A Mathematicians Ways to Travel Eberhard Malkowsky This

A Mathematician’s Ways to Travel Eberhard Malkowsky

This talk deals with some answers you may expect from a mathematician to your simple question: How do I get to B?

The Most Sensible Way of Travelling If you want to go from A to B in a plane then it is natural to follow the straight line segment that joins A and B, because of the following important properties of straight lines. i. Invariance of angles with intersecting lines ii. Identically vanishing curvature iii. Minimum distance between A and B.

The Manhattan Approach If the streets are straight lines intersecting at right angles (Cartesian coordinate system) then you start at A with an uniquely defined angle.

Travelling Long Distances The Manhattan approach can only be applied locally. If you travel long distances you have to take into account that earth is not a plane but a sphere, as a fairly good approximation. As no straight lines exist on a sphere you cannot tavel along them. But we may stick to the principle of constant angle with coordinate lines. A loxodrome is a curve on a surface that intersects each coordinate line in one family at a constant angle.

A Loxodrome on a Plane in Polar Coordinates Logarithmic spirals are the loxodromes of the polar coordinates in a plane. They intersect each of the concentric circles at a constant angle. Since each of the radial rays is orthogonal to the circles the angle between the rays and the logarithmic spiral is also constant.

A Loxodrome on a Sphere A loxodrome on a sphere intersects each meridian at a constant angle. It also intersects each parallel at a constant angle.

From A to B Along a Loxodrome on a Sphere If the longitude and latitude of A are both equal to 0, and the longitude and latitude of B are equal to x and y, respectively, then a choice for the constant angle between the loxodrom and the parallels is

Many Loxodromes Lead From A to B In the Manhattan case in the plane one and only one angle will lead you from A to B. In the case of loxodromes on a sphere any choice of the infinite numbers of angles will take you from A to B.

Consequences of Wrong Choice of the Angle • Of course, you will miss B • Continuation of your journey will take you closer and closer to the North Pole, which you will never reach. • But, don’t worry, no matter how long you travel, your path will not exceed the length

Relation Between Loxodromes on a Sphere and Logarithmic Spirals Logarithmic spirals are the loxodromes of the polar coordinates of a plane. They look very similar to loxodromes on a sphere. Both types of curves are closely related by the socalled stereographic projection.

Stereographic Projection

Stereographic Projection of a Loxodrome on a Sphere The stereographic projection of a loxodrome on a sphere is a logarithmic spiral. This is clear since stereographic projection is angle preserving.

A General Loxodrome on a Sphere Varying the angle for loxodromes leads to some nice pictures.

A General Loxodrome on a Sphere

Two Families of General Loxodromes on a Sphere

Loxodromes on a Pseudo-sphere A sphere has constant Gaussian curvature K=1/R. Any surface of revolution with constant Gaussian cuvature is called a pseudo-sphere. In our case K = -1.

Loxodromes on a Catenoid A catenoid is the surface of revolution of a catenary, which is a curve a suspended chain takes.

Loxodromes on a Surface of Revolution

Lines of Constant Slope Travelling along loxodromes depends on the choice of the coordinate lines. In our next attempt we travel at a constant angle with a fixed direction in space. These lines are called lines of constant slope. An appropiate choice for the fixed direction is the earth’s axis.

Lines of Constant Slope on a Sphere Lines of constant slope will not take you to some regions of the earth. The greater the angle to the earth’s axis the smaller is the symmetric region on both sides of the equator for your journey.

A Mathematical Sewing Machine The orthogonal projection of lines of constant slope on a sphere on to a plane orthogonal to the fixed direction is an epicycloid.

Epicycloid - Construction An epicycloid is generated by a point on a circle that rolls along the outside of a circle line.

Another Mathematical Sewing Machine The orthogonal projection of a line of constant slope on a paraboloid of revolution on to a plane orthogonal to the axis of revolution is the involute of a circle.

Lines of Constant Slope on a Surface of Revolution

Lines of Constant Slope on a Torus

Lines of Constant Slope on a Pseudo-sphere

Lines of Constant Slope on a Catenoid

Lines of Varying Slope on a Sphere Varying the angle for a line of constant slope results in nice pictures.

Lines of Varying Slope on a Sphere

Geodesic Lines Experience shows that constant angles are not a satisfactory choice for your long distance journeys. We have not taken into account yet the minimum distance aspect. The shortest way from A to B is along a geodesic line. They share this property with straight lines in a plane. Straight lines also have vanishing curvature. Geodesic lines have vanishing geodesic curvature, that is vanishing components of curvature in the tangent plane of the surface. For these two reasons, geodesic lines are the natural generalization of straight lines on arbitrary surfaces.

Principle Circles Geodesic lines on a sphere are principle circles, that is intersections of the sphere with planes through the centre of the sphere. Flights ideally follow geodesic lines.

Geodesic Line on a Cone, Osculating Plane and Surface Normal Vector The surface normal vectors of a geodesic line always lie in their osculating planes.

Geodesic Line on a Catenoid Geodesic lines on surfaces of revolution may approach a parallel asymptotically.

Geodesic Lines on a Pseudo-sphere

Geodesic Lines on a Surface of Revolution

Geodesic Polar Coordinates on a Surface of Revolution Geodesic polar coordinates on an arbitrary surface are the natural generalization of polar coordinates in a plane.

Conclusion We presented three alternatives for the choice of your route • Loxodromes • Lines of constant slope • Geodesic lines. Doubtlessly they are mathematically correct and scientifically profound, but we suggest, NEXT TIME MAKE SURE YOU DON’T ASK A MATHEMATICIAN.