Finding Shortest Paths on Surfaces

[Winzer]

Ok, so I was eating an apple and I asked myself: If I had two points on the surface and I wanted to find the shortest path between those two points, how would I go about that mathimatically. How about if I had an ellipsoid?
Thanks

 

[matt grime]

The standard method is to use calculus of variations, or something like it.

 

[HallsofIvy]

On a sphere, which an apple approximates, The shortest distance between two points is measured along a great circle. More generally matt grime is correct: a path is a function and "calculus of variations" deals with functions having minimum and maximum properties.

 

[Kummer]

Ok, so I was eating an apple and I asked myself: If I had two points on the surface and I wanted to find the shortest path between those two points, how would I go about that mathimatically. How about if I had an ellipsoid?
Thanks

The two posters before me already said this sort of problem is called "Calculus of Variations". I just wanted to add that given some "object" (I am not going to be formal to be easier to follow). Given an object (a sphere, a torus, a cube ... ) and two points on that object, the shortest path between them is called a geodesic. It is of interest in higher Geometry problems.

 

[Winzer]

The two posters before me already said this sort of problem is called "Calculus of Variations". I just wanted to add that given some "object" (I am not going to be formal to be easier to follow). Given an object (a sphere, a torus, a cube ... ) and two points on that object, the shortest path between them is called a geodesic. It is of interest in higher Geometry problems.

Geodesic, yes I have heard that word before; it involves the shortest path on a curved surface right?

Level wise, where is Calc of variations relative to Calc II and III?

 

[Kummer]

Level wise, where is Calc of variations relative to Calc II and III?

Caculus of Variations is more advanced than Calculus III. It usually involves solving a differential equation (sometimes even a system of differential equations).

Most students learn Calculus of Variations in a "Mathematical Physics"/"Mathematical Methods for Engineers" class. But they only learn a very small part of this theory. Though there are books entirely written on this subject only a few pages are ever necessary for Engineering students. Those are referred to as the http://en.wikipedia.org/wiki/Euler-Lagrange_equation" .

 

[Winzer]

Well I guess this question will have to wait till I cover Mathematical Physics my junoir year.

 

[Kummer]

Well I guess this question will have to wait till I cover Mathematical Physics my junoir year.

Here is the classic Calculus of Variations problem.
-----

In Calculus you study functions (real), basically a function is something that takes a number and turns it into another number. For instance, $$f(x)=x^2$$ means for each number you give in $$x$$ the output number is $$x^2$$.

In Integral Trasforms you study transforms, for example the http://en.wikipedia.org/wiki/Laplace_Transform" . Basically what the Laplace/Fourier transforms are is something that transforms a function into another function. So for example, the Laplace transform on $$f(x)=x$$ is $$F(s)=s^{-2}$$. Which means the function gets turned into another funtion.

In Calculus of Variations you study a similar situation called a functional. And basically a functional is something that turns a function into a number. I will give you an example below and hopefully you will understand.

Here is a classical application of Calulcus of Varations I promised in the beginning of this thread.

Given two distinct points $$(x_1,y_1)$$ and $$(x_2,y_2)$$. What is the shortest path between them? You certainly know the answer is a straight line. But look at how this problem is approach. First we set us an equation involving a functional. Here is what we do. Let $$y=f(x)$$ be any curve between those two points. And the length of that curve is $$\int_{x_1}^{x_2} \sqrt{1+[f'(x)]^2} \ dx$$, this is simply the arc length formula from Calculus II. So we write,
$$I[f(x)] = \int_{x_1}^{x_2}\sqrt{1+[f'(x)]^2} \ dx$$.
This means if $$f(x)$$ is any curve passing through those two points then $$I[f(x)]$$ is the value of this functional. Note, the input is a function but the output is a number.

So the way the problem is stated is:
Given $$I[f(x)]=\int_{x_1}^{x_2}\sqrt{1+[f'(x)]^2} \ dx \mbox{ with }f(x_1)=y_1 \mbox{ and }f(x_2)=y_2$$ minimize the functional.

The conditions $$f(x_1)=y_1 \ f(x_2)=y_2$$ are referred to as the boundary condition, because that assures us the curve passes through those two points.

But the above problem is a typical application of Calculus of Variations which is easily solved with Euler-Lagrange Equation if you read the Wikipedia article.

 

[Winzer]

MMmmm that is a pretty cool way to go about it. Yah functional is new to my vocab since I am in only Calc II . But I can't wait to learn more about those, they sound really interesting.
Thanks

 

[Gib Z]

Halls: When you say "great circle", I'm thinking of a sphere being made up of a revolved circle. We find the circle that goes through both the points we want to find the distance between, they will be points on the circle and we go from there. Am I thinking correctly?

And the geodesic word reminded me of an extremely impractical method, since I hear that word so much of Special Relativity. One could beam a photon from the first point to the next, and it will always follow the geodesic path. We measure how long it took to reach there, and knowing c = 299, 792, 458 m/s, we can work out the distance :) However the will be inaccuracies, as measurement can not be perfect even theoretically (HUP).

 

[HallsofIvy]

As long as the center of a circle, lying on the sphere, is the same as the center of the sphere, then it is a great circle. On the surface of the earth, lines of longitude are great circles, lines of latitude are not.

As far as you "photon measure" is concerned, how are you constraining the photon to follow the curvature of the sphere?

 

[Gib Z]

As long as the center of a circle, lying on the sphere, is the same as the center of the sphere, then it is a great circle.

Yes that was what I was trying to explain :)

As far as you "photon measure" is concerned, how are you constraining the photon to follow the curvature of the sphere?

Quiet you!

 

[matt grime]

Not to mention the idea that we actually do know the speed of the photon is very dodgy too - what you wrote would be the speed in a vacuum, perchance...

 

[Gib Z]

O well that, it shouldn't be too hard to find the refractive index of an apple.

 

[Kummer]

MMmmm that is a pretty cool way to go about it. Yah functional is new to my vocab since I am in only Calc II . But I can't wait to learn more about those, they sound really interesting.
Thanks

It is interesting! In fact most of the advanced mathematical techniques you learn in "Mathematical Physics/Mathematical Methods" are all interesting. But the sad thing is only a little bit is covered on each type of Method. It is understandable why that is done, to save time, but is it much nicer to be exposed to all the techinques from a mathematical point of view.

 

[Winzer]

SO in Math Physics, do they pretty much jump around material wise?

 

[Kummer]

SO in Math Physics, do they pretty much jump around material wise?

Yes. For example, Complex Analysis which is a beautiful subject and an entire semester is devoted to it for math majors is broken down into a few theorems and laws.

 

[Winzer]

Krummer you have been more then helpful but I have one more question:
What course do you get to study about the Riemman Zeta Function?
Just curoius, it looks most interesting, even though I can barly understand it.
That is probably math major stuff right?
I am an engineering major but I have been taking such a keen interest in pure mathematics.

 

[ice109]

Krummer you have been more then helpful but I have one more question:
What course do you get to study about the Riemman Zeta Function?
Just curoius, it looks most interesting, even though I can barly understand it.
That is probably math major stuff right?
I am an engineering major but I have been taking such a keen interest in pure mathematics.

probably number theory since it deals with primes

 

[Kummer]

What course do you get to study about the Riemman Zeta Function?

It is a course in Number Theory. A good understanding Riemann's Zeta Function requires good knowledge of Complex Analysis. It is a type of number theory called Analytic Number Theory, it gets its named because we apply results from analysis (mostly complex) to the study of numbers. Such a course is an advanced Graduate course. And yes, it is a math course, it is never studied in Engineering.

Just curoius, it looks most interesting, even though I can barly understand it.
That is probably math major stuff right?
I am an engineering major but I have been taking such a keen interest in pure mathematics.

http://www.math.wisc.edu/~propp/491/gfologyLinked.pdf" of this free book does explain it a little.

 

[Winzer]

Wow Krummer that is pretty awesome, I started reading it, I will have to finish it.
So after I complete all of calc, diff eq., and linear algebra and I want to continue in mathematics just for myself, I could move on to analysis?

Is there like some kind of chart/list that would be help to look at to see what I could study, something that gives me an order of what to study.

 

[Kummer]

Is there like some kind of chart/list that would be help to look at to see what I could study, something that gives me an order of what to study.

You can read https://www.amazon.com/dp/0452285259/?tag=pfamazon01-20

The true book on the Zeta function is https://www.amazon.com/dp/0486417409/?tag=pfamazon01-20 but I never read it, so I cannot say what I think of it. But I can tell you immediately it is not elementary at all. So if you want a basic chart learn the following in order: Elementary Number Theory, Advanced Calculus, Complex Analysis.