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Curves
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Types of Curves
Maps from R^{1} into R^{2} or R^{3}. |
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Curves in R^{2} vs. R^{3}There are two common types of curves: plane curves, which are maps from R^{1} to R^{2}, and space curves, which are maps from R^{1} to R^{3}. In a somewhat different sense, curves can also be viewed as 1-dimensional subsets of R^{2} or R^{3}. For an example of a curve in R^{2}, see the semicubical parabola; for an example of a curve in R^{3}, see the twisted cubic. Parametric vs. ImplicitAs was mentioned above, curves can be defined as maps or as subsets. Curves which are defined as maps that take a single variable t (called the parameter) and send it to a vector in R^{2} or R^{3} are called parameterized curves. On the other hand, a curve which is described as the set of solutions to an equation (usually in two variables) is said to be defined implicitly. The circle is an excellent example of each. Rectangular vs. PolarFrequently, plane curves are specified in terms of the x and y coordinates of successive points on the curve, as functions of a parameter t. Occasionally, however, it is convenient to specify the distance of a point from the origin (denoted by r) and the angle a line from the point to the origin makes with the positive x axis (denoted by the Greek letter theta). This coordinate system is called polar, while the usual xy system is called rectangular. For an example of a curve in both rectangular and polar coordinates, see the cardioid. Smooth vs. Non-smoothCurves which have derivatives of all orders at a given point are said to be smooth at that point. If all points on a given curve are smooth, then the curve itself is said to be smooth. An example of a smooth curve is the parabola. The absolute value curve, however, is not smooth because its derivative does not exist at the origin. Regular vs. Non-regularRegular curves have nonzero acceleration at all points (i.e., the second derivative of the curve never vanishes.) An example of a regular curve is the circle. The cycloid, however, is non-regular at its cusps. Locally vs. Globally DefinedSome curves are interesting only for their local properties. The clothoid, for example, has a curvature function that varies linearly with the parameter t. Other local properties include torsion, tangent lines, and so forth. Many curves also have global properties, however. For example, a closed curve is either convex or non-convex based on its shape as a whole, completely independent of its local behavior. (A closed curve is said to be convex if, for any two points in the region bounded by the curve, a line between the points remains entirely within the region.) For example, the sine oval is convex, while the nephroid is not. |