Text a curves

Definition and equations of a curve. In ordinary three-dimensional space let us establish a left-handed orthogonal cartesian coordinate system with the same unit of distance for all three axes. In this system any point P has coordinates x, y, z.

A curve may be described qualitatively as the locus of a point moving with one degree of freedom. A curve is also sometimes said to be the locus of a one-parameter family of points or the locus of a single infinity of points.

Definition 1. Let the coordinates x, y, z of а point P be given as single-valued real-valued analytic functions of a real independent variable t on an interval T of t-axis by equations of the form:

x = x (t)_, у = y (t), z = z (t). (1.1)

Further suppose that the functions x(t)_, y(t), z(t) are not all constant on T. Then the locus of the point P, as t varies on the interval T, is a real proper analytic curve C.

Some comments on the foregoing definition will perhaps clarify its meaning. Equations (1.1) аге called the parametric equations of the curve C, the parameter being the variable t. We reserve the right to permit the parameter t to take on complex values. Moreover, one or more of the coordinates x, y, z may, under suitable conditions, be allowed to be complex. The curve С would in this case be called complex, or perhaps, on suitable conditions, imaginary. To say that a curve is proper means that it does not reduce to a single fixed point, as it would do if the coordinates x, y, z were all constant. It is clear that at an ordinary point of a real proper analytic curve, i. e. a point where nothing exceptional occurs, the inequality

x'²+ y'²+ z'² > 0(x) (1.2)

holds. Any point of such a curve where this inequality fails to hold is called singular, although the singularity may belong to the parametric representation being used for the curve defined as a point-locus, or may belong to the curve itself. A curve, or portion of a curve, which is free of singular points may be called nonsingular. Furthermore, we assume that the interval T is so small that

values of the parameter t on the interval T and points (x, y, z) on the curve С are in one-to-one correspondence, so that the parameter t is a coordinate of the corresponding point (x, y, z) on the curve С.

To say that the functions are analytic means, roughly, that they can be expanded into power series. More precisely, this statement means that, at each point t₀ within the interval T, each of these functions can be expanded into a Taylor's series of power of the difference t-t₀ which converges when the absolute value t-t₀ is sufficiently small. It would be possible to study differential geometry under the hypothesis that the functions considered possess only a definite, and rather small number of derivatives; but we assume analyticity in the interests of simplicity. So the word "function" will mean for us "analytic function", and the word "curve" will mean a real proper nonsingular analytic curve unless the contrary is indicated.

Some examples of parametric equations of curves will now be adduced. First of all, the equations (1.1) may be linear, of the form

x = a + lt, y = b + mt, z = c + nt (1.3)

in which a, b, с and l, m, n are constants. Then the curve С is a straight line through the fixed points (a, b, c) and with direction cosines proportional to

l, m, n. If t is the algebraic distance from the fixed point (a, b, c) to the variable point (x, y, z) on the line then l, m, n are the direction cosines of the line and satisfy the equation

l²+ m²+ n²= l (1.4)

As a second example, equations (1.1) may take the form

x = t, у = t², x = t³ (1.5)

The curve С is then a cubical parabola. This is one form of a twisted cubic which can be defined as the residual intersection of two quadric surfaces that intersect elsewhere in a straight line. Finally, if equations (1.1) have the form

x = a cos t , y = a sin t, z = bt (a> 0, b <O) (1.6)

The curve С is a left-handed circular helix, or machine screw. This may be described as the locus of a point which revolves around the z - axis at a constant distance a from it and at the same time moves parallel to the z - axis at a rate proportional to the angle t of revolution. If we had supposed b < 0, then the helix would have been right-handed.

A curve can be represented analytically in other ways than by its parametric equations. For example, it is known that one equation in x, y, z represents a surface, and that two independent simultaneous equations in

x, y, z, say

F(x, y, z) = 0, C(x, y, z) = 0 (1.7)

represent the intersection of two surfaces, which is a curve. Equations (1.7) are called implicit equations of this curve. Sometimes it is convenient to represent a curve by implicit equations, when really the curve under consideration is only part of the intersection of the two surfaces represented by the individual equations.

If the implicit equations (1.7) be solved for two of the variables in terms of the third, say for у and z in terms of x, the result can be written in the form

y = y(x), z=z(x). (1.8)

These equations represent the same curve as equations (1.7), and they, or the equations, which similarly express any two of the coordinates of a variable point on the curve as functions of the third coordinate, are called explicit equations of the curve. Each of equations (1.8) separately represents a cylinder projecting the curve onto one of the coordinate planes. So equations (1.8) are a special form of equations (1.7) for which the two surfaces are projecting cylinders.

If the first of the parametric equations(1.1) of a curve С be solved for t as a function of x, and if the result is substituted in the remaining two of these equations, the explicit equations (1.8) of the curve С are obtained. From one point of view the explicit equations (1.8) of a curve, when supplemented by identity, x = x, are parametric equations

x=x, y=y(x), z=z(x). (1.9)

of the curve, the parameter now being the coordinate x.