Exercises on Space Curves from "Differential Geometry" by K L Wardle

In the attached paper, Space Curve Exercises Wardle Ch2, we provide solutions to the exercises presented after Chapter 2 in "Differential Geometry" by K L Wardle. The exercises provide practice in working with the basic aspects of the Differential Geometry of Space Curves and include several interesting results in their own right. Included (2.18*) are some results that overlap with the previous blog entry on Bertrand Curves. If any typos or errors are found, the author would welcome feedback.

Bertrand Curves -- Theorems from Exercise 14.4-27 A Survey of Geometry Vol.2 Howard Eves

In the attached paper, Bertrand Curve Theorems, we present proofs of several key results about Bertrand curves as presented in Exercise 14.4-27 (a)-(i) in Howard Eves' "A Survey of Geometry Vol 2". The proofs provide a nice introduction to Differential Geometry reasoning applied to these pairs of space curves which share a common principal normal between corresponding points. In two parts of the exercise, (c) and (d), we actually prove a variant of the theorem as stated by Eves. If any typos or errors are found, the author would welcome feedback.

Analysis of Plane Curves - Solutions to Exercises 14.1-(#2 - #40) Howard Eves "A Survey of Geometry" Vol.2

In the papers linked to below, complete solutions to the problems mentioned in the title are provided or, as in the case of two well known problems, references are provided. These problems serve as fruitful applications for classes in calculus including differential equations. No higher mathematics is required. The solutions are divided up over several papers due to the overall length. The solutions error on the side of more detail rather than less.

The main focus of these problems is the study of curves related to given curves. The short section 14.1 in Eves' book gives a very nice overview but as a good teacher he leaves the details to his readers.

There is a glossary of terms provided in the paper Part V. This glossary contains the definitions of the curves studied in these exercises. There are also appendices in some parts to provide background on selected topics that might need a little review.

The following links will cover the exercises as indicated.

1. Exercises 2,4,5,6,7 Part I - Solutions 2,4,5,6,7

2. Exercise 3 on two constructions of the center of curvature

3. Exercises 8,16,24,26,27 on curves related to equiangular spirals

4. Exercises 9,10,11,12,13,14,15 Part II - Solutions 9,10,11,12,13,14,15

5. Exercises 17,18,19,20,21,22,23,25 Part III - Solutions 17,18,19,20,21,22,23,25

6. Exercises 28,29,30,31,32,33,34,35,36 Part IV - Solutions 28,29,30,31,32,33,34,35,36

7. Exercises 37,38,39,40 Part V - Solutions 37,38,39,40

8. More details on the cycloid and its evolute

The Equiangular Spiral

The equiangular spiral is a fascinating and much studied plane curve. In the paper Equiangular Spiral the curve is defined, an important property of the curve is established, and this property is used to show several curves related to an equiangular spiral are congruent equiangular spirals. In particular, several exercises from Howard Eves' "A Survey of Geometry, Vol 2" are addressed (14.1-8,14.1-16,14.1-24, and 14.1-26).

Analysis of the cycloid and its evolute including intrinsic equations

The cycloid is the name of the curve traced out by a point on a circle rolling along a straight line. An evolute of a given curve is the curve comprised of the centers of curvature with respect to the points on the original curve. Most first studies of curves analyze the properties of the parametric equations that define the curves. These parametric equations are defined in terms of some coordinate systems in which the curve is analyzed. To analyze the curve free of any such  coordinate system, we have to describe the curve in terms of its so-called intrinsic properties only. In the case of plane curves the intrinsic equation of a curve is one that defines the arc length (one intrinsic property) as a function of the curvature (another intrinsic property). Such studies are the early explorations of differential geometry.

In the paper The Cycloid and Its Evolute a detailed discussion of the cycloid and its evolute is presented in order to make clear the steps needed to proceed from the parametric to the intrinsic equation for the cycloid and its evolute. The paper is more of an informal "classroom discussion" as it includes details and review materials that might not be included in a more formal presentation of the results obtained.

It should serve as a template to study other curves.

Two constructions for center of curvature at point on a conic [Howard Eves problems 14.1-3 (a) and (b)]

In this entry we present the justifications for two different constructions (compass and straightedge) of the center of curvature at a point on a conic as posed in Problems 14.1-3 (a) and (b) in Howard Eves' A Survey of Geometry Volume Two. The constructions make a nice application of the theory presented in the last entry on curvature at a point. The justifications are given in this attached paper, Constructions of Center of Curvature for Conics.

Curvature at a Point on a Curve

 How might you describe the difference in shape between an ellipse and a circle? One way is to think about how the object is curving (changing direction) at each point. In the case of the circle, the shape or the curvature (rate of change in direction) is exactly the same at every point whereas in the ellipse the shape is not uniform so that at some points its curvature is more or less than at some other points. Think of a basketball versus a football. The basketball has a uniformity of shape that the football does not. At the tips of the football the change in direction per unit length is much greater than at the middle of the ball.

 Given two circles with different radii, the curvature of the smaller circle will be greater than that of the larger circle. For example, if you had two circles drawn on the ground with radii 2 feet and 20 feet, and you moved along the perimeter with a constant tangential velocity of v = 2 feet per second, you would be changing direction much faster on the smaller circle than on the larger circle. In fact, the respective angular velocities w=v/r (rate of change in direction) would be 1 radian per second versus 0.1 radian per second. That is, you would be changing direction 10 times faster around the smaller circle.

 It is especially easy to define curvature in terms of a circle because the circle has constant curvature that is defined completely by the radius of the circle. In particular, we define the curvature at any point on a circle as the inverse of the radius. Thus a circle with small radius has a large curvature and a circle with large radius has a small curvature. Using k to represent curvature, we define k = 1/r as the curvature at any point on the circle of radius r.

 Now what about the ellipse where the curvature varies at different points? How might we define the curvature at some chosen point P on the ellipse? We do it by considering the circle (C,R) with center C and radius R which is tangent to the ellipse at P and has the same curvature as the ellipse at P. We call (C,R) the circle of curvature at P where C and R are called the center of curvature and radius of curvature at P respectively. Thus as we move around the ellipse, (C,R) will vary in size with R being small around the narrow ends of the ellipse and larger around the flatter regions of the ellipse where the curvature is smaller (the radius of curvature being larger).

 In fact, we can apply this definition to any plane curve, not just ellipses. In general, given a plane curve F defined by F={(x,y): y = f(x)}, we define the curvature of F at any point P=(p,f( p)) as the limit, as ∆p goes to zero, of the circle through P and Q=(p+∆p,f(p+∆p))that is tangent to the curve F at P. That is, the limit, as Q approaches P along F, of the circle through P and Q which is tangent to F at P.

 In the paper Curvature Derivation, we have derived the formula for the center and radius of curvature of the circle of curvature at a point P of general curve F using the tools of elementary calculus. At the end of the paper there are some examples worked out and illustrated. Hopefully, the paper will provide the reader with a detailed analytical understanding of and appreciation for the concept of curvature in the plane.