Curves and Surfaces in the Plane and Three-Dimensional Space

Math 162A - Notes Neil Donaldson Winter 2018 Introduction and notation Classical Differential Geometry is the study of curves and surfaces in the plane and three-dimensional space. The word differential indicates that we will be using calculus! Indeed the basic ingredients of differential geometry are multi-variable calculus, linear algebra and differential equations. At grad- uate level, one also requires a great deal of topology, analysis and algebra, but none of this is appears in our course! Our purposes is to use calculus to describe certain properties of curves and surfaces. Of particular interest is the notion of curvature. For curves this will turn out to be fairly simple: curvature is a measure of the ‘bendiness’ of a curve. A straight line should have zero curvature, whereas a circle should have curvature which increases as the radius decreases: a very large circle should have a very small curvature. Zero curvature Small curvature Larger curvature Changing curvature Understanding and quantifying this concept for more complicated curves is our first important goal. For instance, the curvature of the spiral increases as one travels towards the center. The rough idea is as follows: imagining a curve as a roller-coaster along which you travel at a constant speed, the curvature should equal the force you feel, necessary to keep you travelling along the curve. For surfaces, curvature is a more difficult concept. We will hunt for quantities which measure how much a surface appears to be dome- or saddle-shaped. Understanding this will occupy much of the second half of the course. Dome-shaped Saddle-shaped More complicated The third surface is saddle-shaped near the narrow neck and dome-shaped away from it. Euclidean Space Classical differential geometry is usually conducted in Euclidean space En. It is important to distinguish this from Rn. Rn Is the set of n-tuples of real numbers. These can be written as row- or column-vectors, your choice. Thus Rn = (x ,..., x ) : x R 1 n i 2 In this course, Rn is nothing more than a co-ordinate space. It is not assumed to have any addi- tional vector-space structure (no scalar multiplaction or addition of vectors), and certainly no dot product! En is the set Rn together with the usual notion of dot product. We usually write these as column vectors, and denote elements bold face1. For instance, 0 1 x1 3 x = @x2A E 2 x3 This helps us distinguish vectors in Euclidean space from co-ordinates. The dot product allows a b us to measure length and angle. To refresh; if a = 1 and b = 1 are two vectors in E2, a2 b2 then: • The dot product of a and b is a b = a b + a b . · 1 1 2 2 • The length of a is a = pa a. j j · a b • The angle q between a and b satisfies cos q = · . a b j j j j In this class, Rn will always have n = 1 (for curves) or n = 2 (for surfaces). Similarly En will always have n = 2 or 3. If a definition or result is true in both E2 and E3 (and more generally in En), we will just write En. 1Underline x or over-arrow ~x when hand-writing. 2 Examples cos t 1. Let x(t) = where t R. This defines a function x : R E2. The components of x are sin t 2 ! the functions x1(t) = cos t, x2(t) = sin t It is perfectly acceptable to use x and y for the components if you prefer: x(t) cos t x(t) = = y(t) sin t Just make sure you don’t get x and x confused! 0 u 1 2. Let x(u, v) = @ v A where u, v R (or (u, v) R2 if you prefer). This defines a function 2 2 u2 + v2 x : R2 E3. It’s components are ! x(u, v) = u, y(u, v) = v, z(u, v) = u2 + v2 It isn’t important at the moment, but these examples describe, respectively, a circle in the plane and a paraboloid in three-dimensions. 1 Curves 1.1 Parameterized Curves in Euclidean space A in En is a Definition 1.1. parameterized differentiable curve smooth x0(t0) function x : I En of a subinterval I = (a, b) of the real line R ! into En. It’s velocity, or tangent vector at t I is the vector x (t ). In terms 0 2 0 0 of the components, we can write 0 1 x(t0) x10 (t0) B . C x0(t0) = @ . A xn0 (t0) O Smooth means differentiable with continuous derivatives; that the quotient x(t + h) x(t) x0(t) = lim − h 0 h ! exists for every t I and that the function x (t) is continuous. The fact that we are differentiating a 2 0 vector-valued function is of no concern: we are not dividing by a vector in the above quotient! 3 Examples B b a Straight line Fix constant vectors a, b En. The parameterized A − 2 curve defined by x(1) = b x(t) = a + t(b a) − x(0) = a describes the line through the point A and B with position vec- O tors, respectively, a and b. 0cos t1 Helix Let x(t) = @sin tA. The function x : R E3 describes ! t a helix. To help visualize the helix, you can try several tricks. Imaging sitting on top of the z-axis and looking down: you’d only cos t see the horizontal projection of the curve t which de- 7! sin t scribes the unit circle counter-clockwise. The projection onto the z-axis is simply t t, so the curve moves upwards at a constant 7! speed. One could also project onto the xz- or yz-planes. All these images are plotted below y z z 1 3π 3π 2π 2π π π 11 − x 11 1 1 1 − π x − π y − − − Tangent Line Given a parameterized differentiable curve x : I n ! E , we can easily define its tangent line at any fixed t0. This is simply the straight line through the point with position vector x(t0), pointing in the direction of the tangent vector x0(t0). It is itself a parameterized curve, y : R En: ! y(s) = x(t0) + sx0(t0) p For example, the tangent line to the helix at t0 = 3 has equation 0 1 1 0 p3 1 2 2 y(s) = p3 + s −1 @ 2 A @ 2 A p 3 1 4 Corners and Cusps We will almost always think about parameterized curves which y are differentiable everywhere. It is, however, easy to construct 1 curves which are not. Consider the curve t x(t) = , t ( 1, 1) 1 t 2 − − j j 0 101 This has a corner at x(0) = and is not differentiable there. − 1 x Indeed, this is not a parameterized curve in the language of Defi- nition 1.1. A similar difficulty appears with a cusp: for example y 1 t x(t) = p , t ( 1, 1) t 2 − j j is not differentiable at t = 0. 101 − x In either case we call the non-differentiable point a singularity of the curve. Self-intersections Curves which cross themselves might seem difficult, but the pa- y rameterization takes care of everything! The curve in the picture 1 has parameterization sin 2t x(t) = 3 , t [0, 6p) cos t 2 1 1 The curve is differentiable everywhere, even though it intersects − itself. For instance, the curve passes through the origin at both x 3p 9p t1 = 2 and t2 = 2 , with corresponding tangent vectors 2 2 1 3p 9p − x0( ) = − 3 , x0( ) = − 3 2 1 2 1 − The important thing to note about self-intersections is that we always talk about the tangent vector to a parameterized curve at a co-ordinate t = t0. In this example, it is illegal to talk about the tangent vector to the curve at the origin. The parameterization takes care of the fact that there are two distinct tangent vectors at the origin. 5 Speed Definition 1.2. The speed of a curve x(t) is the norm/length of its velocity vector q 2 2 v(t) := x0(t) = x (t) + + x (t) 10 ··· n0 Note that the speed of a curve is a non-negative, scalar function, whereas the velocity is a vector. More- over, for a smooth parameterized curve, the speed must be a continuous function. Examples 0cos t1 1. The helix parameterized by x(t) = @sin tA has speed t q v(t) = ( sin t)2 + (cos t)2 + 12 = p2 − t3 3t2 2. The curve x(t) = has tangent vector x (t) = and speed v(t) = 3t2p1 + 4t6. t6 0 6t5 t 3. The curve x(t) = drawn above, has tangent vector 1 t − j j 8 ! > 1 > if t < 0, <> 1 x0(t) = ! > 1 > if t > 0, :> 1 − and is undefined at t = 0. Its speed is therefore v(t) = p2 for t = 0 and undefined for t = 0. 6 sin 2t 4. The curve x(t) = 3 has tangent vector cos t 2 cos 2t x (t) = 3 3 0 sin t − q 4 2 2t 2 and speed v(t) = 9 cos 3 + sin t. The picture shows the curve and its tangent vector in blue: observe how the speed of the curve (length of the tangent vector) changes as we move round the curve.2 2Click the picture when open in a full-feature viewer such as Acrobat. 6 1.2 Regular curves and the arc-length parameterization If we are going to do differential geometry, we need to be able to differentiate.

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