Parametric Curves. (Com S 477/577 Notes) YanBin Jia. Oct 8, 2015


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1 Parametric Curves (Com S 477/577 Notes) YanBin Jia Oct 8, Introduction A curve in R 2 (or R 3 ) is a differentiable function α : [a,b] R 2 (or R 3 ). The initial point is α[a] and the final point is α[b]. The domain of the curve is the interval [a,b]. A portion of α defined on an interval [c,d] [a,b] is called a curve segment. Example 1. Straight Line linear. Explicitly, the curve The line is the simplest curve in the plane as its coordinate functions are α(t) = p + tv = (x 0 + tv x, y 0 + tv y ), where v 0, (1) is a straight line through the reference point p = α(0) = (x 0, y 0 ) in the direction v = (v x, v y ). Here, t is the signed distance from a point α(t) on the line to p as scaled by v. As shown on the left, the vector from p to a point (x, y) on the line must be either in the direction of (x, y) (v x, v y ) or in its opposite direction. Hence, the cross (x 0, y 0 ) product of the two vectors must be zero, that is, (v x, v y ) (x x 0, y y 0 ) (v x, v y ) = 0. Expansion of the above cross product yields an implicit equation of the line that relates the x and y coordinates of every incident point: v y x v x y v y x 0 + v x y 0 = 0. (2) Example 2. Helix 1 The curve t (a cost, a sint, 0) travels around a circle of radius a > 0 in the xy plane. If we allow this curve to rise (or fall) at a constant rate, we obtain a helix α = (a cost, a sint, bt), where a > 0 and b 0. Example 3. The curve α : R R 3 such that 1 The figure is from [1, p. 16]. α(t) = (e t, e t, 2t) 1
2 shares with the helix in Example 2 the property of rising constantly. However, it lies over the hyperbola xy = 1 in the xy plane instead of a circle. A curve α(t) = (x(t),y(t)) is said to be smooth at t = t 0 if its kth derivative ( ) α (k) (t) = x (k) (t),y (k) (t) exists for any integer k > 0. A piecewise smooth curve α has a domain which is the union of a finite number of subintervals over each of which α is smooth. Example 4. A line α(t) = p + tq is a smooth curve. Here α (t) = q and α (k) = 0 for k > 1. A polygon, on the other hand, is a piecewise smooth curve, where each edge determines a subdomain. y Example 5. Cuspidal cubic The curve α(t) = (t 2, t 3 ) is smooth. We have x α (t) = (2t, 3t 2 ), α (t) = (2, 6t), α (t) = (0, 6), α (k) (t) = 0, k 4. Consider a plane curve α : [a,b] R 2. It is called a closed parametric curve if α(a) = α(b). A point of selfcrossing is a point α(t 1 ) for which there exist finitely many distinct values t 1,...,t n [a,b], n 2, which satisfy α(t 1 ) = α(t 2 ) = = α(t n ), and in the case n = 2, [t 1,t 2 ] [a,b]. Example 6. A circle is closed. The other three curves all have selfcrossings. 2
3 2 Velocity, Speed, and Arc Length Let α(t) be a curve. The velocity vector of α at t is α (t). The speed at t is the length α (t). The meaning is clear if we see α(t) as the location of a moving point at time t. The parametrization α(t) is unitspeed if α (t) = 1 for all values of t. A point where α (t) = 0 is called a cusp on the curve. Example 7. The origin on the cuspidal cubic in Example 5 is a cusp. The curve α(t) is regular if all velocity vectors are different from zero, that is, α (t) 0 for all t. Intuitively, a point moving on the curve with velocity α (t) will never come to a stop or reverse its direction. Example 8. Consider the curve α(θ) = (aθ cos θ, aθ sin θ). It has velocity α (θ) = a(cos θ θ sin θ, sin θ + θ cosθ), and speed α (θ) = a (cosθ θ sin θ) 2 + (sin θ + θ cosθ) 2 = a 1 + θ 2 0. Therefore the parametrization is regular. The velocity and speed depend on its parametrization. Nonregularity at a point may be just a property of the parametrization, and need not correspond to any special feature of the curve geometry. For a different parametrization the curve may have a nonzero velocity at the same point. To formulate the length of α, we note that the portion over [t,t + δt] is nearly a straight line when δt is very small. So the length over [t,t + δt] can be approximated by α(t) α(t + δt) α(t), which again is approximated by α (t) δt. α(t + δt) We divide α up into segments, each of which corresponds to a small increment δt. As δt tends to zero, we will obtain the exact length. The arc length of α from t = a to t = b is thus defined as b a α (t) dt. Example 9. Logarithmic spiral The curve α(t) = (e t cost, e t sin t), has a spiral motion. We obtain that α (t) = ( e t (cost sin t), e t (sin t + cost) ), α (t) = 2e t. 3
4 y x Figure 1: Logarithmic spiral (e t/20 cost, e t/20 sin t) over [0, 50]. Hence the arc length of α starting at α(0) = (1, 0), for instance, is s = t 0 2e u du = 2(e t 1). 3 Reparametrization Let I and J be intervals. Let α : I R 3 be a curve and h a differentiable function. Then the composite function β = α h is a curve called the reparametrization of α by h. β h α β(s) = α(h(s)) J s t I Example 10. Suppose α(t) = ( t, t t, 1 t) on (0, 4). If h(s) = s 2 on (0, 2), then β(s) = α(h(s)) = α(s 2 ) = (s, s 3, 1 s 2 ). The curve α has been reparametrized by h to yield the curve β. At each time s in the interval J, the curve β is at the point β(s) = α(h(s)) reached by the curve α at time h(s) in the interval. Thus β does follow the route of α, but it reaches a given point on the route at a different time than α does. Sometimes one is interested only in the route followed by a curve and not in the particular speed at which it traverses its route. One way to ignore the speed of a curve α is to reparametrize to a curve α which has unit speed α = 1. 4
5 Theorem 1 If α is a regular curve, then there exists a reparametrization α that has unit speed. Proof Consider the arc length function s(t) = t c α (u) du, where c is a number in the domain of α. It then follows that s (t) = α (t) ; namely, the derivative of s is the speed function α (t). Since α is regular, α 0 everywhere; hence ds dt > 0 always holds. By a standard theorem of calculus, the function s has an inverse function t(s), and dt ds = 1 1 = ds α (t). dt Now we let α(s) = α(t(s)) be the reparametrization of α. Then Hence, the speed of α is α (s) = α (t(s)) dt ds. α (s) = α 1 (t(s)) α (t(s)) = 1. The unitspeed curve α is said to have arclength parameterization, since the arc length of α from s = a to s = b, a < b, is just b a. Example 11. Let us consider the helix α = (a cost, a sin t, bt) in Example 2 again. It has velocity Hence Thus α has constant speed: The arc length from t = 0 is then α (t) = ( a sint, a cost, b). α (t) 2 = α (t) α (t) = a 2 sin 2 t + a 2 cos 2 t + b 2 = a 2 + b 2. c = α = a 2 + b 2. s(t) = t 0 c du = ct. Hence, t(s) = s c. Substituting this into the formula for α, we get the unitspeed reparametrization ( ( s α(s) = α = a cos c) s c, a sin s c, bs ). c Although every regular curve has a unitspeed reparametrization, this may be very complicated, or even impossible to write down explicitly, as the following examples show. 5
6 Example 12. The logarithmic spiral α(t) = (e t cost, e t sin t), has speed 2e t > 0. So it is regular. The arc length starting at (1, 0) was found in Example 9 to be s = 2(e t 1). Hence, t = ln( s 2 + 1), so a unitspeed reparametrization of α is given by the rather unwieldy formula α(s) = (( ) ( ( )) ( ) ( ( ))) s s s s cos ln 2 + 1, sin ln Example 13. Twisted cubic 2 This is the space curve given by α(t) = (t, t 2, t 3 ), < t <. We have α (t) = (1, 2t, 3t 2 ), α (t) = 1 + 4t 2 + 9t 4. Since the speed α (t) is not zero everywhere, α is regular. And the arclength starting at α(0) = 0 is t s = 1 + 4u2 + 9u 4 du. The above integral has a horrendous closed form not in terms of familiar functions. 0 4 Tangent and Normal The standard method of studying the geometry normal line of a curve at a point is to attach orthonormal tangent line vectors to the point and see how the directions of t increasing these vectors change as the point moves on the α curve for an infinitesimal distance. We choose (x (t), y (t)) tangent and normal vectors at a regular point. ( y (t), x (t)) Let α(t) = (x(t),y(t)) be a curve. At a regular point α(t) there exists a (nonzero) tangent vector α (t) = (x (t),y (t)). It represents the velocity of the curve at the point. The normal vector ( y (t),x (t)) at α(t) is given by rotating the tangent vector counterclockwise through an angle π 2. Note that (x (t),y (t)) ( y (t),x (t)) = (x (t)) 2 + (y (t)) 2 > 0. If α(t) is a unitspeed curve, then both the tangent vector and the normal vector are unit vectors. By convention they are denoted as T and N, respectively, with the cross product T N = 1. 2 The figure originally appears in [3, p. 14]. 6
7 For a parametric curve we have a tangent line and a normal line at each regular point α(t). The tangent line to the curve at α(t) passes through α(t) and is parallel to α (t) 0. So it has the parametric equation ( ) x(s),y(s) = α(t) + sα (t), s (, ), or equivalently, the algebraic equation ( ) ( ) (x,y) α(t) y (t),x (t) = 0. The normal line at α(t) passes through the point and is parallel to ( y (t),x (t)). So its equations are of the form ( ) ( ) x(s),y(s) = α(t) + s y (t),x (t), s (, ), or equivalently, ( ( ) ) x(s),y(s) α(t) α (t) = 0. Example 14. Crunodal cubic is described as α(t) = ( ) t 2 1, t(t 2 1). Find its tangent and normal lines of the curve at the points t = ±1, 0. We obtain α (t) = (2t, 3t 2 1), α (1) = (2, 2), α ( 1) = ( 2, 2), α (0) = (0, 1), α(±1) = (0, 0). Here α = (0, 0) is referred to as a double point since it is attained at both t = 1 and t = 1. The tangent lines at this double point are respectively and (x, y) = s(1, 1), or equivalently, y = x, The normal lines at the double point are respectively and (x, y) = s( 1, 1), or equivalently, y = x. (x, y) = s( 1, 1), or equivalently, y = x, (x, y) = s( 1, 1), or equivalently, y = x. At t = 0, we have α (0) = (0, 1), and the tangent line at α(0) is The normal line at α(0) is (x, y) = ( 1, 0) + s(0, 1), or equivalently, x = 1. (x, y) = ( 1, 0) + s(1, 0), or equivalently, y = 0. y x 7
8 References [1] B. O Neill. Elementary Differential Geometry. Academic Press, Inc., [2] J. W. Rutter. Geometry of Curves. Chapman & Hall/CRC, [3] A. Pressley. Elementary Differential Geometry. SpringerVerlag London,
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