Analyzing Surfaces using Principal Curvatures and Directions

In the paper An Introduction to Surface Curvature Analysis the analysis of the previous blog entry is generalized to methods that apply to any surface. The paper recounts the fundamental results regarding the computation of the principal curvatures and directions at a point on a surface. It includes several different approaches to these results including derivations using basic definitions, a directional derivative approach, a linear algebra approach, and an eigenvalue/eigenvector approach. The latter is seen to be quite efficient and easy to remember. In bringing these various approaches together in one paper, I hope to save the reader the work involved in sorting through each one independently. The references include the sources I used to gather these approaches.

In the end, just as in the example of the previous blog entry, we set up a coordinate sytem with origin at the point of interest, this time with the principal directions as two axes and the third axis as the normal direction making the trio a right hand system. The key result that the theory establishes is that relative to this coordinate system, the surface can be approximated in a neighborhood of the (new) origin by a mapping z=(x^2 k_1 + y^2 k_2)/2 where k_1 and k_2 are the maximum and minimum curvatures. Thus the surface can be characterized by the signs of the principal curvatures at the point of interest. The theory establishes methods for computing these principal curvatures and their associated principal directions.

The paper includes detailed examples employing the various approaches recounted in the paper. These examples constitute nearly half the paper.

As always, any shortcomings in notation, logic, or lack of clarity are welcomed feedback.

Changing the Frame of Reference to study a Surface Eves Exercise 14.3-5

In the paper Changing the Reference Frame, we solve exercise 14.3-5 from Howard Eves' "A Survey of Geometry, Vol. 2" in a straightforward argument involving the gradient vector as the normal to a surface. A change of coordinates (frame of reference) allows the Taylor expansion at a point to be approximated by the second degree terms of that expansion. The main purpose of the paper is to study an example which showcases the algebraic tools needed to affect a change of coordinate axes in Euclidean space. The key insight is that a translation or rotation of the axes is equivalent to the inverse translation or rotation of the surface point set in the original reference frame. Although the example is a quadratic itself, f(x,y,z) = z + x^2 + y^2 = 0, for illustrative purposes, the result applies to any surface which can be written in the form z - g(x,y)=0 in the new reference frame.

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