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Curves and Surfaces
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Curves
Maps from R^{1} into R^{2} or R^{3}. |
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We all have an intuitive idea of what curves are. A length of string lying on a table and a piece of wire bent in space both trace curves. So does the graph of a function y = f(x), or the boundary of a region like a disk. But while we may have a very good sense of what a curve is, we need a mathematically precise definition from which we can proceed to discuss different varieties of curves with different properties. A good working definition is that a curve is a map from R^{1} into R^{n}. That is, a curve is a function which takes a single number as its input and returns a vector. In the case where n = 2, we call the curve a plane curve. When n = 3, we call the curve a space curve. The above topics divide up the many curves in the library into classes. |