The Distance between Simple Geometric Objects

In this entry we derive expressions for the distance between a point and a line, a point and a plane, and two lines in space. We derive the formulas using vector algebra and the concept of projection which is central to the distance concept. These are basic and well known results and we collect them here to serve as an "extra-reading" project or convenient reference. The results are derived and a few examples are provided in the paper Distance between Geometric Objects. As always, we'll appreciate hearing about any typos or other errata.

Exercises on Space Curves from " A Survey of Geometry, Vol.2" by Howard Eves

In the attached paper, Eves Space Curves Exercises , we provide solutions to the exercises presented after Chapter 14 Section 4 in "A Survey of Geometry, Volume 2" by Howard Eves. The exercises provide practice in working with the basic aspects of the Differential Geometry of Space Curves and include many interesting results in their own right. If any typos or errors are found, the author would welcome feedback.

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.