Graphics Reference
In-Depth Information
8 Vector
Differentiation
8.1 Introduction
Since vectors were discovered, which is only about 150 years ago, a totally new world of
mathematical analysis has emerged through their combination with differential and integral
calculus. Differential vector analysis has provided new ways of describing tangential and normal
vectors to curves and surfaces, as well as providing powerful analytical tools for studying scalar
and vector fields. Integral vector analysis, on the other hand, has been applied to the study
of vector fields used in fluid mechanics and heat transfer problems. Although many of these
techniques are used in advanced aspects of computer graphics, in this introductory text we will
explore only the fundamental concepts of differentiating vectors.
Readers who wish to explore this topic at an advanced level are recommended to study Vector
Analysis for Engineers and Scientists by P. E. Lewis and J. P. Ward [Lewis, 1989]. That topic was
extremely useful in the preparation of this chapter.
8.2 The derivative of a vector
Differential calculus measures the rate of change of a function relative to one or more dependent
variables. For example, the first differential of the polynomial
2x 2
y
=
+
3x
+
4
is
dy
dx =
4x
+
3
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