Geodesics: Minimizing Distance on a Surface

Transcription

1 Geodesics: Minimizing Distance on a Surface Seth Donahue 5/13/13

2 Abstract Geodesics use the principles of differential geometry. The system of differential equations and the analysis of these will be done using a numerical solver because the closed form solutions are quite difficult to procure. The analysis of geodesics will segue into practical applications, such as how planes travel internationally, how game theory works, and how light travels through spacetime.

3 Contents 1 Definitions Curves Surfaces Curvature Ordinary Differential Equations (ODEs) Geodesics The Frenet Frame of Curves 3 3 Introduction to Surfaces 6 4 Curvature 8 5 The Mathematics of Geodesics 9 6 Practical Applications Spheres Ellipsoid Tori Light Light Black Holes List of Figures 1 T,N,B Frame Unit Sphere Equator Longitude Lines Ellipse Torus Torus Torus Torus light cone Model Black hole Star system Star system

4 1 Definitions 1.1 Curves The curve must be defined at every point in the parameterization in R 3. The only curves that will be examined in this project are differentiable in 3-spaces. These curves are a continuous mapping α : I R 3 where I is in the interval such that α(t) = (α 1 (t), α 2 (t), α 3 (t)). These are the coordinates in three space that are defined by the function. They can be parameterized into different forms to make them easier to work with. An example of this is polar coordinates. To find the Frenet frame, the unit tangent, unit normal, and binormal vectors, (T,N,B). From these vectors, all that can be known about the curve at any point will be shown. The Frenet will show the curve is changing through R 3. There are limitless parameterizations for curves; the most helpful for this project are the parameterizations that change rectangular coordinates to polar coordinates. Polar coordinates will be used for many of the calculations of curvature and those specifically of geodesics and for the definition of a surface. 1.2 Surfaces Surfaces are everywhere in the world from simple countertops that can be described mathematically with a plane to more complicated surfaces such as a torus (a donut) or the curvature of space time. For the bulk of this paper, however, the surfaces that will be dealt with will be in R 3, and simple examples of these are spheres, cylinders, and tori. The calculus of the surface will provide the information regarding the surface; therefore this requires the surface to be differentiable. With this idea in mind the surface will be defined as M R 3 and f(u, v) = (f 1 (u, v), f 2 (u, v), f 3 (u, v)). This is the mapping of surface M in R 3. The surface depends on two variables (u, v), so partial differentiation is required, and the surface can start to be described geometrically with the gradient, tangent plane, and normal curvature, which will lead into overall curvature of the surface. The most important part of the analysis of the surface will be the tangent plane and the normal vector to the surface as these will give direct insight to how a geodesic line will act on the surface. 1.3 Curvature Curvature is the way the geometry of a surface can be measured. The equations that define the curvature of the surface can all be found using calculus. They will lead directly to the computations of geodesics, for curvature and geodesics are directly related. 1.4 Ordinary Differential Equations (ODEs) Ordinary Differential Equations are equations that include one or more functions with a single variable that have a derivative in the equation. The differential equations that will be used to model a geodesic are two second-order, 2

5 non-linear homogeneous differential equations or four first-order homogeneous differential equations. ODEs can be be directly solved for and may have closed form solutions; however, most of the geodesics that will be dealt with in this project will be solved for using numerical analysis, because there is not typically a closed-form solution for the system of non-linear ODEs. 1.5 Geodesics Geodesics are a critical piece of differential geometry. A geodesic is a curve on surface M that is the shortest distance between two points as a consequence of the following: α tan = 0; as long as the tangential acceleration of the curve on the surface is 0, then the curve is considered to be a geodesic. The geodesic has a constant speed because the acceleration is 0, so by definition the velocity has to be constant. 2 The Frenet Frame of Curves The Frenet frame is the a way to describe the properties of a curve in space. The frame consists of the unit tangent vector, unit normal vector, and the binormal vector. Each of these vectors will describe the velocity, acceleration, and the rate of change that occurs between the velocity (tangent) and acceleration (normal). These can be defined by vector valued functions in R 3. All of the curves that will be worked with will be parameterized with the variable t, and they will be defined as such: α(t) = x(t) y(t) z(t) We can see that the line is any scalar multiple of this vector d, in R 3 that goes through the given point p. Another example of a curve in R 3 is a helix. The parameterization of the helix is α(t) = cos(t) sin(t) t These vectors form the basis for the helix in R 3. Let the curve be defined as α(t). This is any curve that can be parameterized for all reals in 3-space. The unit tangent vector is the first of the T,N,B frame, and it is defined as: T = α (t) α This is the unit tangent vector. The unit tangent vector is the velocity at any point in time on the given curve. The length of the vector is also normalized to 1 for the simplicity of the graph below. On the curve, the tangent vector only 3

6 touches one point on the curve. This is the definition of the tangent vector in 3- space. This is also known as the displacement, how much the curve is changing at any point. This vector shows the direction the curve is going at that specific point. Curvature is defined as: κ(t) = T (t) Therefore, κ 0, and as α turns more sharply, the greater κ will be. The unit normal vector is by definition perpendicular to the unit tangent vector, and it is defined as: T N = (t) T (t) This is the rate of change of the tangent vector. It is a unit vector like the tangent for easier scaling and for graphical purposes. Another way to define the normal vector along a surface is. N = 1 κ T This is also known as the principal normal vector along the curve α, This is the way that will be used in the Frenet frame for this paper. Then, solving for T κ N = T This gives the first of the three differential equations in the system. The binormal is perpendicular to both the tangent and the normal vectors, and is by definition a unit vector, as it is composed of two unit vectors. It is found by crossing T and N as such: B = T N This can be considered a measure of how much the curve is changing in its direction. The binormal is a unit vector on the curve α by definition as B = T N sin( π 2 ) = (1)(1)(1) = 1. The way that the (T,N,B) moves along the curve α in R 3 will tell exactly how the curve is moving through space at any point in time. The way that the T,N,B frame changes will be described by taking the derivatives of each. From above, the derivative in the tangent direction is known. κn = T By the definition of N, we know this. T, N, and B are all mutually perpendicular, orthonormal, in R 3. Any vector in R 3 is a linear combination of these and the vectors form an orthonormal basis for the curve. Specifically, α = at +bn +cb if a, b, and c, α will be known. Solving for a, b and c: T α = at T + bt N + ct B a = a 1 + b 0 + c 0 4

7 Using the same process, N B = b and B B = c and plugging them into the above equation delivers: α = (T B )T + (N B )N + (B B )B T α = 0, then by the product rule 0 = (T B) = T Ḃ + T B. Then N B = 0 T α = T B = κn B = 0 It is known that B B = 1 so it follows that 0 = (B B) = B B + B B. B B = 0, so we have a single coefficient in the expression of α. This leads to the definition of torsion, which is τ = N B Therefore, in the equation above with 0 in the T and B α = τn This is the differential equation for the binormal. Using the same process for N as was done for B It then follows that N = (T N )T + (N N )N + (B N )B N = κt + τb The full Frenet frame looks like this: T 0 κ 0 T N = κ 0 τ N B 0 τ 0 B To solve this system of first-order differential equations, the easiest way is to use eigenvectors and eigenvalues. In order to get these values and vectors, take (A λi) to find the characteristic polynomial. Solving for λ to get the eigenvalues and the eigenvectors will give the solution, which takes this form: y = e λt v. The is the basis for any curve in R 3, and this curve can be superimposed on a surface, and these will lead into geodesics, but let us first define what a surface is. 5

8 Figure 1: T,N,B Frame 3 Introduction to Surfaces Surfaces are functions of two variables that is denoted f(x, y) (x, y) D. A function of three variables is denoted with this notation: f(x, y, z) (x, y, x) D. An example of a three-variable function. x 2 + y 2 + z 2 = 1 This surface is the unit sphere. Figure 2: Unit Sphere 6

9 These surfaces are often reparameterized so that computationally the calculations will be easier. The example of the sphere above is parameterized as such: cos(u)sin(v) X = sin(u)sin(v) cos(u) Parameterizations such as this will be the norm moving forward, and the typical parameterization is in polar coordinates. Partial derivatives are the key to multivariable calculus, and they are defined in the u or v direction. This is a function defined in the u-direction, by the directional derivative as x u = ( x u, y u, z u ) This is the rate of change in the u-direction. The rate of change in the v-direction as x v = ( x v, y v, z v ) These are the tangent vectors in the u-direction and the v-direction, respectively, using partial derivatives with respect to u. It is critical to think of v as a constant and vice versa when taking these partial derivatives. You are only finding the tangent vector in the direction you are taking the derivative with respect to. Partial derivatives are very important because they will give approximations such as the tangent plane and will lead to other pieces about the geometry of the surface. The tangent plane is the approximation on a surface as to the rate of change of the surface in any direction. In other words, the closer that you zoom into a point on a surface, no matter how curved the surface is, it will resemble more and more a flat surface or plane. For example take the earth; for us as human beings it seems flat. This is because the earth is so massive in comparison to us, that no matter where we stand on the earth, it will seem fairly flat. The closer that you come to the surface of the earth, the more flat the earth will seem. However, when you go up in a spacecraft and leave the earth s atmosphere, you will see that the earth is in fact spherical in shape. Then, as you come back down, the more flat it will seem again as you come closer to a single point on the surface of the earth. The tangent plane is an approximation of how the surface is changing around that point where the plane touches the surface. If the surface is flat at that point, the tangent plane will be accurate in its approximation of a surface. A good example if this a table, the tangent plane will not deviate from the surface. In the example above, with the earth, the tangent plane is not a very good approximation beyond just above the surface or the farther you move away from the point where the tangent plane touches the earth. The true nature of the surface is revealed as you move farther away from it. This is why the tangent plane is only an approximation. The equation of a plane is z z 0 = f x (x 0, y 0 )(x x0) + (f y (x 0, y 0 )(y y 0 )). One will take the partial derivatives and then plug in the respective values to 7

10 get the slope in the proper direction and then plug in those slopes at the point in order to get tangent plane. The tangent plane is important because it tells us about the geometry of the surface and how the surface is changing. This is critical to the study of geodesics. The definition of a geodesic is when α tan = 0; in other words, when the tangential acceleration is 0 and therefore the components of the normal vector stays the same. These are the coefficients for the equation of the tangent plane. This analysis of the normal vector is the point of this paper. If one were to take the tangent plane on the geodesic, the slope of the tangent plane would not change at all as it followed the geodesic around the surface. 4 Curvature Using the traditional notation for finding curvature, where U = the unit normal vector. The vectors x u, x v and U form a basis in R 3. This is because x u x v = U. E = x u x u F = x u x v G = x v x v l = U x uu m = U x uv n = U x vv The second directional derivatives in the directions x uu, x uv, x vv and then the directional derivatives of the normal are as follows: x uu = E u 2E x u E v 2G x v + l U x uv = E v 2E x u G u 2G x v + m U x uu = G u 2E x u + G v 2G x v + n U Recall that U = xu xv x u x v and the directional derivative of this will give the following equations: U u = l E x u m G x v U u = m E x u n G x v These equations will all be used in the geodesic equations in section 5. 8

11 The Gaussian curvature is defined as The mean curvature is defined as H = K = (ln) m2 (EG) F 2 (Gl) + (En) (2F m) 2((EG) F 2 ) The Gaussian curvature is the determinant of the shape operator, and the mean curvature is the trace of the shape operator. The shape operator is defined by calculus to detect changes in the unit normal vector and is a matrix. This is a direct link between the geometry of a surface and linear algebra. The above formulas come from the use of calculus and some very interesting derivations. These equations also start to hint at the ideas of minimal surfaces. The surface is said to be flat if the K(p) = 0 at every p M. This is the definition of a plane; however, it can be found on other surfaces as well. Another example of a surface with 0 gaussian curvature is also a right circular cylinder. When the H(p) = 0, the surface is said to be minimal for p M. 5 The Mathematics of Geodesics The great majority of geodesics do not have a closed form solution, and those that do go through a great deal of difficult calculus and differential equation solving to get the result. What is done to get these curves is numerical analysis of the geodesic equations, a system of differential equations. This system of four non-linear, homogeneous differential equations will be solved to get the equation of the geodesic. The way that geodesics occur, as stated above, is the disappearing 2nd derivative of the curve on the surface M in R 3. α is some curve on the surface M, and the velocity of α on the surface is α. The acceleration is α. There are three vectors that have a relation to this surface: the tangent vector T, where T = α ; and there is U which is the unit normal vector. This is where the curve α has unit speed. The third vector can be found by crossing T and U, T U. These three vectors are mutually perpendicular, just like the vectors in the Frenet frame, and they will form a basis for R 3, meaning that any vector on the surface is a linear combination of T, U and T U. Since α is constrained to the surface, it is then implied that α has components in the T, T U and U directions. This leads to the first differential equation: α = A T + B( T U) + C U Showing the tangent vector going to 0 using the properties of dot and cross products will be important. When examining the equation above, the coefficients are defined as follows: A = α T 9

12 B = α ( T U) C = (α U) U To simplify the tangential and normal components, the tangential component is by definition a unit vector, thus α α = 1. By the chain rule of multi-variable calculus, 2α α = 0 as the derivative of one is zero. Anything that is dotted with 0 is by definition 0, and in this case α = T, thus T α = 0. This shows that for a geodesic the tangential acceleration must be 0. For the curve α on surface M to be a geodesic the α tan = 0. This is important when the geodesic equations are introduced, as only the normal curvature will be explored with the geodesic equations because the tangential vectors are going to be set to equal 0. The surface will be defined by x(u, v) X(u, v) = y(u, v) z(u, v) Thus, a geodesic is defined as α(t) on the X as so x(u(t), v(t)) X(u(t), v(t) = y(u(t), v(t)) z(u(t), v(t)) With the above parameterizations, the curve on the surface will be parameterized as such: α = X( u(t), v(t)) Taking the derivative of α, this equation results α = X du u dt + X dv v dt Taking the second derivative using the chain rule of calculus α = (u 2 ) X uu + 2u v Xuv + X u u + X vu u v + X vv (v 2 ) + X v v This is the equation of a geodesic. Using the formulas from the curvature section of this paper and then simplifying the equation gives: α = X u [u + E u 2E (u ) 2 + E v E u v G u 2E (v ) 2 ]+ X v [v E v 2G (u ) 2 + G u G u v G v 2G (v ) 2 ]+ U[l(u ) 2 + 2mu v + n(v ) 2 ] The first two terms of the equation give the tangential pieces of α. Therefore, for α to be a geodesic, the following equations must be satisfied: u + E u 2E u 2 + E v E u v G u 2E v 2 = 0 10

13 v E v 2G u 2 + G u G u v G v 2G v 2 = 0 Solving for u and v u = E u 2E u 2 E v E u v + G u 2E v 2 v = E v 2G u 2 G u G u v + G v 2G v 2 Taking the equations above and substituting to get the second order system into a first order system with four equations. u 1 = u 2 u 2 = E u 2E u 2 1 E v E u 1v 1 + G u 2E v 2 1 v 1 = v 2 v 2 = E v 2G u 2 1 G u G u 1v 1 + G v 2G v 2 1 Given initial conditions, such as a point on the surface M and a direction, a geodesic can be found on that surface. The theory of differential equations will ensure the existence and uniqueness of a solution, a geodesic, on the surface of M. As stated above, the easiest way to solve this system of differential equations is through the use of a numerical solver, such as Matlab. The examples below are of surfaces with geodesics, and some practical applications will be examined. 6 Practical Applications 6.1 Spheres Geodesics are used in the airline industry for long flights. The shortest distance between two points on the surface of the earth is not a straight line but a geodesic curve. (These can be seen below). There is not a line that is straight per se, but any of these lines are the shortest distance between two points on the surface of the earth. The only line of latitude that is a geodesic is the equator. These curves are the fastest, most efficient routes of travel across the world. This is the easiest-to-understand example of a geodesic. The geodesics on a sphere are known as great circles. On the earth, these are the lines of longitude, the north-south lines that run around the earth as well as the equator as stated above (see Figures 3 and 4). 11

14 Figure 3: Equator Figure 4: Longitude Lines 6.2 Ellipsoid The earth as we know is not a perfect sphere, so the routes that airlines take can be modeled more accurately. These are, of course not completely correct because the plane will not be flying on the curve forever. The FAA must account for the wind and other physical effects as well. The flight path would look like a small section of the curve. In general, however, the planes will follow these curves to reach their destinations in a timely manner. The difference between these and the sphere is that there are not the great circles that go around the ellipse. In fact, some of the geodesic curves look downright strange. 12

15 Figure 5: Ellipse 1 This ellipse has even more deviation from the sphere and this shows how interesting these curves can get. 6.3 Tori We will now look at the torus. This donut shape is very interesting in that there are some simple geodesics that go around the equators on the inside and the outside of the surface as shown below. There are also geodesics that will go around the lesser circles, as shown in Figure 9 below. These are simple circles, and as shown above, they are known to be geodesics as seen on the sphere above. The torus is a circle with a specific radius rotated around the origin; thus, it makes sense that these lesser circles are geodesics. The more complex curve that is shown below is still a geodesic, for it is the shortest path given an initial point and direction. The torus can be used in gaming theory to create maps and apply them to a gaming universe. Space Invaders is a good example of this. Taking your ship around to the right will make it appear on the left side of the screen. This is as if the ship is following the outer or inner equator of the torus. Then, if you were for some reason to leave the top of the screen you would show up again on the bottom of the screen because it is a continuous flat surface. The same would occur if you were to leave the screen at an angle you would come back to the bottom of the screen and continue moving in the same direction. 13

16 Another example of a toroid universe is Conways Game of Life, which can be found online. Figure 6: Torus 1 14

17 Figure 7: Torus 2 Figure 8: Torus 3 15

18 Figure 9: Torus 4 7 Light In spacetime, light travels on geodesics, the shortest paths through spacetime between two points. Many different phenomena occur due to gravity and its interaction with light, gravitational lensing being one of the most interesting. The gravitational lens has been used to peer back into the earlier days of the universe and to see how stars formed billions of years ago. Geodesics can model how light will travel around and in some cases into the gravity well of a star or other dense object that will have sufficient gravity to change the path of the light. Spacetime is curved;it is not straight as we will see in one of the very simple models. This model will show what would actually occur if there were stars of equal mass evenly distributed throughout spacetime on a flat plane. This is model is not accurate, but it will provide a basis so that one may begin to understand what is really happening to light as it travels through the universe. 7.1 Light Here a cone is used to show how light will travel through space as it is emanated from a single point in space. At a specific time anything inside of this light cone will be travelling at less then the speed of light. For example you as a human being are traveling less than than the speed of light; exist, therefore, you inside of this light cone. This cone holds all possible future events of your life. The speed of light is at this point the limit of what we can experience here in this reality. This cone starts with a single event and will then spread out from there. The existence inside of this light cone is in the shape of a hyperboloid with its asymptotes being the cone in three dimensions. There are intersections of your light cones with others and are all the possible interactions with those 16

19 people. The light cone will then travel through spacetime as different actions, are taken which will open up a different set of possibilities for that person while consequently closing other possibilities. All of the space surrounding the cone will not have any effect on the event that occurs at origin. 7.2 Black Holes Figure 10: light cone Black holes pose an interesting problem with light in regard to how it travels through spacetime. Black holes are basically gravitational sinks. They have an incredible amount of mass at a single point, otherwise known as a singularity. Because they are so dense, they warp space time to a single point. Here is a model of how light will travel into the black hole. 17

20 Figure 11: Model Black hole In this image, light can come from either of those two directions, and they will be brought to the bottom of the pseudosphere, or to the black hole. The way that light will travel around stars can also be modelled. This model is extremely simple because it assumes that the mass of each individual star is the same, that the stars are all the same distance from each other, and that they lie in the same plane. These are, of course, ridiculous assumptions to make, as we live in an extremely diverse galaxy with millions of stars, and they all lie with no set order. Light will bend and twist around these stars as it travels through spacetime due to the effect of gravity on the light. The dark blue in the graph is the point where the stars are located within space and where the gravity wells exist in space, while the red shows where the gravity of the stars is the weakest. 18

21 Figure 12: Star system 1 Figure 13: Star system 2 References [1] Differential Geometry : and It s Applications. 2rd ed: (John, Oprea) (MA: Pearson) Print. [2] Differential Equations :. Differential Equations with Boundary Value Problems. 2nd ed. (Polking, John C., Albert Boggess, and David Arnold) Upper Saddle River, NJ: Pearson/ Prentice Hall, Print. [3] Personal Interveiw : (Don Hickethier) [4] Montana Fish, Wildlife and parks : Elk in region 1: (Mt.gov.) Montana Fish Wildlife and Parks, Web. 01 May