Quantifying the Curvature of Curves: An Intuitive Introduction to Di erential Geometry

Transcription

1 Quantifing the Curvature of Curves: An Intuitive Introduction to Di erential Geometr Jason M. Osborne, William J. Cook, and Michael J. Bossé ovember, 07 Abstract In this paper we introduce the reader to a foundational topic of di erential geometr: the curvature of a curve. o make this topic engaging to a wide audience of readers, we develop this intuitive introduction emploing onl basic geometr without calculus and derivatives. It is hoped that this introduction will encourage man more both to consider this mathematical notion and to develop enthusiasm for mathematical studies. Introduction Ride a bike or drive a car. Hopefull, well before ou end up in a ditch, ou will recognize that not all curves in a road are constructed equall: some curves are simpl sharper or more curved than others. he subject of Di erential Geometr leads to the measuring (or quantifing) of curve curvature. Unfortunatel, formall investigating Di erential Geometr at an introductor level requires at least Di erential and Integral Calculus and Linear Algebra. Further, Di erential Equations, ensor Algebra, and Calculus as well as Manifold heor are necessar for more advanced treatments of the subject. However, an intuitive understanding of curve curvature is approachable b high school students and teachers. he main goal of this article is to widen the audience of, and appreciation for, Di erential Geometr b developing the notion of curve curvature in an intuitive manner using rudimentar geometr and a minimal number of equations. Curvature is recognized as one of the most fundamental topics in Di erential Geometr. In fact, M. Spivak devotes an entire volume [?] (vol. II.) to the stud of curvature within a historical framework. his second volume A is a wonderful, modern treatment of the notion of curvature as it evolved from the original investigations of L. Euler to later etensions b C. F. Gauss and B. Riemann. In the tet, [?] Di erential Geometr is approached from the viewpoint of E. Cartan. Aiming to present notions of Di erential Geometr to a wider audience with less mathematical background, B. O eill states his book is an elementar account of geometr of curves and surfaces. It is written for students who have completed standard courses in calculus and linear algebra, and its aim is to introduce some of the main ideas of di erential geometr ([?] page i). his paper seeks to further simplif B C the topic of curvature b making it accessible to students who possess a rudimentar understanding of onl geometr and algebra. Considering onl lines, circles, and ratios, we present an intuitive, geometric understanding of Figure : Curvature Intuitivel. his parabola is most bent at C (the verte) and least this subject. It is our hope that students ma be interested in continuing on to a more rigorous treatment of bent at A. How do we measure this? the subject in the future. Without reading advanced tets, everone has an intuitive understanding of curvature; shapes like straight lines ( ) have no curvature while shapes like the curve ( ) are curved. he idea of curvature is even built into the names of these shapes (straight and curved). Delving deeper, one sees the challenge is not so

2 Figure : he ormal Vector. he vector predicts the direction in which the vector will turn. ote that at inflection points, denoted b J, the normal vector abrubtl changes direction. otice that during the first bend in the curve in Figure??, the vector is turning down as the magnetic ball moves from left to right, and, correspondingl, the normal vector is pointing down. Inthenetbend, the vector turns up and so the normal vector points up. here are points on a curve where the normal vector is not strictl defined. At these points, denoted b in Figure?? and called inflection points, there are two vectors perpendicular to and the choice for can therefore be made b simpl flipping a coin. hat is, the vectors will abruptl change direction from pointing down/up to up/down at these points of inflection. Curves of Constant Curvature or urning (Lines and Circles) ow that we know how and work together to indicate how a plane curve turns, we can sa that: At each point, curve curvature quantifies the degree of deviation of the curve from a straight line. he change in is used to precisel quantif this deviation. Intuitivel, if a curve has a sharp turn and so is changing rapidl, then the curvature should have a large value. In contrast, if a turn is gradual, and so is changing slowl, then curvature should be small. ote that since and change together we can quantif these deviations using just as well as. When discussing lines, studing the change in natural, while in the case of circles, is a better choice. When a curve has a constant curvature value from one location to another, one can imagine the curve as being constructed from a series of uniforml bent components. he first eample of a curve of Figure : A Line Has Zero Curvature. constant curvature is a line. he line ( ) can be viewed as an object comprised of a series of the smaller straight line components ( ) each with no curvature. otice that in Figure?? the direction vector does not turn. Since does not change, the curvature should have a constant value 0. Recall that when does not change, is not uniqel determined. In Figure?? we made a choice of pointing up; but pointing down could have been equall valid. A circle is another eample of a curve with constant curvature. Like a to train set which has been

3 he Osculating Circle of a Plane Curve and Curve Curvature Equation (??) relates the curvature of a circle to its radius, and is the ke to developing a tool that can be applied to curves of non-constant curvature. his tool is called the osculating circle Radius of Osculating Circle is R =.0007 Curvature of Curve is /R = Figure : An Osculating Circle. hecircle of radius is the best circular approimation of this function at this location. Ecept at points where the curve is perfectl flat, there is a circle of some radius that osculates (which means, kisses) the curve (see Figure??). he osculating circle can be viewed as the best circular approimation of the curve at that instant. his means that this circle is of the perfect radius so as to conform to the shape of the curve at that point. As shown in Figure??, this circle s radius can change based on the location it kisses. hat is, for the circle to conform to the contours of the curve it must be able to change its shape (and therefore, its radius) accordingl. In fact, Figure?? shows a tacit relationship between the radius of the osculating circle and the curvature of the curve. At locations where the curve has a large curvature, for eample at the point where the left most circle kisses the curve, the radius of the osculating circle is small. he opposite is true at the point where the right most circle kisses the curve; the curve is flatter and so the radius of the osculating circle is larger. he middle circle kisses a point where the curve appears to be nearl flat (almost line-like), and therefore the radius of the osculating circle is much larger. hese observations strongl suggest that we define the curvature of a curve at a given point as simpl the reciprocal of the radius of the osculating circle at that point. At a point where the curve is perfectl flat, we define the curvature to be 0. If we tr to draw an osculating circle at such a point, we can never make the radius large enough for the circle to conform to the curve. For the circle to conform, it would have to have an infinite radius or, in other words, the best approimation at such a point is a line and not a circle. As its radius grows, a circle limits to a line. Likewise, the reciprocal of its radius limits to 0. hink back to the curve ou created b bending our long piece of straight wire. Your curve will most likel not have constant curvature since ou will have decided to make a more interesting shape than a line or a circle. And while our curve, on the whole, is not a line or a circle, ou can think of our curve as being Figure 7: A Sine Curve with Osculating Circles. At each point, an osculating constructed from a series circle has the perfect radius so as to conform to the curve. ote the inverse of various sized straight relationship; big circles correspond with little curvature and vice-versa. line segments and circular bends. At flat points the curvature is defined to be 0, while at an other point the curvature of the curve is defined to be the curvature of its osculating circle. hat is, as a mathematical formula, we define the curvature apple and apple = r of generic curve as 0 at flat points where r is the radius of the osculating circle. ()

4 7 Eample: he Archemedean Spiral As a final eample, we analze the Archemedean spiral using the same techniques as in the previous eample. A C B (a) he Archemedean Spiral. 0 (b) Osculating Circles of Archemedean Spiral Are Decreasing in Radius Radius of Osculating Circle is R =.97 Curvature of Curve is /R = 0.9 Radius of Osculating Circle is R =.99 Curvature of Curve is /R = 0. (c) Osculating Circles of Fresnel Spiral Are Also Decreasing in Radius Radius of Osculating Circle is R = 0.78 Curvature of Curve is /R =.7 Figure : Archemedean Spiral vs. Fresnel Spiral. Do the radii of the osculating circles decrease in the same wa? (a) (Point A, r =.9) (b) (Point B, r =.0) (c) (Point C, r = 0.) Figure : Curvature Values at A, B, and C. Finding osculating circles at these three points and computing their reciprocals gives A ( =.), B ( k = 0.), and C ( =.). t (,) (.0 (. (. (.98 (.00 (.00 (. (. (.9.0 ). ). ).8 ). ).0 ).9 ). ).7 ) r k t Figure : Archemedean Spiral Data he curvature of this spiral grows non-linearl with time. 9

5 We now switch into visual interactive mode. 8 D Plane Curves Visual Interactive Radius of Osculating Circle is R =.9 Here isofwhat weisknow D Plane Curves: Curvature Curve /R = about 0.89 here is no reason these results cannot be etended to D Space Curves. he vector points in the direction of the curve. he vector is perpendicular to. Osculating or Kissing Circles are the best circular approimation of the curve at a point. he Fresnel spiral has osculating circles of smaller and smaller radius... Radius of Osculating Circle is R =. Curvature of Curve is /R =. -...so the curvature gets and larger.circle is R =.997 Radius oflarger Osculating Curvature of Curve is /R = with time on the curve. just a bit more andthe his is the Fresnel spiral. Its curvature increases linearl Move the point he radius of osculating circle is about R=/ and the curvature is about /R=/ radius gets bigger so the Sine curve gets flatter. - he Sine curve is the most curved it can be here. he radius of this osculating circle is R=9/ he curvature of the curve is /R=/ he radius of this circle is R= he curve curvature is /R= he Sine curve is consecutivel flatter at these points b/c the radius of the osculating circles are consecutivel bigger radii: r=, /,, / curvatures:, /,/, / - 0

6 9 D Circular Heli Visual Interactive Here is what we can sa in D. he osculating circle still works for a circular heli of constant curvature A D Circular Heli... with an osculating circle See. Ecept for perspective, the osculating circles are actuall circles. It is the Fresnel sprial again, but D... he spiral gets tighter and tighter......and bigger... Each osculating circle is of the same radius, so the circular heli has constant curvature. he circular heli is the D equivalent of the D circle. he are both are constant curvature.