Geometric Transformations for 3D Modeling

Geometric Transformations for 3D Modeling, Second Edition

Michael E. Mortenson


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Delivering the theory and application of geometric transformations is the comprehensive mission of Geometric Transformations for 3D Modeling by Michael E Mortenson. The 380 page hardcover volume is the second edition. It provides a thorough introduction. Written from the standpoint of transformation mathematics, the content is accessible by students, teachers and professionals in the field. It is also ideal for practitioners in engineering, math and physics beyond the manufacturing industry.


The first three chapters provide needed abstract foundation material and theory to cue up actual transformations in subsequent chapters. The author's approach reduces the clutter of theoretical derivation in the remaining text. Mortenson introduces operational and application-centered tools and concepts as the need dictates.


The volume assumes that readers are already familiar with analytic geometry and first-year calculus. Readers should also possess basic working knowledge of matrix and vector algebra. With these prerequisites, Geometric Transformations for 3D Modeling will appeal to anyone working in 3D and geometric modeling, computer graphics, animation, robotics and kinematics.



  • Explores and develops the subject in substantially greater breadth and depth than other books
  • Offers improved understanding of transformation theory, the role of invariants, uses of various notation systems and the relations between transformations
  • Describes how geometric objects may change position, orientation, or even shape when subjected to mathematical operations (while properties characterizing their geometric identity and integrity remain unchanged)
  • Presents eigenvalues, eigenvectors, and tensors in an easy to understand way
  • Contains revised and improved figures. Many illustrations use color to highlight important features
  • Exercises are included in nearly all chapters, with answers found at the end of the book

Michael E. Mortenson, a graduate of the UCLA School of Engineering, writes and does independent research in 3D geometric modeling and computer-aided design. He is a former research scientist with a major aerospace corporation and the author of several highly successful textbooks.



  • What Is Geometry
  • History
  • Geometric Objects
  • Space
  • Geometry Is…
  • E Pluribus Unum – Transformation and Invariance

Theory of Transformations

  • Functions, Mappings, and Transformations
  • Linear Transformations
  • Geometric Invariants
  • Isometries
  • Similarities
  • Affinities
  • Projectivities
  • Topological Transformations

Vector Spaces

  • Introduction to Linear Vector Spaces
  • Basis Vectors
  • Eigenvalues and Eigenvectors
  • Tensors

Rigid-Body Motion

  • Translation
  • Rotation
  • Composite Motion
  • Kinematics

Reflection and Symmetry

  • Central Inversion
  • Reflections in the Plane
  • Reflections in Space
  • Summary of Reflection Matrices
  • Symmetry Basics
  • Symmetry Groups
  • Ornamental Groups
  • Polygonal Symmetry and Tiling
  • Polyhedral Symmetry

More Linear Transformations

  • Isotropic Dilation
  • Anisotropic Dilation
  • Shear
  • Projective Geometry
  • Parallel Projection
  • Central Projection
  • Map Projections
  • Display Projection

Nonlinear Transformations

  • Linear and Nonlinear Equations
  • Inversion in a Circle
  • Curvilinear Coordinate Systems
  • Deformations
  • Answers to Selected Exercises