Professor John Vince

Mathematics for Computer Graphics (4th Edition)

by John Vince

ISBN 978-1-4471-6289-6, Springer-Verlag, 2013

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This book is not for mathematicians! They would probably raise their hands in horror about the level of mathematical rigour I have employed, or probably not employed! This book is for people working in computer graphics who know that they have to use mathematics in their day-to-day work, and don't want to get too embroiled in axioms, truths, and Platonic realities.

This book originally appeared as part of Springer's excellent Essential series, and was revised to include chapters on analytical geometry, barycentric coordinates, and worked examples. The third edition included a new chapter on geometric algebra, which I have written about in my books Geometric Algebra for Computer Graphics and Geometric Algebra: An Algebraic System for Computer Games and Animation.In this fourth edition, I have reviewed the entire book and included chapters on differential and integral calculus, which I have written about in Calculus for Computer Graphics.

Whilst writing this book I have borne in mind what it was like for me when I was studying different areas of mathematics for the first time. In spite of reading and rereading an explanation several times it could take days before "the penny dropped" and a concept became apparent. Hopefully, the reader will find the explanations useful in developing their understanding of these specific areas of mathematics, and enjoy the sound of various pennies dropping!

Calculus for Computer Graphics
by John Vince
ISBN 978-1-4471-5465-5, Springer-Verlag, 2013

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Calculus is one of those subjects that appears to have no boundaries, which is why some calculus books are so large and heavy! So when I started writing this book, I knew that it would not fall into this category: it would be around 200 pages long and take the reader on a gentle journey through the subject, without placing too many demands on their knowledge of mathematics.

The objective of the book is to inform the reader about functions and their derivatives, and the inverse process: integration, which can be used for computing area and volume. The emphasis on geometry gives the book relevance to the computer graphics community, and hopefully will provide the mathematical background for professionals working in computer animation, games and allied disciplines to read and understand other books and technical papers where differential and integral notation is found.

Matrix Transforms for Computer Graphics
by John Vince
ISBN 978-1-4471-4320-8, Springer-Verlag, 2012

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Matrix transforms are ubiquitous within the world of computer graphics, where they have become an invaluable tool in a programmer’s toolkit for solving everything from 2D image scaling to 3D rotation about an arbitrary axis. Virtually every software system and hardware graphics processor uses matrices to undertake operations such as scaling, translation, reflection and rotation. Nevertheless, for come newcomers to the world of computer games and animation, matrix notation can appear obscure and challenging.

Matrices and determinants were originally used to solve groups of simultaneous linear equations, and were subsequently embraced by the computer graphics community to describe the geometric operations for manipulating two- and three-dimensional structures. Consequently, to place matrix notation within an historical context, I provide readers with some useful background to their development, alongside determinants.

Although this book assumes that the reader is familiar with everyday algebra and the solution of simultaneous linear equations, it does not expect any prior knowledge of matrix notation. The book includes chapters on matrix notation, determinants, matrices, 2D transforms, 3D transforms and quaternions, and includes many worked examples to illustrate their practical use. After reading this book, readers should be completely familiar with matrix notation and their application within the world of computer games and animation.

Quaternions for Computer Graphics

by John Vince
ISBN 978-0-85729-759-4, Springer-Verlag, 2011



Sir William Rowan Hamilton was a genius, and will be remembered for his significant contributions to physics and mathematics. The Hamiltonian, which is used in quantum physics to describe the total energy of a system, would have been a major achievement for anyone, but Hamilton also invented quaternions, which paved the way for modern vector analysis.

Quaternions are one of the most documented inventions in the history of mathematics, and this book is about their invention, and how they are used to rotate vectors about an arbitrary axis. Apart from introducing the reader to the features of quaternions and their associated algebra, the book provides valuable historical facts that bring the subject alive.

Quaternions for Computer Graphics introduces the reader to quaternion algebra by describing concepts of sets, groups, fields and rings. It also includes chapters on imaginary quantities, complex numbers and the complex plane, which are essential to understanding quaternions. The book contains many illustrations and worked examples, which make it essential reading for students, academics, researchers and professional practitioners.

Rotation Transforms for Computer Graphics

by John Vince
ISBN 978-0-85729-153-0, Springer-Verlag, 2011

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Rotation transforms are used everywhere in computer graphics from rotating pictures in editing software, to providing an arbitrary view of a 3D virtual environment. Although the former is a trivial operation, the latter can be a challenging task.

Rotation Transforms for Computer Graphics covers a wide range of mathematical techniques used for rotating points and frames of reference in the plane and 3D space. It includes many worked examples and over 100 illustrations that make it essential reading for students, academics, researchers and professional practitioners.The book includes introductory chapters on complex numbers, matrices, quaternions and geometric algebra, and further chapters on how these techniques are employed in 2D and 3D computer graphics. In particular, matrix and bivector transforms are developed and evaluated to rotate points in a fixed frame of reference, and vice versa.

Mathematics for Computer Graphics (3rd Edition)

by John Vince

ISBN 978-1-84996-022-9, Springer-Verlag, 2010

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This book is not for mathematicians! They would probably raise their hands in horror about the level of mathematical rigour I have employed, or probably not employed! This book is for people working in computer graphics who know that they have to use mathematics in their day-to-day work, and don't want to get too embroiled in axioms, truths, and Platonic realities.

This book originally appeared as part of Springer's excellent Essential series, and was revised to include chapters on analytical geometry, barycentric coordinates, and worked examples. The third edition included a new chapter on geometric algebra, which I have written about in my books Geometric Algebra for Computer Graphics and Geometric Algebra: An Algebraic System for Computer Games and Animation

Whilst writing this book I have borne in mind what it was like for me when I was studying different areas of mathematics for the first time. In spite of reading and rereading an explanation several times it could take days before "the penny dropped" and a concept became apparent. Hopefully, the reader will find the explanations useful in developing their understanding of these specific areas of mathematics, and enjoy the sound of various pennies dropping!

Geometric Algebra: An Algebraic System for Computer Games and Animation

by John Vince

ISBN 978-1-84996-022-9, Springer-Verlag, 2009


Due to some strange decisions taken at the end of the nineteenth century, the true power of vectors has never been exploited. Nevertheless, for over a century, mathematicians, engineers, scientists, and more recently programmers, have been using vectors to solve an extraordinary range of problems. However, today, we can discover the true potential of oriented, lines, planes and volumes in the form of geometric algebra. As such geometric elements are central to the world of computer games and computer animation, geometric algebra offers programmers new ways of solving old problems. 

The first two chapters review the products for real, complex and quaternion structures, and any non-commutative qualities that they possess. Chapter three reviews the familiar scalar and vector products and introduces the idea of ‘dyadics’, which provide a useful mechanism for describing the features of geometric algebra. Chapter four introduces the geometric product and defines the inner and outer products, which are employed in the following chapter on geometric algebra. Chapters six and seven cover all the 2D and 3D products between scalars, vectors, bivectors and trivectors. Chapter eight shows how geometric algebra brings new insights into reflections and rotations, especially in 3D. Finally, chapter nine explores a wide range of 2D and 3D geometric problems followed a concluding tenth chapter.

Filled with lots of clear examples, full-colour illustrations and tables, this compact book provides an excellent introduction to geometric algebra for practitioners in computer games and animation.

Geometric Algebra for Computer Graphics

by John Vince

ISBN 978-1-8462-8996-5, Springer-Verlag, 2008

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Since its invention, geometric algebra has been applied to various branches of physics such as cosmology and electrodynamics, and is now being embraced by the computer graphics community where it is providing new ways of solving geometric problems. It took over two-thousand years to discover this algebra, which uses a simple and consistent notation to describe vectors and their products.

The first five chapters review the algebras of real numbers, complex numbers, vectors, and quaternions and their associated axioms, together with the geometric conventions employed in analytical geometry. As well as putting geometric algebra into its historical context, I provide chapters on Grassmann's outer product and Clifford's geometric product, followed by the application of geometric algebra to reflections, rotations, lines, planes and their intersection. The conformal model is also covered, where a 5D Minkowski space provides an unusual platform for unifying the transforms associated with 3D Euclidean space.

Filled with lots of clear examples and useful illustrations, this compact book provides an excellent introduction to geometric algebra for computer graphics.

Vector Analysis for Computer Graphics

by John Vince

ISBN 978-1-84628-803-6, Springer-Verlag, 2007

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Vector analysis is relatively young in the history of mathematics, however, in the short period of its existence it has become a powerful and central tool in describing and solving a wide range of geometric problems, many, of which, arise in computer graphics. These may be in the form of describing lines, surfaces and volumes, which may touch, collide, intersect, or create shadows upon complex surfaces.

Vector Analysis for Computer Graphics provides a complete introduction to vector analysis, especially within the context of computer graphics. The author shows why vectors are useful and how it is possible to develop analytical skills in manipulating the vector algebra. Each topic covered is placed in the context of a practical application within computer graphics.

The book is divided into eleven chapters covering the mathematical foundations of vector algebra and its application to lines, planes, intersections, rotating vectors, vector differentiation, projections, rendering and motion.

Mathematics for Computer Graphics (2nd Edition)

by John Vince

ISBN 978-1-84628-034-6, Springer-Verlag, 2006


Geometry for Computer Graphics

by John Vince

ISBN 978-1-85233-834-2, Springer-Verlag, 2005



Geometry is the cornerstone of computer graphics and computer animation, and provides the framework and tools for solving problems in two and three dimensions. This may be in the form of describing simple shapes such as a circle, ellipse or parabola, or complex problems such as rotating objects about an arbitrary axis.

Geometry for Computer Graphics draws together a wide variety of geometric information that will provide a sourcebook of facts, examples and proofs for students, academics, researchers and professional practitioners.

Introduction to Virtual Reality

by John Vince

ISBN 978-1-85233-739-1, Springer-Verlag, 2004


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Essentially, virtual reality is about the navigation and manipulation of 3D computer-generated environments. A VR user is able to navigate by walking, running or even flying through a virtual environment and explore viewpoints that would be impossible in the real world. But the real benefit of VR is the ability to touch, animate, pick up and reposition virtual objects and create totally new configurations. Key topics: The origins of VR, How VR works, and how VR is being used.


Essential Mathematics for Computer Graphics

by John Vince

ISBN 978-1-85233-380-5, Springer-Verlag, 2001


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In ten chapters you will rediscover - and hopefully discover for the first time a new way of understanding - the mathematical techniques required to solve problems and design computer programs for computer graphic applications. Each chapter explores a specific mathematical topic and takes you forward into more advanced areas until you are able to understand 3D curves and surface patches, and solve problems using vectors.


Essential Computer Animation

by John Vince

ISBN 978-1-85233-141-2, Springer-Verlag, 2000



I tell you about the principles used in the powerful software products currently available on the market; the terms and processes involved; and in an easy-to-understand way, with no complicated math.


So if you want to learn more about 3D computer animation without being swamped by complex mathematics, then read this book and have fun creating your own animated programs.



Essential Virtual Reality

by John Vince

ISBN 978-1-85233-012-5, Springer-Verlag, 1998



Essential Virtual Reality tells you what is and isn’t VR! I provide a potted history of VR and explain in easy-to-understand terms what computers is and how it is integral to VR systems. You’ll see how important it is to understand the part human factors have to play in creating a good VR system (sound, sight, touch and balance) and take a look at a working VR system.

3-D Computer Animation

by John Vince

ISBN 978-0-20162-756-6, Addison-Wesley, 1992


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3D Computer Animation presents a clear and accessible guide to modern techniques for three-dimensional computer animation. Richly illustrated, it describes the major techniques and algorithms used to produce animated computer graphics in television, advertising, film special effects and flight simulators.


3-D Computer Animation

by John Vince

ISBN 978-0-20162-756-6, Pearson Education, 1992


3D Computer Animation presents a clear and accessible guide to modern techniques for three-dimensional computer animation. Richly illustrated, it describes the major techniques and algorithms used to produce animated computer graphics in television, advertising, film special effects and flight simulators.


Computer Graphics

by John Vince

ISBN 978-0-85072-303-8, The Design Council, 1992


A comprehensive, state-of-the-art introduction to computer graphics in 2D design, Stunningly illustrated with outstanding examples from leading international design exponents, this is the essential reference for anyone working with computer graphics in design.


The Language of Computer Graphics

by John Vince

ISBN 978-1-85454-147-5, ADT Press, 1990



The Language of Computer Graphics is an invaluable reference book for anyone whose work or research involves computer graphics.

Directory of Computer Graphics

by John Vince

ISBN 978-0-86353-116-3, Online Publications, 1988


This second edition of The International Computer Graphics Directory lists companies manufacturing and supplying computer graphics hardware, software, systems and services, together with details of the products and services they offer.

Computer Graphics for Graphic Designers

by John Vince

ISBN 978-0-86187-472-9, Frances Pinter, 1985


This book introduces readers to the principles of computer graphics and includes: computer principles, 2D computer graphics, peripherals, 2D monochrome applications, colour systems, frame stores,3D computer graphics, modelling, shading and realism, turnkey systems.


A Dictionary of Computer Graphics

by John Vince

ISBN 978-0-86187-623-7, Frances Pinter, 1984


A dictionary of terms employed in computer graphics.