Este libro da una visión valiosa en principios para robótica, visión por computador, diseño, manufactura, cinemática y dinámica desde un punto de vista práctico. Al mismo tiempo, mantiene un contacto con las propiedades matemáticas decisivas para los objetos y transformaciones tratadas.
- Title
- Copyright
- Contents
- 1 Introduction
- 2 Basic Concepts. Groups
- 2.1 Functions
- 2.1.1 Properties of Functions
- 2.1.2 Composition of Functions
- 2.2 Binary Operations and Groups
- 2.2.1 Binary Operations
- 2.2.2 Groups
- 2.3 Square Matrices
- 2.3.1 Matrix Invertibility
- 2.3.2 Properties of the Inverse and Transposed Matrix
- 2.4 The General Linear Group GL(n, R)
- 2.5 The Positive Linear Group (GL+(n, R))
- 2.6 The Orthogonal Group O(n)
- 2.7 The Special Orthogonal Group SO(n)
- 2.8 The Group (X, ∘)
- 2.9 Transformations
- 2.9.1 Proposed Exercise
- 2.9.2 Linear Transformations
- 2.10 Affine Transformations
- 2.10.1 Affine Transformations in R2
- 2.10.2 Proposed Exercise
- 2.11 Summary
- 3 Property Invariance under Geometric Transformations
- 3.1 Introduction
- 3.2 Property Preservation in General Transformations
- 3.2.1 Colinearity Preservation
- 3.2.2 Distance Preservation
- 3.2.3 Volume Preservation
- 3.2.4 Angle Preservation
- 3.2.5 Orientation Preservation
- 3.2.6 Origin Preservation
- 3.2.7 Example. Plane Reflection
- 3.3 Property Preservation in Affine Functions
- 3.3.1 Discusion. Affine and Linear Transformations in 3D
- 3.4 Example. A Linear Transformation in R2
- 3.5 Example. Non-Linear Transformations R2 → R2
- 3.6 Proposed Exercise. Non-Linear Transformations R2 → R2
- 3.6.1 Area-Preservation. Proof
- 3.6.2 Area-Preservation. Programming
- 3.7 Proposed Exercise. Affine Transformations R2 → R2, Aff(2, R)
- 3.8 Solved Exercise. Non-affine Transformations R2 → R2
- 3.9 Homogeneous Coordinates
- 3.9.1 Definition
- 3.9.2 Rationale for Homogeneous Coordinates
- 3.9.3 Transformations in Homogeneous Coordinates
- 3.9.4 Proposed Exercises
- 3.10 Coordinate Systems
- 3.10.1 Definition. Coordinate Systems
- 3.10.2 Definition. Right Handed Canonical Coordinate System in R3
- 3.10.3 Proposed Exercise
- 3.10.4 Solved Exercise
- 4 Rigid Transformations in R3
- 4.1 Definition. Rigid Transformations
- 4.2 Pure Translations
- 4.3 Pure Rotations
- 4.3.1 Rotations about the Principal Axes
- 4.3.2 Proposed Exercises
- 4.4 Eigenvalues and eigenvectors of R ∈ SO(3)
- 4.4.1 Eigenvalues and Eigenvector of Matrices SO(3)
- 4.4.2 Trasformation Sequences
- 4.4.3 Solved Exercise. Transformation Sequences
- 4.4.4 Proposed Exercises. Rotations about Main Axis
- 4.4.5 Rotations about Arbitrary Axis. Quaternion
- 4.5 General Rigid Transformation Using Quaternions
- 4.6 Solved and Proposed Exercises
- 4.6.1 Solved Exercise. Quaternion
- 4.6.2 Proposed Exercise. Rigid Transformations
- 4.6.3 Proposed Exercise. Flight Simulator
- 5 Non-Rigid Transformations and Functions
- 5.1 Non-Rigid Affine Transformations
- 5.1.1 Scalings
- 5.1.2 Reflections
- 5.1.3 Shears
- 5.2 Pseudo-affine Geometric Functions. Parallel Projections
- 5.2.1 Orthogonal Parallel Projections
- 5.2.2 Non-orthogonal Parallel Projections
- 5.3 Non-Linear Non-Invertible Functions. Perspective Projections
- 5.3.1 Perspective of a Point
- 5.3.2 Perspective of a Line
- 5.3.3 Perspective and Partition of the Lines in R3
- 5.3.4 Proposed Exercise. Perspective Projection
- Bibliography
- Academic Acknowledgments
- Contracubierta
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