Geometry and Motion, MA 134 Week 1

Geometry and Motion, MA 134 Week 1 Mario J. Micallef Spring, 2007 Warning. These handouts are not intended to be complete lecture notes. They should be supplemented by your own notes and, importantly, by relevant books. 1 Curves and their Parameterisation We shall start with some familiar curves in the plane. Then, in 1.3, we shall give definitions that encapsulate features that are common to these curves and, indeed, should be common to all curves. 1.1 Equation of a line in the plane You are undoubtedly familiar with the Cartesian equation y = mx + c of the line L of slope m and y-intercept c; L is the subset of 2 defined by L := {(x, y) y = mx + c, x }. More generally, given two real numbers a and b, not both zero, the line L := {(x, y) 2 ax + by = c} (1.1) is orthogonal to the vector (a, b) at a distance c / a 2 + b 2 from the origin. (You should be able to prove these facts; if you cannot, please consult your supervisor.) You will also be familiar with the Vector form of the equation of a line. A line L n through r 0 in the direction of a 0 is {r(t) := r 0 + ta t }. (1.2) Digression. Definition of direction of a nonzero vector. Given a nonzero vector r = (x 1,..., x n) n \ {0}, the direction of r is the unit vector ˆr := r r where r is variously called the norm, length or magnitude of r and is defined as Thus, if r 0, r := 12 nx xi! 2 i=1 r = r ˆr which is the mathematical way of saying that a (nonzero) vector is a quantity which has magnitude and direction. End of digression and definition.. emarks (i) a in (1.2) does not have to be a unit vector. (ii) When n = 2, (1.2) has an advantage over the form y = mx + c in that it allows vertical lines by taking a = (0, 1). (iii) When n = 2, how do we go between (1.1) and (1.2)? More precisely, what is the formula that relates a, b, c in (1.1) to r 0 = (x 0, y 0 ), a = (a 1, a 2 ) in (1.2)? There are, in fact, many such formulas; can you see why? Convince yourself that one which works is: c r 0 := a 2 (a, b), a = ( b, a). + b2 1

1.2 Conic sections These are the curves obtained by intersecting a (double) cone in 3-space by a plane. 1.2.1 Equation of a circle C, centred at (a, b), of radius The Cartesian equation of C follows from the geometric definition of C as the set of points whose distance from (a, b) is : C := {(x, y) (x a) 2 + (y b) 2 = 2 }. Therefore ( ) x a 2 ( ) y b 2 + = 1 and so, there exists t [0, 2π) such that cos t = x a y b and sin t =, yielding C = {(a + cos t, b + sin t) t [0, 2π)}. (1.3) This is the parametric representation of C, and t is the parameter associated to each point of C. The choice of parameterisation is not unique: we could, somewhat perversely, have said that there exists τ [0, (2π) 1/3 ) such that cos τ 3 = x a and sin τ 3 = y b, and then C = {(a + cos τ 3, b + sin τ 3 ) τ [0, (2π) 1/3 )}. Note that (1.2) is the parametric representation of L. A precise definition of parameterisation will be given in 1.3. 1.2.2 Ellipse Geometric definition The ellipse is the set of points P in the plane such that the sum of the distances F 1 P and F 2 P of P from two given points F 1 and F 2 (called foci) is constant. Of course, F 1 = F 2 gives a circle. To aw an ellipse, place pins at F 1, F 2. Loop a piece of string around F 1 and F 2 and trace out the ellipse with pencil at P, keeping the string taut: Equation Place F 1 at ( c, 0) and F 2 at (c, 0). If P = (x, y) is a point on the ellipse then, the geometric definition of the ellipse yields: (x + c) 2 + y 2 } {{ } F 1 P + (x c) 2 + y }{{} 2 = 2a F 2 P where a := 1 2 ( F 1 P + F 2 P ) > c (by the triangle inequality). So: (x + c) 2 + y 2 = 4a 2 + (x c) 2 + y 2 4a (x c) 2 + y 2 i.e. 4xc = 4a 2 4a (x c) 2 + y 2. 2

It follows that (xc a 2 ) 2 = a 2( (x c) 2 +y 2) which, on expanding both sides, rearranging and cancelling, yields x 2 (a 2 c 2 ) + a 2 y 2 = a 2 (a 2 c 2 ) i.e. x 2 a 2 + y2 b 2 = 1 where b := a 2 c 2. We can now parameterise the ellipse in the same way we parameterised the circle: there exists t [0, 2π) such that x = a cos t, y = b sin t, i.e. What is the geometric significance of t? ellipse = {(a cos t, b sin t) t [0, 2π)}. 1.2.3 Hyperbola This is the set of points P in the plane for which the absolute value of the difference of the distances F 1 P and F 2 P of P from two given points F 1 and F 2 (again called foci) is constant. The equation of a hyperbola can be derived in the same way as for an ellipse, except that F 1 P F 2 P is now set equal to 2a and a < c. The equation is then x 2 a 2 y2 b 2 = 1, where b := c 2 a 2. The hyperbola has two branches, one where F 1 P F 2 P = 2a and one where F 2 P F 1 P = 2a. The first branch can be parameterised by making use of the identity sec 2 t tan 2 t 1 which enables us to set x := a sec t, y := b tan t, t ( π/2, π/2). 3

For the second branch we set: x := a sec t, y := b tan t, t ( π/2, π/2). The lines y = b a x and y = b ax are asymptotes of the hyperbola. The parameter t does not have a simple geometric interpretation. Digression. Hyperbolic trigonometric functions The functions cosh and sinh are defined by: cosh t := et + e t Verify that they have the following properties: (i) cosh 2 t sinh 2 t 1, 2 and d d (ii) cosh t = sinh t, sinh t = cosh t, (iii) cosh 2t = cosh 2 t + sinh 2 t, sinh 2t = 2 cosh t sinh t. Sketch the graphs of cosh and sinh and also of tanh t := sinh t cosh t web) to check your answers. End of digression sinh t := et e t, t. 2 cosh t and coth t :=. Consult an appropriate book (or the sinh t An alternative parameterisation of the right branch of the hyperbola is obtained by setting x := a cosh t, y := b sinh t, t. 1.3 Definitions of curve and parameterisation Compare the vector equation of L in (1.2) and the parameterisation (1.3) of C: In both cases we have a set of the form L = {(x 0 + ta 1, y 0 + ta 2 ) t } where a = (a 1, a 2 ) C = {(a + cos t, b + sin t) t [0, 2π)}. {(x(t), y(t)) t I} where I is an interval. The same is true for the ellipse and the hyperbola. We are therefore led to the following definition of a curve and its parameterisations: Definition of a curve such that C n is a curve (or path) if there exists a continuous map r : I n C = {r(t) t I}, where I is an interval. Definition of a parameterisation The mapping t r(t): I n is called a parameterisation of C. It consists of n functions, x 1 (t),..., x n (t), the components of r, of one variable, t. More definitions; nomenclature The parameterisation r is regular if it is differentiable and 0 t I. The curve C is regular if it can be parameterised by a regular parameterisation. A regular curve has a tangent line at each of its points whose direction at r(t) is that of. In particular, a regular curve cannot have sharp corners. An exercise on Examples Sheet 1 shows that a curve with a differentiable parameterisation may have sharp corners if is allowed to be 0. We shall only consider regular curves. 4

A parameterisation orients a curve in the direction of increasing t. If I = [a, b], the curve is called closed if r(a) = r(b): A closed curve is also called a loop. A closed curve is regular if 0 t I and (a) = (b). The curve is embedded or simple if r is injective, i.e., t 1 t 2 r(t 1 ) r(t 2 ) or {t 1, t 2 } = {a, b} where a and b are the endpoints of I. Thus a simple/embedded curve has no self-intersections. A Jordan curve is a curve in the plane which is both simple and closed. emarks (i) A parameterisation can be thought of as motion of a particle along the curve. r(t) is then the position of the particle at time t. is the velocity of the particle and is its speed. (ii) Let ϕ: J I be a continuous bijection between the intervals I and J. parameterisation of C then so is r ϕ := r ϕ, r ϕ : J n. If r : I n is a 5

Thus the same curve can be parameterised in infinitely many different ways because there are infinitely many choices of such bijections ϕ. From the mechanical point of view this corresponds to the fact that the speed of a particle along a given path can vary in an arbitrary manner. 1.4 Elementary curve sketching It is important to be able to sketch the curves defined by elementary parameterisations. Note that the slope of the tangent line to a regular curve in the plane at r(t) = (x(t), y(t)) is given by: Examples 1. r(t) = (t 2, t 3 ). dy dx = dy/ dx/. 2. The logarithmic spiral, r(t) = e t (cos t, sin t). 3. r(t) = (1 + e t )(cos t, sin t). 6

1.5 Length of Curves 1.5.1 Discovering the definition of length Let C be a curve in n which is parameterised by r : [a, b] n. Chop [a, b] into n equal segments. Let t n := b a n and t n = b. We can then approximate: and let t i := a + i t n, 0 i n so that t 0 = a But r(t i+1 ) r(t i ) (t i) t n, and so, n 1 length of C r(t i+1 ) r(t i ). i=0 i=0 n 1 length of C = lim n (t i) t n = So it seems reasonable to define l(c) as follows. b a. Definition of length of a curve Let I be an interval whose end-points are a and b, a < b and let r : I n be a differentiable parameterisation of a curve C n. The length of C, l(c), is then defined by: b l(c) :=. In mechanics language, this says that distance travelled = speed. But l(c) should not depend on the parameterisation of C. We verify this. a Proposition Let r : [a, b] n and r ϕ : [h, k] n be two parameterisations of C where r ϕ = r ϕ and ϕ: [h, k] [a, b] is a differentiable bijection so that u [h, k], ϕ (u) > 0. Then k h φ b du du =. a Proof 7

1.5.2 Examples Example 1 Find the length of the graph of y = x 2 between x = 0 and x = 2. Solution Example 2 The length of an ellipse C: r(t) = (a cos t, b sin t), t [0, 2π] a = 2 sin 2 t + b 2 cos 2 t 2π l(c) = a 2 sin 2 t + b 2 cos 2 t = 0 2π 0 (a 2 b 2 ) sin 2 t + b 2 This is a so-called elliptic integral. Elliptic integrals cannot be expressed in terms of elementary fuctions. Go to the 3rd year course on elliptic curves to learn more about these integrals! 8