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    A Catalog of Special Plane Curves
    A Catalog of Special Plane Curves
    A Catalog of Special Plane Curves
    Ebook299 pages1 hour

    A Catalog of Special Plane Curves

    By J. Dennis Lawrence

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    About this ebook

    Among the largest, finest collections available—illustrated not only once for each curve, but also for various values of any parameters present. Covers general properties of curves and types of derived curves. Curves illustrated by a CalComp digital incremental plotter. 12 illustrations.
    LanguageEnglish
    PublisherDover Publications
    Release dateDec 31, 2013
    ISBN9780486167664
    A Catalog of Special Plane Curves

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      Book preview

      A Catalog of Special Plane Curves - J. Dennis Lawrence

      CURVES

      CHAPTER 1

      PROPERTIES OF CURVES

      1.1. Coordinate Systems

      Of the various systems for representing plane curves,* six will be discussed in this section. Each system has its own advantages and difficulties, as indicated in Table 1. Most of the succeeding chapters will use the parametric representation within a Cartesian coordinate system; direct Cartesian representation and polar representation will occur somewhat less frequently; and pedal, bipolar, and intrinsic systems will be of infrequent use.

      The first three systems of representation mentioned in Table 1 are well known, and require only a few comments. Coordinate axes for the Cartesian and parametric systems are a pair of perpendicular lines, the abscissa (x-axis) and the ordinate (y-axis). Coordinate axes for the polar system consist of a point (the pole) and a ray from this point (the axis). In this work, the polar pole and axis will be assumed to fall on the Cartesian origin and positive x-axis, respectively.

      TABLE 1. Representation of Curves

      In the parametric system, coordinates of a curve are expressed independently as functions of a single variable, t, such as

      Hereafter, t, f, and g, with or without subscripts and/or superscripts, will be reserved for the parametric representation of some curve. The vector equation

      will also be frequently used, as are vector representations of points

      Parametric methods are most important for our purposes, and will be heavily relied on.

      The Cartesian and polar coordinate systems are basically point concepts; given any point P, there is one and only one set of coordinates (x, y) or (r, θ) for P. Pedal coordinates, however, are basically dependent on curves, and a point P may have many different pedal coordinates (r, p), depending on the curve under consideration.

      Let 0 be a fixed point (the pedal point, or pole) lying at the origin, and let C be a differentiate curve (i.e., its tangent exists). At a point P on C whose pedal coordinates are desired, construct the tangent line L to C. The pedal coordinates of P (with respect to C and 0) are the radial distance r from 0 to P and theperpendicular distance p from 0 to L (Figure 2). For a different curve C1 through P, r is, of course, the same, but p may very well be different. Furthermore, if C does not have a tangent at P (e.g., if P is an isolated point, or cusp), then the pedal coordinates of P do not exist. Some examples of pedal equations are given in Table 2.

      Let 01, and 02 be two fixed points (the poles) a distance 2c apart. The line segment L = 0102 is termed the base line, and the bisector of L is then known as the center. The bipolar coordinates of a point P are the distances r1 and r2 from 01 and 02, respectively. Now, 01, 02, and P form a triangle, so r1, r2, and c must satisfy the inequalities

      Further, since r1, r2, and c are all assumed to be positive, any equation in bipolar coordinates describes a locus that is symmetric about L; conversely, a locus that is not symmetric about some line cannot have a bipolar equation. Examples of bipolar equations are given in Table 3.

      It is sometimes desirable to write the equation of a curve in such a manner as to be independent of certain coordinate transformations. The type of transformations being considered are those that preserve length and angle; all the transformations discussed below within and between coordinate systems fit this definition. Such a transformation will also preserve area, arc length, curvature, number of singularities, etc. Whewell introduced a system involving arc length s and tangential angle ø, while Cesáro gave a system involving arc length s and radius of curvature ρ. Since ρ dø = ds, by definition, it is evident that these are related. Examples of both are given in Table 4.

      Figure 2. Pedal Coordinates

      As an example of these systems of representation, consider the straight line, with Cartesian equation

      There are two frequently used parametric forms of this equation. Let P1 = (x1., y1.) and P2 = (x2 y2) be two points. The two-point form of the linear equation may be written (in vector notation) as

      TABLE 2. Pedal Equations

      TABLE 3. Bipolar Equations

      TABLE 4. Intrinsic Equations

      while the one-point form is

      with r = (λ, µ) being a parameter dependent on the slope of the line. This latter form may be connected with 1.1.4 by the relations

      In polar coordinates, the line has equation

      where p is the distance from the origin to the line and α is the angle between the axis and this perpendicular (Figure 3). This may be related to 1.1.4 by

      yielding

      Figure 3. Polar Equation of a Line

      1.2. Angles

      Three angles are of importance in coordinate geometry: the slope angle ø, the radial angle θ, and the tangential-radial angle ψ. Since the angle ν between lines is of frequent usage, it is also described here. Definitions are in terms of a point P0 = (x0 , y0 ) on a line L; comparable definitions for a curve may be obtained using the tangent line (if it exists).

      Let a Cartesian-coordinate system be given. The slope angle ø of a line L is defined to be the angle formed by L and the x-axis, taken clockwise from L (Figure 4); if L is parallel to the x-axis, then θ is taken to be zero. The radial angle θ is the angle between the radial line R between 0 and P0 and the x-axis, taken clockwise from R. Finally, ψ is defined as the angle between R and L, taken clockwise from L. Note that all three angles are in the interval (0, π).

      Slope. The slope m of a line L is defined to be the tangent of angle ø. Using the standard equations 1.1.4, 1.1.6, and 1.1.8 for the line, m is found to be

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