Chapters
The Tractrix and Similar Curves
Author: Walter Gander, Stanislav Bartoň, Jiří Hřebíček
We generalize the classical tractrix problem (Gottfried Wilhelm Leibnitz, 17th century: "Given a watch attached to a chain, what is the orbit in the plane described by the watch as the endpoint of the chain is pulled along a straight line?") to compute the orbit of a toy pulled by a child, and then we compute the orbit of a dog which attacks a jogger. We use MATLAB for numerical solving two similar system of differential equations and show also how the motions can be visualized by MATLAB.
We offer the first chapter in the full form to show the nature of our book. It is available in the following formats:
Trajectory of a Spinning Tennis Ball
Author: František Klvaňa
This chapter shows how to describe and visualize the variant of a motion of tennis ball in the air. Both Maple and MATLAB solutions are discussed. We start with the simplest model in the vacuum, then we modify it to a model of a tennis ball moving in the air and finally to the model of a spinning tennis ball in the air. We assume the conditions near earth surface.
The Illumination Problem
Author: Stanislav Bartoň, Dominik Gruntz
Let us consider two lights on a horizontal road, given the heights of the lamps and their illumination powers. Suppose that we also know the distance between them and that there are no more lamps around. The goal is to find a point X between the two lamps which is minimally illuminated. We are also looking for the optimal heights of the lamps to have the best illumination on the whole road. The problem is solved using Maple.
Orbits in the Planar Three-Body Problem
Author: Dominik Gruntz, Jörg Waldvogel
The planar three-body problem is the problem of describing the motion of three point masses in the plane under their mutual Newtonian gravitation. It is a popular application of numerical integration of system of ordinary differential equations since most solutions are too complex to be described in terms of known functions. Maple and MATLAB can be used efficiently to construct and display numerical solutions of the planar three-body problem. First we will do straightforward numerical integration in MATLAB. Although for most initial conditions this approach will quickly produce an initial segment of the solution, it will usually fail at a sufficiently close encounter of two bodies, owing to the singularity at the corresponding collision. Therefore we use a set of variables that amounts to automatically regularizing each of the three types of close encounter whenever they occur. Owing to the complexity of the transformed equations of motion, the Hamiltonian formalism will be used for deriving these equations. Then Maple's capability of differentiating algorithms (automatic differentiation) will be used to generate the regularized equations of motion.
The Internal Field in Semiconductors
Author: František Klvaňa, Jan Pešl
Suppose we have a semiconductor of a given length, which is doped with a concentration of electrically active impurities (acceptor and donor). Let it be connected to a given external potential. The goal is to find the external potential inside the semiconductor as a function of the position between the ends. As a background we use Boltzmann's statistics for electrons and holes. The mathematical description leads to nonlinear Poisson equation. We show the solution using simple numerical algorithm implemented in MATLAB and even more simple solution in Maple using the new Maple capabilities for solving boundary value problems.
Some Least Squares Problems
Author: Walter Gander, Urs von Matt
This chapter considers some least squares problems, that arise in quality control in manufacturing using coordinate measurement techniques. In mass production machines are producing parts and it is important to know if the output satisfies the quality requirements. Therefore some sample parts are usually taken out of the production line, measured carefully and compared with the nominal features. If they do not fit the specified tolerances, the machine may have to be serviced and adjusted to produce better parts. We show how to compute several least squares fits -- especially for fitting lines, rectangles and squares in the plane and fitting hyperplanes in space.
The Generalized Billiard Problem
Author: Stanislav Bartoň
Given a billiard table and two balls on it, from which direction should the first ball be struck, so that it rebounds off the rim of the table, and then impacts the second ball? This is a very simple mathematical problem for rectangle tables and has be solved also for circular tables and some other specific curves. In this chapter we solve that problem in general, for any parametrically described billiard table, using the generalized reflection method and the shortest trajectory method. The computation is done in Maple and if the curve is simple enough that the analytic solution exists, then the analytic solution is given; the numerical approximation is done only if it is really needed.
Mirror Curves
Author: Stanislav Bartoň
The generalized billiard problem brings another interesting mathematical problem: Given a starting point and applying the generalized reflection method, we will get a curve of mirrored points (function of the tangent point, where the ball hits the rim). This chapter studies these curves using Maple and then solves the inverse problem - to compute the starting points curve from the mirror curve and given tangent point. The solution can be obtained numerically using Maple, but for some cases also the analytical solution using geometrical tricks is shown.
