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.

Solutions to Exercises of section 12.6 in A Survey of Geometry

In a blog entropy of 20 June 2019 we talked about cross ratios in the Gauss plane, the topic of section 12.6 of Howard Eves' A Survey of Geometry. One thing that distinguishes this text is the wonderful set of exercises the author includes. They give the reader the chance to deepen their understanding of the theory presented in the book while extending the results covered by the author. In many cases, readers don't have time to explore these carefully selected exercises in full.

Although the real fun and true value of exercises is in doing them oneself, not everyone has the time to do so. Since I enjoy working out such things and in this case did so, I am including this link to them  Solutions to Exercises of section 12.6 in A Survey of Geometry  in case someone might like to learn more about cross ratios. There a few results in this set which are pretty useful results in themselves. My favorites were the Constructions in 12.6-14 and the Theorems that follow from Schick's Theorem in 12.6-16.

I hope that there are no typos or other such nuisances and apologize in advance if there are. If there are questions or concerns about any solution, I would welcome comments in that regard.

Entropy

If you look up the meaning of "Entropy" with a google search you will likely be no more sure of what it is after reading the multitude of descriptions offered. That is not to say they aren't all true, it is just that it is not completely clear what the English words mean. Here are ten examples:

 1. The entropy of an object is a measure of the amount of energy which is unavailable to do work.

 2. Entropy is a measure of the number of possible arrangements the atoms in a system can have.

 3. Entropy is a measure of uncertainty or randomness.

 4. Physics: a thermodynamic quantity representing the unavailability of a system's thermal energy for conversion into mechanical work, often interpreted as the degree of disorder or randomness in the system.

 5. Lack of order or predictability; gradual decline into disorder. "a marketplace where entropy reigns supreme"

 6. Entropy is a measure of the energy dispersal in the system. We see evidence that the universe tends toward highest entropy many places in our lives. A campfire is an example of entropy. The solid wood burns and becomes ash, smoke and gases, all of which spread energy outwards more easily than the solid fuel.

 7. Entropy is a measure of the random activity in a system. The entropy of a system depends on your observations at one moment. How the system gets to that point doesn't matter at all.

 8. Entropy, the measure of a system's thermal energy per unit temperature that is unavailable for doing useful work. Because work is obtained from ordered molecular motion, the amount of entropy is also a measure of the molecular disorder, or randomness, of a system.

 9. Here are two examples:  Low entropy: A carbon crystal structure at a temperature near absolute zero. ... High entropy: A box filled with two elements in their gaseous state, both of which are noble gases, heated to a very high temperature, with the gas "not very dense".

 10. Entropy is one of the consequences of the second law of thermodynamics. The most popular concept related to entropy is the idea of disorder. Entropy is the measure of disorder: the higher the disorder, the higher the entropy of the system. ... This means that the entropy of the universe is constantly increasing.

 So, if you are like me, after reading such prosaic descriptions of what entropy is, you are looking for a mathematical description that satisfies. In particular, something precise, something that you know how to measure. In fact, part of the difficulty with understanding entropy in lay terms is that it is a mathematical construction. Physics uses mathematical constructions to help explain things. Electric fields are an example. What is an electric field? Describing the theory of electricity (and magnetism) in terms of Electric (Magnetic) fields gives us a model we can "picture" in our minds that helps to describe the properties of electrons moving under the influence of uneven charges. Another example of a mathematical construct is energy. What is energy? It is often described as an "ability to do work". That is, energy (whatever it is) is expended in doing work. It is conserved in its many forms. That is, it cannot be created or destroyed. But what is it? A mathematical construct.

 The point is, sometimes, as in the cases of electric fields, energy, and entropy, there are mathematical functions of well known entities which satisfy certain useful properties. By giving names to these functions, we can then use them in our study of those entities. Such mathematical functions are not like the real things that exist in nature like atoms, but they are functions of such real things which prove useful. When we then go back to describe these concepts in non-mathematical terms, they lose the exactness of their true definition. If you really want to understand a mathematical construction, you have to understand the mathematics that gave rise to it.

 I undertook this quest to try to understand what entropy is. I read a book by Enrico Fermi, listened to some lectures from the Khan Academy, reviewed some Calculus, and in the end started to get an idea of what lies beneath the descriptions of entropy such as those above.

 Here's the bottom line: There are two ways to describe the mathematical entity called entropy. The classic version arose out of studies about heat and work. The statistical one arose out of statistical mechanics.

 Classic: The change in entropy of a system when it transforms from equilibrium state A to equilibrium state B is the sum over all the heat sources T of ratios Q/T where Q is the amount of heat the system absorbs from a source at temperature T.

 Statistical: The entropy of a thermodynamical system in equilibrium state A is proportional to the logarithm of the number N of dynamical states that give rise to that thermodynamical state where the constant of proportionality is the ratio of the gas constant R to Avogadro's number A.

 Okay, so now we know what entropy is!! You can read all about it in a summary paper I wrote after my study. Although there is nothing new in the paper, I have tried to iron out a few of the wrinkles and fill in a few gaps I ran across in my readings to make it a little smoother reading for you. The paper is pretty self-contained if you have studied a little chemistry and calculus.

Oriented Angles and Segment Lengths

In Volume 2, Chapter 12 of Howard Eves' "A Survey of Geometry" he introduces the basic notions of the Geometry of Complex Numbers. Everyone gets a small dose of this in high school but not a lot. Oftentimes the next time one runs into complex numbers is a Complex Analysis course in an undergraduate or graduate course. After reading Eves' treatment in his "Survey" I wish I had had this deeper dig into this topic to better appreciate the analytic course. That is, a deeper understanding of the algebraic and geometric properties of the Inversive plane gives one a better appreciation for the properties of analytic functions.

Like many topics in mathematics, the modern approach in school is often to emphasize breadth at the cost of depth in introducing new concepts. The result is that one can achieve advanced placement standing in Calculus with a very elementary depth of knowledge about plane Geometry. Often concepts are presented without an appreciation for the historical development of that concept. This can make it hard for any but the most advanced to fully grasp the subtleties of the things they learn. Unless students are presented with and then work through a wide assortment of problems they will never run against the "special cases" that make them fully appreciate the results they are learning.

In Eves' Chapter 12, section 6, he looks at the cross ratio of four points in the inversive plane. The cross ratio is shown to have a useful connection to homographies which basically account for all circular transformations in the plane (i.e. transformations that map circles into circles). In fact, the Moebius theorem says that a circular transformation must be a homography or an antigraphy. A homography is a transformation of the inversive plane onto itself defined as z' = (az+b)/(cz+d) where (ad-bc) is not zero. An antigraphy is just the product of a homography and a reflection in the real line. (Replace z with its conjugate.)In his Theorem 12.6.9, Eves establishes the result that the homography determined by the three distinct pairs of finite corresponding points Z1,Z1'; Z2,Z2'; and Z3,Z3' can be expressed in terms of the cross ratio (z'z1',z2'z3') = (zz1,z2z3) where (ab,cd) = (a-c)(b-d)/(a-d)(b-c).

The point of this entry is much simpler than than the (motivating) discussion above. It turns out that in the cross ratio discussion of section 6, Eves draws on the idea of oriented angles and segment measures in a way that is very useful but possibly a little confusing if one is not tuned into the details of the argument. To fine tune the reader's attention I have included a short paper that lays out in detail the definitions of oriented angles and segment measures to make the arguments on cross ratio representations most clear. For anyone interested the paper is Oriented Angles and Segment Lengths.

Why did I win so many tennis matches against an opponent with whom I felt so evenly matched point by point?

I played tennis for nearly two decades with a friend who shared the same tennis spirit with me. He was a reliable (on time and eager to meet) partner and we played on public courts nearly all through the year. Only injuries, rain or temperatures below 25 would keep us off the courts. In the early days I was a beginner and he patiently played with me as I developed my skills. But over the final few years, even though we felt evenly matched point by point, I seemed to win a surprising number of matches (best of three). In fact, even the three set match became somewhat of a rarity. So the question became, how evenly matched were we point by point? From a mathematical point of view we could try to answer the question a posteriori. That is, by noting my winning percentage in terms of sets, deduce what my per point advantage might be.

In the paper Tennis Probabilities we calculate the probabilities of winning a game, set, match given the probability of winning any given point. The paper includes a table of results from which one can work in reverse. Thus, if I was winning 81.5% of the sets we played, I could read from the table that my probability of winning any one point must be .55, etc. This 55% chance of winning a point is not much better than 50-50 so it never felt like a big advantage......but the numbers tell a different story!

Circle Inversions and Applications

There were a lot of geometric discoveries made in the 19th century that not many people know much about. By "not many" I mean not only those who end their mathematical studies in high school or even college, but even many mathematicians who spend their limited time and energy in other areas. Thanks to some though we do have some excellent texts that capture some of these results...one being Dan Pedoe's "Geometry A Comprehensive Course". In the last entry (Dec 7, 2017) we gave a detailed answer to an introductory exercise from Pedoe's book which opened up a whole universe of interrelated circles. Since then, having delved further into this book, this blogger learned about a 19th century geometric tool that gives insight not only regarding that problem we analyzed using primitive means, but offers a way to solve many problems in a way that we might just not be able to get our heads around using only primitive tools. This "new" tool is that of "inversion in a circle". These inversions transform circles and lines in the planes into other circles and lines while keeping the angles of intersection of any two such figures (and thus tangency and orthogonality).

In the paper Circles: Inversions and Applications , we have presented an introduction to this topic, proved the necessary results needed to establish the tools and worked many examples and exercises using the tools. Taken in its entirety it will give the reader a firm grasp of inversion as a tool to use in everyday geometric situations.

This blogger wants to send kudos to Dan Pedoe for his excellent book of which this is but a small slice.

It is a shame that the standard mathematical curriculum jumps away from geometry so soon when there is so much more to explore. I have run into many people who say they loved Geometry but began to lose interest in subsequent courses (how might that be possible?). Although the Greeks did amazing work in Geometry it should be no surprise that the subsequent centuries would add to that heritage. Well, here at oriolescience we like to offer up a bit of what you may have missed.

One solution we explain in detail in the paper is the Steiner Problem. We give the theoretical solution and work out a detailed example. This in itself is worth the reader's time and attention.

Exercise 0.9 from Dan Pedoe's Geometry A Comprehensive Course

The following problem was posed as Dan Pedoe's Exercise 0.9:

If (z-a)/(z-b) = r(cos(θ) + i sin(θ), show that the curves r = constant, and θ = constant, for fixed complex numbers a and b and varying Z, are circles of orthogonal coaxal systems, the one which has the points a and b as limiting points, and the other of circles which pass through these two points.

Since this problem occurred in an introductory chapter of the text, my expectation was that it was going to be a straightforward application of his quick review on complex numbers. It proved to be something more interesting.

In fact, establishing the part when r is constant is pretty straightforward once you recognize that the set of points whose distance to two fixed points is a constant ratio r is in fact a circle.

But the case for fixed θ proved the more interesting/complicated.

I recommend that anyone who has read this much should try to work it out before reading the solution linked below. I would love to see another approach to the solution if you find one.

For anyone who is interested, here is a link to a paper which provides what I hope will be a pretty self-contained solution intended for a reader who has studied senior high school level math. It includes a little background material for anyone who might be a little rusty.

Note: One point not explicitly mentioned in the paper is that when θ is held constant and r varies, θ presents itself as the angle between the perpendicular bisector of the segment ab and the line joining the center of the so-called θ-circle and the point a (or b). Thus the changing θ-circles can be visualized for θ in [0,π/2].