CHAPTER TWO: THE GEOMETRY OF CURVES

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1 CHAPTER TWO: THE GEOMETRY OF CURVES Thi material i for June 7, 8 (Tueday to Wed.) 2.1 Parametrized Curve Definition. A parametrized curve i a map α : I R n (n = 2 or 3), where I i an interval in R. We can write α(t) = (x 1 (t), x 2 (t),..., x n (t)). Here t I i called the parameter. Sometime, we emphaize the parameter by writing t α(t) = (x 1 (t), x 2 (t),..., x n (t)). When n = 2, it i called a plane curve, and n = 3, it i called a pace curve. Remark: (1) {α(t), t I} R n i called the trace of α; (2) Different parametrized curve may have the ame trace. Definition The tangent vector (or velocity vector) of α at the point α(t ) i defined to be α (t ), and α (t ) i the peed. To ee the reaon for thi terminology, note that the vector α(t + δ) α(t) δ i parallel to the chord joining the two point α(t) and α(t + δ) of the image C of α. We expect that, a δ tend to zero, the 1

2 chord become parallel to the tangent to C at α(t). Hence, the tangent hould be parallel to α(t + δ) α(t) lim δ δ = α (t). Definition. A parametrized curve α(t), t I i aid to be regular if α (t) for all t I. Example of Plane Curve: The traight line paing through two point a and b i i parametrized by t (1 t)a + tb. The domain of t can be [, 1], (, ) or (1, ), etc. Note that when t = the particle pae through the point a, when when t = 1 the particle pae through the point b. The circle in R 2 with center (x, y ) with radiu r ha a parametrization t (x, y ) + r(co t, in t), i.e., α(t) = (x + r co t, y + r in t). If the curve i the graph of a function f(x), i.e. the curve i y = f(x), x I, then we can parametrize thi curve a α(t) = (t, f(t)), t I (i.e. we let x = t). The tandard ellipe with equation x2 a + y2 = 1 i parametrized 2 b2 by t (a co t, b in t). 2

3 A tandard parabola with equation y 2 = 4ax i parametrized by t (, ) (at 2, 2at). The parametrized curve given by α(t) = (co 3 t, in 3 t) i called atroid. Click here to ee the graph of atroid and click here to ee the graph of atroid and other famou plane curve. Example of Space Curve: A helix (coiling about the y-axi) ha a parametrization α(t) = (R in t, bt, R co t), t (, + ). ARC-LENGTH Definition. The arc-length of a curve α(t) at t [a, b] i length(α) = b a α (t) dt. That i, the ditance an article travel-the arc-length of it trajectory- i the integral of it peed. Example 1. For the unit circle α(t) = (co t, in t), t 2π, we have α (t) = ( in t, co t), o α (t) = 1. Hence the are-length of the unit circle i S = 2π α (t) dt = 2π dt = 2π. 3

4 Example 2. Let α(t) = (t, t 2 /2, ). Then α (t) = (1, t, ). So α (t) = 1 + t 2. Hence, the arc-length of the parametrized curve from α() to α(t) i (t) = t α (u) du = t 2.2 Change of parameter: Re-parametrization 1 + u2 du = 1 2 (t 1 + t 2 +ln(t+ 1 + t 2 )). We ay that β : J R 3 i a reparametrization of α(t) : I R 3 provided there exit a mooth bijection θ : I J uch that α(t) = β(θ(t)). For intance β( t) = (co(2 t), in(2 t)), t π, i a reparametrization of α(t) = (in t, co t), t 2π, with θ : [, 2π] [, π] given by θ(t) = t/2. The geometric quantitie aociated to a curve do not change under reparametrization. Thee include length and the curvature which will be defined later. ARC-LENGTH PARAMETRIZATION. Theorem. If α i a regular curve. Then α may be reparametrized to have unit-peed (or equivalently, by arc-length), i.e. it peed i identically one. 4

5 Proof. Let α : [a, b] R 3. Define (t) = t a α (u) du = length function from α(a) to α(t). The, by the fundamental theorem of calculu (ee your Calculu book!), (t) = α (t) > which implie that (t) i invertible ( by the invere function theorem) and let t = t() be the invere function (defined on certain interval). Then the reparameterization β() := α(t()) ha velocity vector β () = α (t()) dt d = α (t()) 1d dt = α (t()) (t) = α (t()) α (t()). = β () = 1,. Thi how that β() i a unit-peed curve. The above proce alo howed u how to reparametrize a (non- arc-length-parameterized) curve α(t) uing the arc-length parameter o it become unit-peed. We ummarize the the following three tep: Step for arc-length parametrization: Step 1: Calculate (t) = t a α (u) du. 5

6 Step 2: Calculate t = t(), the invere function of = (t). Note that theoretically the function t = t() exit, but it i in general hard (ometime it i impoible) to find it a an explicit form. Step 3: Plug t = t() into α(t) to get β() = α(t()). Example. Conider the circular helix α(t) = (a co t, a in t, bt). Parametrize α(t) in term of arc-length parameter. Solution. Step 1: Calculate (t) = t a α (u) du. Since α (t) = ( a in t, a co t, b), α (t) = a 2 + b 2. (t) = t α (u) du = t a2 + b 2 du = t a 2 + b 2. Step 2: We calculate t = t(), the invere of = (t). From (t) = t a 2 + b 2 we get the invere function t = t() = a2 + b 2. 6

7 Step 3: Subtitute t = β() = α(t()) = α = a 2 +b 2 back to α to get a2 + b 2 a co a2 + b 2, a in a2 + b 2, b a2 + b 2. Hence we get an arc-length parametrization of the circular helix a b β() = a co a2 + b 2, a in a2 + b 2, a2 + b 2. It i eay to check that β () 1. 7