An Easier Derivation of the Curvature Formula from First Principles

An easier derivation of the curvature formula from first principles

Robert Ferguson Florida, USA bobferg13@comcast .net

Introduction

he radius of curvature formula is usually introduced in a university calculus Tcourse. Its proof is not included in most high school calculus courses and even some first-year university calculus courses because many students find the calculus used difficult (see Larson, Hostetler & Edwards, 2007, pp. 870– 872). Fortunately, there is an easier way to motivate and prove the radius of curvature formula without using formal calculus. An additional benefit is that the proof provides the coordinates of the centre of the corresponding circle of curvature. Understanding the proof requires only what advanced high school students already know: e.g., algebra, a little geometry about circles, and the intuition that both a circle and a curve have a slope and curvature. The proof is suitable for first-year university and advanced high school students. In terms of algebra, students need to know how to solve two simultaneous linear equations with two unknowns. In terms of geometry, students need to know that: 1. The radius of a circle drawn to a point of tangency between the circle and a tangent line is perpendicular to the tangent line. 2. If two separate lines are tangent to a circle at two different points, the lines drawn perpendicular to the tangent lines at their points of tangency intersect each other at the circle’s centre. 3. Each perpendicular line’s segment from its point of tangency to the point of intersection is a radius. The paper’s development of the radius of curvature formula can be used as an insightful application of the mathematics advanced high school students already know. Australian Senior Mathematics Journal vol. 32 no. 2

16 An easier derivation of the curvature formula from first principles Australian Senior Mathematics Journal vol. 32 no. 2 17

) 3 ), . 0 0 y y , , 0 ) and This 0 1 y ). ). 0 , 0 (x , y 0 = f ). approaches x 1 ). 1 1 , y 1 , y 1 ) and (x 0 , y 0 ) and whose radius is the radius of ) and whose radius is the radius 0 y , 0 ). ) is defined to be the distance from (x ) is defined ) is defined to be the circle whose centre ) is defined to be the circle whose 0 0 0 y y , y , , 0 0 0 )) consisting of points in the Cartesian plane. plane. Cartesian the in points of consisting )) (x f , ) may approach a limiting point as x ) may approach a limiting point as y , ). 0 , y 0

2 fixed. (x 0 ). x , y line. Students can gain intuition by considering a circle. Students will readily intuition by considering a circle. Students can gain is the centre of curvature at (x it by (x curvature at (x matches the given curvature. the centre of curvature, to (x the line perpendicular to the tangent to the curvethe line perpendicular to the tangent at (x to infinity. of inflection. Find the point of intersection of these two perpendicular lines and denote Find the point of intersection of Keeping This will be true for most functions of interest, except at particular points, e.g., critical points or points This will be true for most functions of interest, except at particular points, e.g., critical points Twice differentiable can be presented to advanced high school students as ‘smooth’. differentiable can be presented to advanced high school students Twice a straight is exception An obvious of interest. functions most for intersection of a point will be There In general: Consider the line perpendicular to the tangent to the curveConsider the line perpendicular at (x Given a curvature,Given a that only one circle radius, hence is only one there Consider two close points on the curve,Consider two close points on the (x The curvature of a circle usually is defined as the reciprocal of its radius The circle of curvature at (x The radius of curvatureThe radius at (x This point is defined to be the curve’s centre of curvatureThis point is defined to be the curve’s at (x A circle’s curvature from infinity to zero as its radius varies from zero varies A circle’s A circle’s curvature monotonically decreasing function of its radius. is a A circle’s (the smaller the radius, the greater the curvature). • • • • • • • • • accept that a circle’s curvature two points on the Considering any is constant. a circle’s accept that function gives a curvea gives function (x follow naturally. 3 point on the circle, the radius of curvaturepoint on the circle, the radius of the circle is the at any point on circle, the perpendiculars to the tangents at these two points will meet at the to the tangents at these circle, the perpendiculars students so circle, the on points two any for true is This circle. the of centre on the curve. the radius and circle of curvature Definitions and formulae for 2 radius of the circle, and the circle of curvatureradius of the circle, and the circle itself. is the circle will readily accept that the circle’s centre is the centre of curvature for any will readily accept that the circle’s Here is the procedure for finding the centre of curvatureHere is the procedure at any point (x Consider a curveConsider function y differentiable given by a twice The intuition behind the procedure is that: 1 The intuition behind the procedure The procedure of curvature the radius for finding 18 Australian Senior Mathematics Journal vol. 32 no. 2 Ferguson The algebra The intersectionofthe twoperpendicularlinesapproximatesthecurve’s Using thepropertythatslopeofalineperpendiculartoanotheris Using thepoint-slopeformofalineanddenotingitsslopebym Consider a curve givenby: Consider acurve Consider two points on the curve, (x Consider twopointsonthecurve, (y

line at(x line to the curve at(x line tothecurve the negativeinverseofthatline’s slope,thelineperpendiculartotangent centre of curvature. As centre of curvature. of the circle of curvature that matches exactly the curve’s curvature atthe thatmatchesexactlythecurve’s curvature of thecirclecurvature point (x presumed nottobeproblematic,e.g.,criticalpointsorofinflection. Similarly, thecorresponding tangentandperpendicularlinesatthepoint are: and • • • • A curve’s curvature between two points approaches its curvature at a point between two points approaches its curvature A curve’s curvature The radius of the absolute area minimizing circle approaches the curve’s ofacirclethatminimisestheabsoluteareabetween The curvature This circle’s radiuscanbeviewedasapproximatingthecurve’s radiusof For any two close points on a curve, atleastonecircleminimises the For anytwoclosepointsonacurve, radius of curvature asthetwopointsapproacheachother.radius ofcurvature curve’s curvature asthetwopointsapproacheachother.curve’s curvature approachesthe andthecirclebetweentwoclosepointsoncurve curve curvature. as thetwopointsapproacheachother. andthecirclebetweentwopoints. absolute areabetweenthecurve 0 , y 0 , y 0 ). 0 ) is: 0 , y x 1 0 approaches ) is: ( ( y y –y –y − − y y 1 0 0 1 y ) =m ) =m ) ) =f = = x 0 − − (x , 0 , the intersection becomes the centre 1 0 ( ( y (x (x x x ) (1) 0 ) and(x m m − − –x –x 1 0 x x 1 0 ) ) 1 0 ) (4) ) (2)

1 , y 1 ). Thetwopointsare 0 , thetangent (3) (5) An easier derivation of the curvature formula from first principles Australian Senior Mathematics Journal vol. 32 no. 2 19 ), 0 (8) (6) (7) (9) (10) (14) (12) (13) (11) (15) – x (x 0 denotes 0 = m denotes the 0 – y ), the tangent line

), where n ) y ) ) 0 )

0 , 0 0 0 ) ) 0 y x x x 0 0 1 – x ), e.g., y − ), where m − − x x 0

− 0 m 0 1

1 1 ) y 1 − − y

(x m ) ) x 0 x , 0 x ( 1 ) ( ( 0 0 0 x ( x 0 x ( y – x + x ) ) ( + 1 m − x 1 1 ) − − − ⎞ ⎟ ⎠ ) 1 0 + − m m 1 ) = n 1 0 (x ]. x = x m ) 0 1 y 1 ( x 1 x 1 − m ) ( 0 ( x ) − + 0 ( 0 − m x m 1 + , x ⎞ ⎟ ⎠ − ) 1 1 y 1 ⎞ ⎟ ⎠ 0 − 1 x 1 m 1 ), respectively. Since the function is ), respectively. − x 1 ) – m m m ( ( ( − 0 m ) ( ) = m 1 m 0 1 m m 0 y m 0 1 ⎛ ⎜ ⎝ − x − x 0 + m + y , − x − ( + 0 0 ( = ) − 1 = = m 0 0 − 1 0 1 = m – y ) + ( ⎞ ⎟ ⎠ ) x m 1 x m m 1 x ( 0 ) 0 1 0 ⎞ ⎟ ⎠ ( ⎛ ⎜ ⎝ ( y 1 ⎛ ⎜ ⎝ − 1 x 0 m 0 − m y m = ). 1 − m x − = 0 m ), e.g., (m m − = ) − 0 ( = 1 0 ) 1 ) x 0 y y ) − y 0 − m , y ( ) and (x 1 x ( ( 0 , 0 1 1 y m x 0 − = − x y − 1 = ⎛ ⎜ ⎝ m − − , ) ) ) − ) x = 1 ⎛ ⎜ ⎝ y 0 0 ( 0 x 0 ) ) x ( ( y x x x 0 0 0 ( 0 y − − − x − m m − y − x x x ( ( ( ( y x ( ( − − ) term in equation (15) is the difference in the original curve’s ) term in equation (15) is the difference in the original curve’s 0 – m 1 ) is twice differentiable, (y (x f a function (the curve) has a slope at each point; just as the tangent line to the original curve at (x is a good approximation to the original curve near (x the slope (rate of change) of the slope function is a good approximation to the slope (rate of change) of the slope function is a good approximation the slope function near (x these slopes can be viewed as another function; to the slope function at (x One way of helping students gain intuition about the original function and One way of helping students gain intuition about the original function The (m • • • slopes at the two points (x Since Substitute equation (14) into equation (3) to obtain Substitute from equation (3) into equation (5). (3) into equation from equation Substitute curve’s slope somewhere in the intervalcurve’s [x twice differentiable, the function that represents its slope (its derivative) is to point out that: the function that represents its slope (its derivative) is to point out

The intersection is the solution of equations (3) and (5). of equations is the solution The intersection 20 Australian Senior Mathematics Journal vol. 32 no. 2 Ferguson (x As

Equations (19) and (20) give the x where equations (19)and(20)thepoint(x or circle corresponding to the radius of curvature at(x circle correspondingtotheradiusofcurvature is denoted by formulas, below, foundincalculus textbooks. Substitute equationfrom(16)intoequations(14)and(15). –x The radius of curvature, R The radiusofcurvature, For studentsthathavenothadcalculus,noteinthetermm y x 1 "(x approachesx 0 ) and(y n ), sothattheycanseeequation(22)isequivalent tothevarious denotestheslopefunction’s [x slopesomewhereintheinterval R –y d d x = y or 0 ⎝ ⎜ ⎛ ) approach: 1 0 + f , alltheslopesapproachtheirvaluesat(x '(x ⎝ ⎛ dx d dx dy 2 ) or y 2 R ⎠ ⎞ 2 2 ( ⎠ ⎟ ⎞ ( x = = = ( y x , isthedistancebetweenpoint(x 3 2 m ( − ( m '(x ( ( − y x 1 1 y x = 0 R 2 − x − + − 0 − ) and the term n ( 0 ) ( y 1 m m x = ) y 1 0 n 2 0 = 0 + 0 = 0 0 ) 2 + 0 2 ) ) ( and y ) − m ) = 2 1 − ( 3 = = f m + 0 + 2 f ( m ( " n n 1 ) ( 0 ' m 1 ( 2 ( 0 y 0 + ( ( x x 0 + 1 ( 0 x n + 2 − , y n m 1 ) ) n 1 ) m coordinates of the centre of the + n 0 ) ( 3 2 + y m − 0 2 1 0 m 0 0 2 ). ) m x 1

+ ) ) 3 2 m n 2 ) 0 0 2

m 1 0 )

2 = ) n )

0 2

0 )

( is denoted by 2 1 0 +

, y ( y y 0 " ). ' ( ( x x ) ) ) 2 ) 3 2

0 , , d d x y 2 y 2 y ) givenby 0 ), hence or 0 f , (22) (21) (16) (19) (18) (17) (20) (23) "(x x 1 ]. 0 )

An easier derivation of the curvature formula from first principles Australian Senior Mathematics Journal vol. 32 no. 2 21 ) y , that are one- ) points from changes from y x , ranging from 0.5 to 0.0001. from 0.5 to ranging used to compute the slopes the to compute used x x ∆ ), and observing how they approach y , Figure 1 Figure and the points on this curve at (0.5, 0.125) and 3 ) and the changes in ) and the changes = x 0 x ). Figure 1 plots the function and the (x and the the function plots Figure 1 ). and a plot of the function. – y 1 x values. Figure 1 also contains a plot of the radius of curvature values. Figure 1 also contains a plot x (22) without calculus by evaluating the slopes (derivatives) and (derivatives) slopes the evaluating by calculus without (22) , ).125 + ∆ ).125 , x ), plotting the resultant points, (x ), plotting the resultant ∆ 0 + The iterations for the centre of the circle of curvature are shown by the – x 1 small values for (x for values small square markers, which run from upper left to lower right as ∆ and changes in slopes. Additional insight can be gained by evaluating (x slopes. Additional insight can be and changes in a limiting point that is the centre of the circle of curvaturethat is the centre of the circle of a limiting point to corresponding and (20) evaluated with calculus of (0.1094, 0.6458). The iterations for the and (20) evaluated with calculus of (0.1094, 0.6458). The iterations found from equations (14) and (15) for a decreasing sequence of values for (14) and (15) for a decreasing found from equations for each value of ∆ Example equations (14) and (15) for several values of several for (15) and (14) equations equation spreadsheet and suitably (second derivatives) using an Excel changes in slopes tenth of the ∆ the curve’s radius of curvature.the curve’s radius of curvature are shown by the triangular markers, which run from upper 0.5 to 0.0001. The points converge to the coordinates given by equations (19) 0.5 to 0.0001. The points converge to the coordinates given by equations First year university and advanced high school students can evaluate students can school advanced high university and First year Consider the function y The slopes at the two points are computed using changes in x The slopes at the two points are (x (0.5 Teaching the radius of curvature the radius formula Teaching right to lower left as ∆x changes from 0.5 to 0.0001. The points converge to the radius of curvature given by equation (23), about 0.6510. Ferguson

Conclusion

This paper presented an easier way to motivate and prove the radius of curvature formula that uses mostly high school mathematics and no formal calculus. An additional benefit is that the proof provides the coordinates of the centre of the corresponding circle of curvature. Understanding the proof requires what high school students already know: algebra, a little geometry about circles, and the intuition that both a circle and a curve have a slope and curvature. The proof is suitable for first-year university and advanced high school students. Advanced high school students can find a function’s radius of curvature and the centre of the corresponding circle of curvature without use of calculus by evaluating the required slopes and changes in slopes along a curve using an Excel spreadsheet and suitably small distances between two points on the curve. Additional insight can be gained by doing so for a decreasing sequence of values for the distance between the points, plotting the resulting sequence of radii of curvature and their centres of the corresponding circles of curvature, and observing how they approach a limit.

Reference

Larson, R., Hostetler, R. P. & Edwards, B. H. (2007). Calculus: Early transcendental functions (4th ed.). Boston, MA: Houghton Mifflin. Australian Senior Mathematics Journal vol. 32 no. 2