Intrinsic geometry of surfaces let s and s be regular surfaces in 3space. The differential geometry of regular curves on a regular timelike surface emin ozyilmaz and yusuf yayli department of mathematics, ege university, bornova, izmir, 35100, turkey department of mathematics, ankara university, dogol cad. Chapter 19 basics of the differential geometry of curves. Therefore, b d t n is a constant vector the normal to p0, and so b0 d n d 0. In differential geometry the study of a curve mainly concerns a neighbourhood of a regular point. Prerequisites are kept to an absolute minimum nothing beyond first courses in linear algebra and multivariable calculus and the most direct and straightforward approach is used. Differential geometry of curves and surfaces, prentice hall, 1976 leonard euler 1707 1783 carl friedrich gauss 1777 1855. The gauss map s orientable surface in r3 with choice n of unit normal. Basics of euclidean geometry, cauchyschwarz inequality. A smooth vector field w defined along a smooth curve. The name of this course is di erential geometry of curves and surfaces.
Equivalently, we say that is an immersion of i into r3. Regular smooth curves are among the main objects in di erential geometry. Suppose and are reparametrization of the same curve. Elementary differential geometry andrew pressley download. A regular smooth curve has a welldefined tangent line at each point, and the map is onetoone on a small. General definition of curvature using polygonal approximations foxmilnors theorem. Let us consider a segment of a parametric curve between two points and as shown in fig. Belton lancaster, 6th january 2015 preface to the original version. First, a possible motivation for the allowability conditions of a curve is presented.
Secondly, the basic results of the differential geometry of curves are summarized and organized. Math 501 differential geometry professor gluck february 7, 2012 3. The advantages of the parametrization by arc length are as follows. Any regular curve can be parametrized by arc length. The covariant derivative d dt is the portion of the acceleration d dt t which is tangent to s. The notion of surface we are going to deal with in our course can be intuitively understood as the object obtained by a potter full of phantasy who takes several pieces of clay. Chern, the fundamental objects of study in differential geometry are manifolds. Curve, frenet frame, curvature, torsion, hypersurface, fundamental forms, principal curvature, gaussian curvature, minkowski curvature, manifold, tensor eld, connection, geodesic curve summary. I wrote them to assure that the terminology and notation in my lecture agrees with that text. This lecture is a bit segmented it turns out i have 5 parts covering 4. Its length can be approximated by a chord length, and by means of a taylor expansion we have. Math 501 differential geometry herman gluck tuesday february 21, 2012 4. Loosely speaking, the curvature of a curve at the point p is partially due to the fact that the curve itself is curved, and partially because the surface is curved. A di erentiable parametrised curve in rn is a c1map n.
If the particle follows the same trajectory, but with di. There are many great homework exercises i encourage. What is the natural or good parametrization for a space curve. The image ni in r is the corresponding geometric curve. Give the assumption which mu st hold for torsion to be wellde ned, and state the fundamental theorem for curves i n r 3. It is based on the lectures given by the author at e otv os. The curved line is the first species of quantity, which has only one dimension, namely length. Let be a smooth curve on the regular surface s, with velocity vector field t. The fundamental concept underlying the geometry of curves is the arclength of a. Regular smooth curves are among the main objects in differential geometry. All page references in these notes are to the do carmo text. Definition of curves, examples, reparametrizations, length, cauchys integral formula, curves of constant width. I, the arc length of a regular parameterized smooth curve.
Differential geometry of curves and surfaces chapter 1 curves. Differential geometry of curves and surfaces manfredo p. R3 h h diff i bl a i suc t at x t, y t, z t are differentiable a. That is, the distance a particle travelsthe arclength of its trajectoryis the integral of its speed.
Irm these are the types of maps that will arise most frequently. The derivative 0t is called the tangent of at the point t and l z i j 0tjdt 1 is the arclength of. The purpose of this course is the study of curves and surfaces, and those are, in gen. Differential geometry of curves and surfaces 326 pages. Recall that smooth means infinitely differentiable, i. In this video, i introduce differential geometry by talking about curves. Geometry is the part of mathematics that studies the shape of objects. Calculus of euclidean maps 2 distance function on irn. Parameterized curves definition a parameti dterized diff ti bldifferentiable curve is a differentiable map i r3 of an interval i a ba,b of the real line r into r3 r b. Curves and surfaces are the two foundational structures for differential geometry. If is an arc length parametrized curve, then is a unit vector see 2. A curve is regular if all of its points are regular. We would like the curve t xut,vt to be a regular curve for all regular. Curves in space are the natural generalization of the curves in the plane which were discussed in chapter 1 of the notes.
This is the definition that appeared, more than 2000 years ago in euclids elements. In this study, we consider timelike regular surface in minkowski space as y. Differential geometry 177 where h is the path length and is the angle subtended by the tangent with the x axis. The fundamental concept underlying the geometry of curves is the arclength of a parametrized curve. For example, the positive xaxis is the trace of the parametrized curve. R 3 be another regular smooth curve with curvature k and. Chapter 20 basics of the differential geometry of surfaces.
Suitable for advanced undergraduates and graduate students of mathematics, this texts prerequisites include an undergraduate course in linear algebra. Basics of the differential geometry of curves upenn cis. Classical differential geometry ucla department of mathematics. The curve t t3, t2 in the plane fails to be regular when t 0. S s is an isometry if for all points p s and tangent vectors w1, w2 tps we have. The name geometrycomes from the greek geo, earth, and metria, measure.
In mathematics, a curve also called a curved line in older texts is an object similar to a line which does not have to be straight intuitively, a curve may be thought as the trace left by a moving point. The following conditions are equivalent for a regular curve qt. Using the arclength, we can integrate functions defined over the curve. A regular space curve a, b r 3 is a helix if there is a. Any reparametrization of a regular curve is regular proof. We thank everyone who pointed out errors or typos in earlier versions of this book. Elementary differential geometry presents the main results in the differential geometry of curves and surfaces suitable for a first course on the subject. For a point on a curve defined by the general equation 1 to be regular, it. Since regular curves always have a unitspeed parametrization, let. The treatment begins with a chapter on curves, followed by explorations of regular surfaces, the geometry of the gauss map, the intrinsic geometry of surfaces, and global differential geometry. Isometries of euclidean space, formulas for curvature of smooth regular curves. This book is an introduction to the differential geometry of curves and surfaces, both.
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