Intrinsic Geometry

Today I wanted to discuss the geometry of curves and surfaces.

Curves, Curvature and Normals

First let us consider a curve r(s) which is parameterised by s, the arc length.

Now, t(s) = Screen Shot 2017-02-14 at 8.05.47 AM.png is a unit tangent vector and so t2 = 1, thus t.t = 1. If we differentiate this, we get that t.t‘ = 0, which specifies a direction normal to the curve, provided t‘ is not equal to zero. This is because if the dot product of two vectors is zero, then those two vectors are perpendicular to each other.

Let us define t’ = Kwhere the unit vector n(s) is called the principal normal and K(s) is called the curvature. Note that we can always make K positive by choosing an appropriate direction for n.

Another interesting quantity is the radius of curvature, a, which is given by

a = 1/curvature

Now that we have n and t we can define a new vector x n, which is orthonormal to both t and n. This is called the binormal. Using this, we can then examine the torsion of the curve, which is given by

T(s) = –b’.n

Image result for binormal tangent

Intrinsic Geometry

As the plane is rotated about n we can find a range

Screen Shot 2017-02-14 at 8.16.01 AM.pngwhereScreen Shot 2017-02-14 at 8.16.08 AM.png and Screen Shot 2017-02-14 at 8.16.14 AM.png are the principal curvatures. Then

Screen Shot 2017-02-14 at 8.17.34 AM.png

is called the Gaussian curvature.

Gauss’ Theorema Egregium (which literally translates to ‘Remarkable Theorem’!) says that K is intrinsic to the surface. This means that it can be expressed in terms of lengths, angles, etc. which are measured entirely on the surface!

For example, consider a geodesic triangle on a surface S.

Screen Shot 2017-02-14 at 8.20.37 AM.png

Let θ1, θ2, θ3 be the interior angles. Then the Gauss-Bonnet theorem tells us that

Screen Shot 2017-02-14 at 8.22.10 AM.png

which generalises the angle sum of a triangle to curved space.

Let us check this when S is a sphere of radius a, for which the geodesics are great circles. We can see that Screen Shot 2017-02-14 at 8.16.08 AM.png=Screen Shot 2017-02-14 at 8.16.14 AM.png= 1/a, and so K = 1/a2, a constant. As shown below, we have a family of geodesic triangles D with θ1 = α, θ2 = θ3 = π/2.

screen-shot-2017-02-14-at-8-26-15-am

Since K is constant over S,

Screen Shot 2017-02-14 at 8.27.09 AM.png

Then θ1 + θ2 + θ3 = π + α, agreeing with the prediction of the theorem.

M x

Brachistochrone Curve Problem

A Brachistochrone Curve is the curve that would carry a bead from rest along the curve, without friction, under constant gravity, to an end point in the shortest amount of time.

Source: storyofmathematics

The Brachistochrone problem was one of the earliest problems posed in calculus of variations. The solution, which is a segment of a cycloid, was found individually by Leibniz, L’Hospital, Newton and both the Bernoulli’s.

The following solutions are taken from wikipedia.

Johann Bernoulli’s Solution

Johann Bernoulli used Fermat’s principle that “the actual path between two points taken by a beam of light is the one which is traversed in the least time” in order to derive the brachistochrone curve. He did this by considering the path that a beam of light would take in a medium where the speed of light increases due to a constant vertical acceleration equal to g.

Due to the conservation of energy, v={\sqrt  {2gy}}, where y is the vertical distance. Furthermore, the law of refraction gives us a constant (vm) of the motion for a beam of light in a medium of variable density:

{\frac  {\sin {\theta }}{v}}={\frac  {1}{v}}{\frac  {dx}{ds}}={\frac  {1}{v_{m}}}

Rearranging this gives us

v_{m}^{2}dx^{2}=v^{2}ds^{2}=v^{2}(dx^{2}+dy^{2})

which can be manipulated to give

dx=\frac{v\, dy}{\sqrt{v_m^2-v^2}}

If we assume that the beam, with coordinates (x,y) departs from the origin and reaches a maximum speed after falling a vertical distance D:

v_{m}={\sqrt  {2gD}}

we can rearrange the equation to give us the following:

dx={\sqrt  {{\frac  {y}{D-y}}}}dy

which is the differential equation of an inverted cycloid generated by a circle of diameter D, as required.

Jakob Bernoulli’s Solution

Jakob Bernoulli’s approach was to use second order differentials to find the condition for the least time.  The differential triangle formed by the displacement along the path, the horizontal displacement and the vertical displacement is a right-handed triangle, therefore:

ds^2=dx^2+dy^2

Differentiating this gives

2ds\ d^{2}s=2dx\ d^{2}x

{\frac  {dx}{ds}}d^{2}x=d^{2}s=v\ d^{2}t

Consider the follow diagram:

Path function 2.PNG

The horizontal separation between paths along the central line is d2x.

The diagram gives us two separate equations:

d^{2}t_{1}={\frac  {1}{v_{1}}}{\frac  {dx_{1}}{ds_{1}}}d^{2}x

d^{2}t_{2}={\frac  {1}{v_{2}}}{\frac  {dx_{2}}{ds_{2}}}d^{2}x

For the path of the least time, these times are equal hence their difference is equal to zero.

d^{2}t_{2}-d^{2}t_{1}=0={\bigg (}{\frac  {1}{v_{2}}}{\frac  {dx_{2}}{ds_{2}}}-{\frac  {1}{v_{1}}}{\frac  {dx_{1}}{ds_{1}}}{\bigg )}d^{2}x

Consequently, the condition for the least time is

{\frac  {1}{v_{2}}}{\frac  {dx_{2}}{ds_{2}}}={\frac  {1}{v_{1}}}{\frac  {dx_{1}}{ds_{1}}}

as required.

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Hope you enjoy, M x

How Scandalous!

Although mathematics is often considered a ‘bland’ subject, (I, however, completely disagree with this!), throughout the years there have been many scandals. Here are my top 5 mathematics scandals!

Alan Turing Trial

Turing was oDr-Alan-Turing-2956483n of the mathematical geniuses of the 20th century, working in the areas of cryptology and computer science. In World War II, he worked at Bletchley Park and played a major role in breaking the German codes.

However, he was a homosexual, and at that time this was illegal in Great Britain. After being charged in 1952, he pleaded guilty and as a consequence was stripped of his security clearance and put under hormone treatments. He became deeply unhappy and, sadly, Turing committed suicide by poison apple just two years later at the age of 41.

The British government only officially pardoned him for “the appalling way he was treated” in 2013.

Andre Bloch Murders

I was pers6a00d834523c1e69e20147e2a982e3970bonally unfamiliar with this story, but it’s pretty shocking so I’d thought I’d share it with you!

Bloch was a French mathematician, who was active for 31 years and is best known for his contributions to complex analysis. However, he spent all these 31 years in a mental institute. Why? In 1917, when he was on leave from World War I, he killed his brother, his aunt and his uncle. He told one of his mathematician colleagues that he committed these murders as an act to rid his family line of people afflicted with mental illness. Crazy right!

Newton vs Leibniz

This story is a classic.

Most of you know ‘calculus’ – we all studied it at some point in secondary school. It is the study of the infinite and infinitesimal and is one of the most amazing tools offered to a student in mathematics. Well, Isaac Newton and Gottfired Leibniz strongly disagreed on who deserved credit for its discovery – they both wanted full credit! The war between Newton and Leibniz was ugly and they battled it out via the letters and journals of the day, each accusing the other of plagiarism. The funny thing is historical documents now seem to reveal that both men made their discoveries independently and nearly simultaneously – they both deserved credit!

newton-leibniz_eng

Burning of the Library of Alexandria

The_Burning_of_the_Library_at_Alexandria_in_391_ADThis library, which was built around the 3rd century BC, was the house of many academic wonders, including a wealth of discoveries in mathematics. In this library were the works of Euclid, Archimedes, Eratosthenes, Hipparchus and many other notable mathematicians. Although, there aren’t many details of the fire, it’s clear that the destruction of the library was a major setback to academics of the time.

Hippasus’ Murder

Hippasus.jpgHippasus was part of the Pythagorean society (the people who discovered the infamous theory about right angled triangles: a2+b2=c2). The Pythagorean society is known for their secrecy and, in the 5th century, when Hippasus managed to prove that the square root of 2 was irrational, it is said that he was going to reveal this to the public at large, and so the society drowned him at sea. However, there are some questions about the details of the legend of Hippasus – don’t take this to be fact!

And now for a false scandal: The Nobel Prizefoto_de_alfred_nobel

Why is there no Nobel Prize for mathematics? The famous rumour is that this is because Alfred Nobel’s wife was having an affair with a mathematician. This mathematician would have been one of the potential first winners of the Nobel Prize for mathematics. Mr. Nobel, therefore, didn’t set up a prize for mathematics so that he couldn’t win! However, Alfred Nobel was never actually married… This is discussed in detail in the book Mathematical Scandals.