Examples of Curves
Section 5.1 starts from the basic notion of a regular curve. Then we investigate cycloidal curves and other remarkable parametric curves, as well as curves given implicitly as level curves of functions in two variables. The level sets are useful in solvin
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Examples of Curves
Section 5.1 starts from the basic notion of a regular curve. Then we investigate cycloidal curves and other remarkable parametric curves, as well as curves given implicitly as level curves of functions in two variables. The level sets are useful in solving problems with conditional extrema. We study the variational calculus for functions of one variable and employ it to derive Euler’s spiral. In Section 5.2 we study some fracR tal curves. Section 5.3 discusses the basic MATLAB capabilities for plotting space curves, surveys the parallel and perspective projections, and presents curves with shadows on planar, cylindrical, or spherical displays. Section 5.4 introduces helix-type curves on surfaces of revolution and studies curves obtained by the intersection of pairs of surfaces.
5.1 Plane Curves Intuitively, according to Euclid, a curve is the trajectory of a moving point, a boundary of a surface, a one-dimensional figure. The mathematically correct definition of a curve is based on notions from topology, but it starts from the key notion of an elementary curve, which can be imagined as an interval I = (a, b) (or a segment I¯ = [a, b]) of the line after continuous deformation. A set γ in Rn is called a curve if it can be covered by a finite or countable number of elementary curves. One can distinguish self-intersecting, simple (i.e., without self-intersections; for example, graphs), closed, and connected curves. There are many methods of classifying curves. One method is to divide them into algebraic and transcendental curves. An algebraic (plane) curve is given by a polynomial equation P(x, y) = 0. Its degree n = deg P is called the order of the curve. Curves of order n = 2 are studied in analytical geometry. The first classification of curves of R V. Rovenski, Modeling of Curves and Surfaces with MATLAB , Springer Undergraduate Texts in Mathematics and Technology 7, DOI 10.1007/978-0-387-71278-9 5, c Springer Science+Business Media, LLC 2010
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5 Examples of Curves
order n = 3 was obtained by Newton. The case n > 3 is more difficult. But among easily obtained curves there are many that are nonalgebraic, for example, the cycloid and the spiral of Archimedes; we study them by using parametric or implicit equations or by plotting them in polar coordinates (see Section 1.4). The names of curves, like geographical names, contain much interesting information. We may group curves according to the meaning of their names: By the name of a scientist — Dinostratus’ quadratrix, conchoid of Nicomedes, Pascal’s limac¸on, lemniscate of Bernoulli, cissoid of Diocles, etc. By the method of construction — caustics, equidistant (parallel) curves, pedal curves, evolutes, evolvents, etc. By an important property — trisectrices, quadratrices, tractrices, etc. By an essential property of their shape — astroid (star-shaped), deltoid (Greek letter Δ ), cardioid (heart-shaped), nephroid (kidney-shaped), etc. By historical factors — ellipse, parabola, hyperbola. By the structure of the formula — (semi)cubi
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