Geometric Algebra for Computer Science (Revised Edition): An by Leo Dorst

By Leo Dorst

Geometric Algebra for laptop technological know-how (Revised Edition) provides a compelling replacement to the constraints of linear algebra.

Geometric algebra (GA) is a compact, time-effective, and performance-enhancing solution to symbolize the geometry of 3D gadgets in computing device courses. This e-book explains GA as a ordinary extension of linear algebra and conveys its value for 3D programming of geometry in pix, imaginative and prescient, and robotics. It systematically explores the techniques and strategies which are key to representing easy gadgets and geometric operators utilizing GA. It covers intimately the conformal version, a handy technique to enforce 3D geometry utilizing a 5D illustration area. various drills and programming workouts are important for either scholars and practitioners. A spouse site comprises hyperlinks to GAViewer, a software that may let you engage with some of the 3D figures within the ebook; and Gaigen 2, the platform for the instructive programming routines that finish each one chapter.

The booklet might be of curiosity to execs operating in fields requiring advanced geometric computation corresponding to robotics, special effects, and desktop video games. it's also be excellent for college students in graduate or complex undergraduate courses in machine technology.

  • Explains GA as a traditional extension of linear algebra and conveys its importance for 3D programming of geometry in pics, imaginative and prescient, and robotics.
  • Systematically explores the recommendations and methods which are key to representing effortless items and geometric operators utilizing GA.
  • Covers intimately the conformal version, a handy technique to enforce 3D geometry utilizing a 5D illustration house.
  • Presents potent techniques to creating GA an essential component of your programming.
  • Includes a variety of drills and programming workouts worthy for either scholars and practitioners.
  • Companion site comprises hyperlinks to GAViewer, a software that might let you engage with some of the 3D figures within the booklet, and Gaigen 2, the platform for the instructive programming routines that finish every one bankruptcy.

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Additional resources for Geometric Algebra for Computer Science (Revised Edition): An Object-Oriented Approach to Geometry (The Morgan Kaufmann Series in Computer Graphics)

Example text

Na¨ıve implementations run slow, because of the size of the algebra (32-D for the basis of the conformal model of a 3-D Euclidean space). 5 THE STRUCTURE OF THE CHAPTERS 19 In the third part of this book, we give a computer scientist’s view of the algebraic structure and describe aspects that are relevant to any efficient implementation, using its multiplicative and sparse nature. We end with a simple ray tracer to enable comparison of computational speeds of the various methods in a computer graphics application.

1. As with the rotation in item 2, there is obvious structure preservation: the reflection of the rotated circle is the rotation of the reflected circle (in the reflected line). We can even reflect the rotor to become R ≡ π exp(␾ L∗ /2)/π = exp(−␾ (−πL∗ /π)/2), which is the rotor around the reflected line, automatically turning in the opposite orientation. 7. Programming. 2. The outcome is plotted immediately through the calls to the multivector drawing function draw(). And since it has been fully specified in terms of geometric entities, one can easily change any of them and update the picture.

The outer product of three points then represents an oriented circle, and the outer product of four points an oriented sphere. 4(e,f). 1 showed. It is very satisfying that there is one abstract product underlying such diverse constructions. However, these varied geometrical interpretations can confuse the study of its algebraic properties, so when we treat the outer product in Chapter 2 and the rest of Part I, we prefer to focus on the vector space model to guide your intuitive understanding of geometric algebra.

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