorithmic way. The authors start with the fundamentals in 2D and 3D, then move on to higher dimensions, expanding on the fundamentals and introducing new topics.

Produktdetails

• Verlag: Taylor & Francis Ltd
• Seitenzahl: 590
• Erscheinungstermin: 26. August 2024
• Englisch
• Abmessung: 235mm x 191mm
• Gewicht: 453g
• ISBN-13: 9781032021911
• ISBN-10: 1032021918
• Artikelnr.: 71235481

Autorenporträt

Gerald Farin (deceased) was a professor in the School of Computing, Informatics, and Design Systems Engineering (CIDSE) at Arizona State University. He received his doctoral degree in mathematics from the University of Braunschweig, Germany. His extensive experience in geometric design started at Daimler-Benz. He was a founding member of the editorial board for the journal Computer-Aided Geometric Design (Elsevier), and he served as co-editor in chief for more than 20 years. He published more than 100 research papers. Gerald also organized numerous conferences and authored or edited 29 books. This includes his much read and referenced textbook Curves and Surfaces for CAGD and his book on NURBS. In addition to this book, Gerald and Dianne co-authored The Essentials of CAGD, Mathematical Principles for Scientific Computing and Visualization both also published by AK Peters/CRC Press. Dianne Hansford, received her Ph.D. from Arizona State University. Her research interests are in the field of geometric modeling with a focus on industrial curve and surface applications related to mathematical definitions of shape. Together with Gerald Farin (deceased), she delivered custom software solutions, advisement on best practices, and taught on-site courses as a consultant. She is a co-founder of 3D Compression Technologies. She is now lecturer in the School of Computing, Informatics, and Design Systems Engineering (CIDSE) at Arizona State University, primarily teaching geometric design, computer graphics, and scientific computing and visualization. In addition to this book, Gerald and Dianne co-authored The Essentials of CAGD, Mathematical Principles for Scientific Computing and Visualization both also published by AK Peters/CRC Press.

Inhaltsangabe

1. Descartes' Discovery. 1.1. Local and Global Coordinates: 2D. 1.2. Going
from Global to Local. 1.3. Local and Global Coordinates: 3D. 1.4. Stepping
Outside the Box. 1.5. Application: Creating Coordinates. 1.6. Exercises.
2. Here and There: Points and Vectors in 2D. 2.1. Points and Vectors. 2.2.
What's the Difference? 2.3. Vector Fields. 2.4. Length of a Vector. 2.5.
Combining Points. 2.6. Independence. 2.7. Dot Product. 2.8. Application:
Lighting Model. 2.9. Orthogonal Projections. 2.10. Inequalities. 2.11.
Exercises. 3. Lining Up: 2D Lines. 3.1. Defining a Line. 3.2. Parametric
Equation of a Line. 3.3. Implicit Equation of a Line. 3.4. Explicit
Equation of a Line. 3.5. Converting Between Line Forms. 3.6. Distance of a
Point to a Line. 3.7. The Foot of a Point. 3.8. A Meeting Place: Computing
Intersections. 3.9. Application: Closest Point of Approach. 3.10.
Exercises. 4. Changing Shapes: Linear Maps in 2D. 4.1. Skew Target Boxes.
4.2. The Matrix Form. 4.3. Linear Spaces. 4.4. Scalings. 4.5 Reflections.
4.6. Rotations. 4.7. Shears. 4.8. Projections. 4.9. Application: Free-form
Deformations. 4.10. Areas and Linear Maps: Determinants. 4.11. Composing
Linear Maps. 4.12. More on Matrix Multiplication. 4.13. Matrix Arithmetic
Rules. 4.14. Exercises. 5. 2 x 2 Linear Systems. 5.1. Skew Target Boxes
Revisited. 5.2. The Matrix Form. 5.3. A Direct Approach: Cramer's Rule.
5.4. Gauss Elimination. 5.5. Pivoting. 5.6. Unsolvable Systems. 5.7.
Underdetermined Systems. 5.8. Homogeneous Systems. 5.9. Kernel. 5.10.
Undoing Maps: Inverse Matrices. 5.11. Defining a Map. 5.12. Change of
Basis. 5.13. Application: Intersecting Lines. 5.14. Exercises. 6. Moving
Things Around: Affine Maps in 2D. 6.1. Coordinate Transformations. 6.2.
Affine and Linear Maps. 6.3. Translations. 6.4. Application: Animation.
6.5. Mapping Triangles to Triangles. 6.6. Composing Affine Maps. 6.7.
Exercises. 7. Eigen Things. 7.1. Fixed Directions. 7.2. Eigenvalues. 7.3.
Eigenvectors. 7.4. Striving for More Generality. 7.5. The Geometry of
Symmetric Matrices and the Eigendecomposition. 7.6. Quadratic Forms. 7.7.
Repeating Maps. 7.8. Exercises. 8. 3D Geometry. 8.1. From 2D to 3D. 8.2.
Cross Product. 8.3. Lines. 8.4. Planes. 8.5. Scalar Triple Product. 8.6.
Application: Lighting and Shading. 8.7. Exercises. 9. Linear Maps in 3D.
9.1. Matrices and Linear Maps. 9.2. Linear Spaces. 9.3. Scalings. 9.4.
Reflections. 9.5 Shears. 9.6. Rotations. 9.7. Projections. 9.8. Volumes and
Linear Maps: Determinants. 9.9. Combining Linear Maps. 9.10. Inverse
Matrices. 9.11. Application: Mapping Normals. 9.12. More on Matrices. 9.13.
Exercises. 10. Affine Maps in 3D. 10.1. Affine Maps. 10.2. Translations.
10.3. Mapping Tetrahedra. 10.4. Parallel Projections. 10.5. Homogeneous
Coordinates and Perspective Maps. 10.6. Application: Building Instance
Models. 10.7. Exercises. 11. Interactions in 3D. 11.1. Distance Between a
Point and a Plane. 11.2. Distance Between Two Lines. 11.3. Lines and
Planes: Intersections. 11.4. Intersecting a Triangle and a Line. 11.5.
Reflections. 11.6. Intersecting Three Planes. 11.7. Intersecting Two
Planes. 11.8. Creating Orthonormal Coordinate Systems. 11.9. Application:
Camera Model. 11.10. Exercises. 12. Gauss for Linear Systems. 12.1. The
Problem. 12.2. The Solution via Gauss Elimination. 12.3. Homogeneous Linear
Systems. 12.4. Inverse Matrices. 12.5. LU Decomposition. 12.6.
Determinants. 12.7. Least Squares. 12.8. Application: Fitting Data from a
Femoral Head. 12.9. Exercises. 13. Alternative System Solvers. 13.1. The
Householder Method. 13.2. Vector Norms. 13.3. Matrix Norms. 13.4. The
Condition Number. 13.5. Vector Sequences. 13.6. Iterative Methods:
Gauss-Jacobi and Gauss-Seidel. 13.7. Application: Mesh Smoothing. 13.8.
Exercises. 14. General Linear Spaces. 14.1. Basic Properties of Linear
Spaces. 14.2. Linear Maps. 14.3. Inner Products. 14.4. Gram-Schmidt
Orthonormalization. 14.5. QR Decompositon. 14.6. A Gallery of Spaces. 14.7.
Least Squares. 14.8. Application: Music Analysis. 14.9. Exercises. 15.
Eigen Things Revisited. 15.1. The Basics Revisited. 15.2. Similarity and
Diagonalization. 15.3. Quadratic Forms. 15.4. The Power Method. 15.5.
Application: Google Eigenvector. 15.6. QR Algorithm. 15.7. Eigenfunctions.
15.8. Application: Inuenza Modelling. 15.9. Exercises. 16. The Singular
Value Decomposition. 16.1. The Geometry of the 2 x 2 Case. 16.2. The
General Case. 16.3. SVD Steps. 16.4. Singular Values and Volumes. 16.5. The
Pseudoinverse. 16.6. Least Squares. 16.7. Application: Image Compression.
16.8. Principal Components Analysis. 16.9. Application: Face
Authentication. 16.10. Exercises. 17. Breaking It Up: Triangles. 17.1.
Barycentric Coordinates. 17.2. Affine Invariance. 17.3. Some Special
Points. 17.4. 2D Triangulations. 17.5. A Data Structure. 17.6. Application:
Point Location. 17.7. 3D Triangulations. 17.8. Exercises. 18. Putting Lines
Together: Polylines and Polygons. 18.1 Polylines. 18.2. Polygons. 18.3.
Convexity. 18.4. Types of Polygons. 18.5. Unusual Polygons. 18.6. Turning
Angles and Winding Numbers. 18.7. Area. 18.8. Application: Planarity Test.
18.9. Application: Inside or Outside? 18.10. Exercises. 19. Conics. 19.1.
The General Conic. 19.2. Analyzing Conics. 19.3. General Conic to Standard
Position. 19.4. The Action Ellipse. 19.5. Exercises. 20. Curves. 20.1.
Parametric Curves. 20.2. Properties of Bézier Curves. 20.3. The Matrix
Form. 20.4. Derivatives. 20.5. Composite Curves. 20.6. The Geometry of
Planar Curves. 20.7. Application: Moving along a Curve. 20.8. Exercises.
A. Applications. B. Glossary. C. Selected Exercises Solutions.
Bibliography.