A Treatise on the Differential Geometry of Curves and Surfaces
496
A Treatise on the Differential Geometry of Curves and Surfaces
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| ISBN-13: | 9780486159461 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 03/28/2013 |
| Series: | Dover Books on Mathematics |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 496 |
| File size: | 32 MB |
| Note: | This product may take a few minutes to download. |
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A Treatise on the Differential Geometry of Curves and Surfaces
By Luther Pfahler Eisenhart
Dover Publications, Inc.
Copyright © 2004 Dover Publications, Inc.All rights reserved.
ISBN: 978-0-486-15946-1
CHAPTER 1
CURVES IN SPACE
1. Parametric equations of a curve. Consider space referred to fixed rectangular axes, and let (x, y, z) denote as usual the coördinates of a point with respect to these axes. In the plane z = 0 draw a circle of radius r and center (a, b). The coördinates of a point P on the circle can be expressed in the form
(1)
x = a + r cos u, y = b + r sin u, Z = 0,
where u denotes the angle which the radius to P makes with the positive x -axis. As u varies from 0° to 360°, the point P describes the circle. The quantities a, b, r determine the position and size of the circle; whereas u determines the position of a point upon it. In this sense it is a variable or parameter for the circle. And equations (1) are called parametric equations of the circle.
A straight line in space is determined by a point on it, P0(a, b, c), and its direction- cosines α, β, γ. The latter fix also the sense of the line. Let P be another point on the line, and let the distance P0P be denoted by u, which is positive or negative. The rectangular coördinates of P are then expressible in the form
(2)
x = a + ua, y = b + uβ, z = c + uy.
To each value of u there corresponds a point on the line, and the coördinates of any point on the line are expressible as in (2). These equations are consequently parametric equations of the straight line.
When, as in fig. 1, a line segment PD, of constant length a, perpendicular to a line OZ at D, revolves uniformly about OZ as axis, and at the same time D moves along it with uniform velocity, the locus of P is called a circular helix. If the line OZ be taken for the z-axis, the initial position of PD for the positive x-axis, and the angle between the latter and a subsequent position of PD be denoted by u, the equations of the helix can be written in the parametric form
(3)
x = a cos u, y = a sin u, z = bu,
where the constant b is determined by the velocity of rotation of PD and of translation of D. Thus, as the line PD describes a radian, D moves the distance b along OZ.
In all of the above equations u is the variable or parameter. Hence, with reference to the locus under consideration, the coördinates are functions of u alone. We indicate this by writing these equations
(4)
x = f1(u), y = f2(u), z = f3(u).
The functions f1, f2, f3 have definite forms when the locus is a circle, straight line or circular helix. But we proceed to the general case and consider equations (4), when f1, f2, f3 are any functions whatever, analytic for all values of u, or at least for a certain domain. The locus of the point whose coördinates are given by (4), as u takes all values in the domain considered, is a curve. Equations (4) are said to be the equations of the curve in the parametric form. When all the points of the curve do not lie in the same plane it is called a space curve or a twisted curve; otherwise, a plane curve.
It is evident that a necessary and sufficient condition that a curve, defined by equations (4), be plane, is that there exist a linear relation between the functions, such as
(5)
af1 + bf2 + cf3 + d = 0,
where a, b, c, d denote constants not all equal to zero. This condition is satisfied by equations (1) and (2), but not by (3).
If u in (4) be replaced by any function of v, say
(6)
v = φ(u),
equations (4) assume a new form,
(7)
x = F1(v), y = F2(v), z = F3(v).
It is evident that the values of x, y, z, given by (7) for a value of v, are equal to those given by (4) for the corresponding value of u obtained from (6). Consequently equations (4) and (7) define the same curve, u and v being the respective parameters. Since there is no restriction upon the function φ, except that it be analytic, it follows that a curve can be given parametric representation in an infinity of ways.
2. Other forms of the equations of a curve. If the first of equations (4) be solved for u, giving u = φ(x), then, in terms of x as parameter, equations (7) are
(8)
x = x, y = F2(x), z = F3(x).
In this form the curve is really defined by the last two equations, or, if it be a plane curve in the xy -plane, its equation is in the customary form
(9)
y = f(x).
The points in space whose coördinates satisfy the equation y = F2(x) lie on the cylinder whose elements are parallel to the z-axis and whose cross section by the xy-plane is the curve y = F2(x). In like manner, the equation z = F8(x) defines a cylinder whose elements are parallel to the y-axis. Hence the curve with the equations (8) is the locus of points common to two cylinders with perpendicular axes. Conversely, if lines are drawn through the points of a space curve normal to two planes perpendicular to one another, we obtain two such cylinders whose intersection is the given curve. Hence equations (8) furnish a perfectly general definition of a space curve.
In general, the parameter u can be eliminated from equations (4) in such a way that there result two equations, each of which involves all three rectangular coördinates. Thus,
(10)
Φ1(x, y, z) = 0, Φ2(x, y, z) = 0.
Moreover, if two equations of this kind be solved for y and z as functions of x, we get equations of the form (8), and, in turn, of the form (4), by replacing x by an arbitrary function of u. Hence equations (10) also are the general equations of a curve. It will be seen later that each of these equations defines a surface.
It should be remarked, however, that when a curve is defined as the intersection of two cylinders (8), or of two surfaces (10), it may happen that these curves of intersection consist of several parts, so that the new equations define more than the original ones.
For example, the curve defined by the parametric equations
(i)
x = u, y = u2, z = u2,
is a twisted cubic, for every plane meets the curve in three points. Thus, the plane
ax + by + cz + d = 0
meets the curve in the three points whose parametric values are the roots of the equation
cu8 + bu2 + au + d = 0.
This cubic lies upon the three cylinders
y = x2, z = x8, y8 = z2.
The intersection of the first and second cylinders is a curve of the sixth degree, of the first and third it is of the sixth degree, whereas the last two intersect in a curve of the ninth degree. Hence in every case the given cubic is only a part of the curve of intersection — that part which lies on all three cylinders.
Again, we may eliminate u from equations (i), thus
(ii)
xy = z, y2 = xz,
of which the first defines a hyperbolic paraboloid and the second a hyperbolic- parabolic cone. The straight line y = 0, z = 0 lies on both of these surfaces, but not on the cylinder y = x2. Hence the intersection of the surfaces (ii) consists of this line and the cubic. The generators of the paraboloid are defined by
x=a, z = ay; y=b, z=bx;
for all values of the constants a and b. From (i) we see that the cubic meets each generator of the first family in one point and of the second family in two points.
3. Linear element. By definition the length of an arc of a curve is the limit, when it exists, toward which the perimeter of an inscribed polygon tends as the number of sides increases and their lengths uniformly approach zero. Curves for which such a limit does not exist will be excluded from the subsequent discussion.
Consider the arc of a curve whose end points m0, ma, are deter mined by the parametric values u0 and a, and let m1m2...., be intermediate points with parametric values u1, u2, ... The length lk of the chord [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
By the mean value theorem of the differential calculus this is equal to
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
and the primes indicate differentiation.
As defined, the length of the arc m0ma is the limit of [summation]lk, as the lengths [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] tend to zero. From the definition of a definite integral this limit is equal to
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Hence, if s denotes the length of the arc from a fixed point (u0) to a variable point (u), we have
(11)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
This equation gives s as a function of u. We write it
(12)
s = φ(u),
and from (11) it follows that
(13)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
which we may write in the form
(14)
ds2 = dx2 + dy2 + dz2.
As thus expressed ds is called the element of length, or linear element, of the curve.
In the preceding discussion we have tacitly assumed that u is real. When it is complex we take equation (11) as the definition of the length of the arc.
If equation (12) be solved for u in terms of s, and the result be substituted in (4), the resulting equations also define the curve, and s is the parameter. From (11) follows the theorem:
A necessary and sufficient condition that the parameter u be the arc measured from the point u = u0is
(15)
f'21 + f'22 + f'23 = 1.
An exceptional case should be noted here, namely,
(16)
f'21 + f'22 + f'23 = 0.
Unless f'1, f'2, f'3 be zero and the curve reduce to a point, at least one of the coördinates must be imaginary. For this case sis zero. Hence these imaginary curves are called curves of length zero, or minimal curves. For the present they will be excluded from the discussion.
Let the arc be the parameter of a given curve and s and s + e its values for two points M(x, y, z) and M1(x1, y1, z1). By Taylor's theorem we have
(17)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where an accent indicates differentiation with respect to s.
Unless x', y', z' are all zero, that is, unless the locus is a point and not a curve, one at least of the lengths x1 - x, y1 - y, z1 - z is of the order of magnitude of e. If these lengths be denoted by δx, δy, δz, and e by δ8, then we have
√δx2 + δy2 + δz2 = δ8 + l2,
where l2 denotes the aggregate of terms of the second and higher orders in δ8. Hence, as M1 approaches M the ratio of the lengths of the chord and the arc MM1 approaches unity; and in the limit we have ds2 = dx2 + dy2 + dz2.
4. Tangent to a curve. The tangent to a curve at a point M is the limiting position of the secant through M and a point M1 of the curve as the latter approaches M as a limit.
In order to find the equation of the tangent we take s for parameter and write the expressions for the coördinates of M1 in the form (17). The equations of the secant through M and M1 are
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
If each member of these equations be multiplied by e and the denominators be replaced by their values from (17), we have in the limit as M1 approaches M
(18)
X - x/x' = Y - y/y' = Z - z/z'.
If α, β, γ denote the direction-cosines of the tangent in consequence of (15), we may take
(19)
α = x', β = y', γ = z'.
When the parameter u is any whatever, these equations are
(20)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
They may also be written thus :
(21)
α = dx/ds, β = dy/ds, γ = dz/ds.
From these equations it follows that, if the convention be made that the positive direction on the curve is that in which the parameter increases, the positive direction upon the tangent is the same as upon the curve.
A fundamental property of the tangent is discovered by considering the expression for the distance from the point M1, with the coördinates (17), to any line through M. We write the equation of such a line in the form
(22)
X - x/a = Y - y/b = Z - z/c,
where a, b, c are the direction-cosines.
The distance from M1 to this line is equal to
(23)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Hence, if MM1 be considered an infinitesimal of the first order, this distance also is of the first order unless
a/x' = b/y' = c/z',
in which case it is of the second order at least. But when these equations are satisfied, equations (22) define the tangent at M. Therefore, of all the lines through a point of a curve the tangent is nearest to the curve.
5. Order of contact. Normal plane. When the curve is such that there are points for which
(24)
x"/x' = y"/y' = z"/z',
the distance from M1 to the tangent is of the third order at least. In this case the tangent is said to have contact of the second order, whereas, ordinarily, the contact is of the first order. And, in general, the tangent to a curve has contact of the nth order at a point, if the following conditions are satisfied for n = 2, ..., n - 1, and n:
(25)
x(n)/x(n-1) = y(n)/y(n-1) = z(n)/z(n-1).
When the parameter of the curve is any whatever, equations (24), (25) are reducible to the respective equations
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
(Continues...)
Excerpted from A Treatise on the Differential Geometry of Curves and Surfaces by Luther Pfahler Eisenhart. Copyright © 2004 Dover Publications, Inc.. Excerpted by permission of Dover Publications, Inc..
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Table of Contents
I. Curves in SpaceII. Curvilinear Coordinates on a Surface. Envelopes
III. Linear Element of a Surface. Differential Parameters. Conformal Representation
IV. Geometry of a Surface in the Neighborhood of a Point
V. Fundamental Equations. The Moving Trihedral
VI. Systems of Curves. Geodesics
VII. Quadrics. Ruled Surfaces. Minimal Surfaces
VIII. Surfaces of Constant Total Curvature. W-Surfaces. Surfaces with Plane or Spherical Lines of Curvature
IX. Deformation of Surfaces
X. Deformation of Surfaces. The Method of Weingarten
XI. Infinitesimal Deformation of Surfaces
XII. Rectilinear Congruences
XIII. Cyclic Systems
XIV. Triply Orthogonal Systems of Surfaces
Index