Shape-optimizing problems abound both in the natural and the man-made worlds. Certain angles in a honeycomb are constructed by bees to minimize surface area for a given volume. Engineers designing the shape of a car, or the wing of an airplane, use sophisticated techniques to find a shape with good wind resistence, or airlift, properties.
Our problem will be to find the shape of the trajectory that minimizes the time traveled by an object moving from point A to point B under the influence of gravity. (With A being higher placed than B.) With certain assumptions the problem has already been solved in the time of Newton. With other assumptions, for example on the resistence between the object and the surface, numerical approximations will have to be used.
We will try out various ways of constructing such numerical approximations, because we don't know yet which will work the best. We will learn a lot about mathematical objects such as different classes of interpolating polynomials and cycloids. The representation, manipulation and optimization of such objects on a computer will be investigated. And we will encounter shape-optimizing solutions in unexpected places around us.
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