12.4. Curvature. Introduction. Prerequisites. Learning Outcomes

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1 Curvature 1.4 Introduction Curvature is a measure of how sharpl a curve is turning as it is traversed. At a particular point along the curve a tangent line can be drawn; this line making an angle ψ with the positive ais. Curvature is then defined as the magnitude of the rate of change of ψ with respect to the measure of length on the curve - the arc length s. That is Curvature = ds In this section we eamine the concept of curvature and, from its definition, obtain more useful epressions when the equation of the curve is epressed either in Cartesian form = f() or in parametric form = (t) = (t). We show that a circle has a constant value for the curvature, which is to be epected, as the tangent line to a circle turns equall quickl irrespective of the position on the circle. For ever other curve, other than a circle, the curvature will depend upon position, changing its value as the curve twists and turns. Prerequisites Before starting this Section ou should... Learning Outcomes After completing this Section ou should be able to... 1 understand the geometrical interpretation of the derivative be able to differentiate standard functions 3 be able to use the parametric description of a curve understand the concept of curvature be able to calculate curvature if the curve is defined in Cartesian form or in parametric form

2 1. Curvature Curvature is a measure of how quickl a tangent line turns on a curve. For eample, consider a simple parabola, with equation =. Its graph is shown in the following diagram. R P Q It is obvious, geometricall, that the tangent lines to this curve turn more quickl between P and Q than along the curve from Q to R. Itisthe purpose of this section to give, a quantitative measure of this rate of turning. If we change from a parabola to a circle, (centred on the origin, of radius 1) we can again consider how quickl the tangent lines turn as we move along the curve. It is immediatel clear that the tangent lines to a circle turn equall quickl no matter where on the circle ou choose to consider. However, if we consider two circle with the same centre but different radii: Q Q P ψ P ψ It is again obvious that the smaller circle bends more tightl than the larger circle and we sa it has a larger curvature. Athletes who run the 00 metres find it easier to run in the outside lanes (where the curve turns less sharpl) than in the inside lanes. On the two circle diagram (above) we have drawn tangent lines at P and P ;both lines make an angle ψ with the positive ais. We need to measure how quickl the angle ψ changes as we HELM (VERSION 1: March 18, 004): Workbook Level 1

3 move along the curve. As we, move, from P to Q (inner circle) or from P to Q (outer circle) the angle ψ changes b the same amount. However, the distance traversed on the inner circle is less than the distance traversed on the outer circle. This suggests that a measure of curvature is: curvature is the magnitude of the rate of change of ψ with respect to the distance moved along the curve We shall denote the curvature b the greek smbol κ (kappa). So ds where s is the measure of arc-length along a curve. This rather odd-looking derivative needs converting to a more familiar form if the equation of the curve is given in the form = f(). As a preliminar we note that ds = d d ds = /( ) ds d d We now obtain epressions for the derivatives d Consider the following diagram. and ds d in terms of the derivatives of f(). δ δs δ ψ Small increments in the and directions have been denoted b δ and δ, respectivel. The hpotenuse on this small triangle is δs which is the change in arc-length along the curve. From Pthagoras theorem: δs = δ + δ so ( ) δs =1+ δ ( ) δ so that δ δs δ = 1+ ( ) δ δ In the limit as the increments get smaller and smaller, we write this relation in derivative form: ds d = 1+ ( ) d d 3 HELM (VERSION 1: March 18, 004): Workbook Level 1

4 However as = f() isthe equation of the curve we obtain ( ) ds df d = 1+ =(1+[f ()] ) 1/ d We also know the relation between the angle ψ and the derivative df d : so differentiating again: Inverting this relation: d = and so, finall, the curvature is given b ds = d df d = tan ψ d f d = sec ψ d =(1+tan ψ) d =(1+[f ()] ) d f () (1+[f ()] ) /( ) ds = d f () (1 + [f ()] ) 3/ Ke Point At each point on a curve, with equation = f(), the tangent line turns at a certain rate. A measure of this rate of turning is the curvature κ f () (1 + [f ()] ) 3/ Obtain the curvature of the parabola =. First calculate the derivatives of f() f() = df d = d f d f() = df = d f = d d HELM (VERSION 1: March 18, 004): Workbook Level 1 4

5 Now find an epression for the curvature f () [1+[f ())] ] 3/ = [1 + 4 ] 3/ Finall, plot the curvature κ as a function of κ κ This picture confirms what we have alread argued: close to = 0the parabola turns sharpl. (near =0the curvature κ is, relativel, large.) Further awa from =0the curve is more gentle (in these regions κ is small). In general, the curvature κ, isafunction of position. However, from what we have said earlier, we epect the curvature to be a constant for a given circle but to increase as the radius decreases. This can now be checked directl. Eample Find the curvature of the circle =(a ) 1/ (this is the equation of the upper half of a circle centred at the origin of radius a). 5 HELM (VERSION 1: March 18, 004): Workbook Level 1

6 Solution Here f() =(a ) 1 df d = (a ) 1 d f d = a (a ) 3 1+[f ()] =1+ a = r a a (a ) 3/ [ ] a 3/ = 1 a a For acircle, the curvature is constant. The value of κ (at an particular point on the curve, i.e. at a particular value of ) indicates how sharpl the curve is turning. What this result states is that, for a circle, the curvature is inversel related to the radius. The bigger the radius, the smaller the curvature; precisel what, as we have argued above, we should epect.. Curvature for parametricall defined curves An epression for the curvature is also available if the curve is described parametricall: = g(t) = h(t) t 0 t t 1 We remember the basic formulae connecting derivatives d d = ẏ ẋ d ẋÿ ẏẍ = d ẋ 3 where, as usual ẋ d Then dt, ẍ d etc. dt f () {1+[f ()] } 3/ = ẋÿ ẏẍ [ ẋ 3 1+ ( ) ẏ ] 3/ ẋ = ẋÿ ẏẍ [ẋ +ẏ ] 3/ HELM (VERSION 1: March 18, 004): Workbook Level 1 6

7 Ke Point The formula for the curvature in parametric form is ẋÿ ẏẍ [ẋ +ẏ ] 3/ First find ẋ, ẏ, ẍ, ÿ An ellipse is described parametricall b the equations =cos t = sin t 0 t π Obtain an epression for the curvature κ and find where the curvature is a maimum or a minimum. ẋ = ẏ = ẍ = ÿ = Now find κ ẋ = sin t ẏ = cos t ẍ = cos t ÿ = sin t Find maimum and minimum values of κ b inspection ma min denominator is ma when t = π/. This gives minimum value of 1/4, denominator is min when t =0. This gives maimum value of ẋÿ ẏẍ [ẋ +ẏ ] = sin t+ cos t 3/ [4 sin t+cos t] = 3/ [1+3 sin t] 3/ minimum value of κ 1 maimum value of κ 1 7 HELM (VERSION 1: March 18, 004): Workbook Level 1