Vector Analysis for Computer Graphics
In my last book, Geometry for Computer Graphics, I employed a mixture of algebra and vector analysis to prove many of the equations used in computer graphics. At the time, I did not make any distinction between the two methodologies, but slowly it dawned upon me that I had had to discover, for the first time, how to use vector analysis and associated strategies for solving geometric problems. I suppose that mathematicians are taught this as part of their formal mathematical training, but then, I am not a mathematician! After some deliberation, I decided to write a book that would introduce the beginner to the world of vectors and their application to the geometric problems encountered in computer graphics. I accepted the fact that there would be some duplication of formulas between this and my last book; however, this time I would concentrate on explaining how problems are solved. The book contains eleven chapters: The first chapter distinguishes between scalar and vector quantities, which is reasonably straightforward. The second chapter introduces vector repres- tation, starting with Cartesian coordinates and concluding with the role of direction cosines in changes in axial systems. The third chapter explores how the line equation has a natural vector interpretation and how vector analysis is used to resolve a variety of line-related, geometric problems. Chapter 4 repeats Chapter 3 in the context of the plane.
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a x b algebra angle apply associated b x c begin calculating Cartesian chapter complex number components compute condition consider containing coordinates Cramer's rule create defined describe determinant direction direction cosines discover distance dot product edge equals example Expanding expression Figure geometric given gives graphics identical illustrates interpolated intersection introducing length line equation line intersecting line segments located magnitude means multiply negative normal vector Note object obtain obvious orientation origin parallel parametric perpendicular plane position vector problem quaternion reference reflected relative represented result reveal rotated scalar scenario shown in Fig shows simple solved sphere Substituting surface transform triangle unit vector vector product volume zero