x¡( n) represent derivatives with respect to u and o( hn) is a term such that ² Then we can express xi( u0 + h) as follows in a Taylor development: It is sufficient for this purpose to postulate that there exists at a point P of the curve, where u = u0, a set of finite derivatives xi( n+1)( u0), n sufficiently large. We also suppose that in the given interval of u the functions xi( u) are single-valued and continuous, with a sufficient number of continuous derivatives (first derivatives in all cases, seldom more than three). (1–1) represent a straight line parallel to a coordinate axis.
The functions xi( u) are not all constants. A left-handed helix can never be superimposed on a right-handed one, as everyone knows who has handled screws or ropes. This sense of the helix is independent of the choice of coordinates or parameters it is an intrinsic property of the helix.
1–2a) when b is negative it is left-handed (Fig. When b is positive the helix is right-handed (Fig. This curve lies on the cylinder x² + y² = a² and winds around it in such a way that when u increases by 2 π the x and y return to their original value, while z increases by 2 πb, the pitch of the helix (French: pas German: Ganghöhe).
Its plane can be taken as z = 0 and its equation can then be written in the form: This equation represents a line passing through the point ( ai) with its direction cosines proportional to bi. Where ai, bi are constants and at least one of the bi ≠ 0. A straight line in space can be given by the equation We use the notation P( xi) to indicate a point with coordinates xi.ĮXAMPLES. The equation of the curve then takes the form We also denote ( x, y, z) by ( x1, x2, x3), or for short, xi, i = 1, 2, 3. We select the coordinate axes in such a way that the sense OX → OY → OZ is that of a right-handed screw. It is often convenient to think of u as the time, but this is not necessary, since we can pass from one parameter to another by a substitution u = f(v) without changing the curve itself. The rectangular coordinates ( x, y, z) of the point can then be expressed as functions of a parameter u inside a certain closed interval:
We can think of curves in space as paths of a point in motion. In this attractive, inexpensive paperback edition, it belongs in the library of any mathematician or student of mathematics interested in differential geometry. The result was to further increase the merit of this stimulating, thought-provoking text - ideal for classroom use, but also perfectly suited for self-study. Struik has enhanced the treatment with copious historical, biographical, and bibliographical references that place the theory in context and encourage the student to consult original sources and discover additional important ideas there.įor this second edition, Professor Struik made some corrections and added an appendix with a sketch of the application of Cartan's method of Pfaffians to curve and surface theory. Written by a noted mathematician and historian of mathematics, this volume presents the fundamental conceptions of the theory of curves and surfaces and applies them to a number of examples. A selection of more difficult problems has been included to challenge the ambitious student. The text features an abundance of problems, most of which are simple enough for class use, and often convey an interesting geometrical fact. It is aimed at advanced undergraduate and graduate students who will find it not only highly readable but replete with illustrations carefully selected to help stimulate the student's visual understanding of geometry. Elementary, yet authoritative and scholarly, this book offers an excellent brief introduction to the classical theory of differential geometry.
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