Linkages are structures which consist of rigid bars (line segments) that are hinged together at the endpoints (vertices) of the bars so that they can rotate with respect to one another. Your finger or a robot arm can be thought of as a linkage. The serious mathematical study of linkages began in the 19th century. The inspiration for interest in linkages was problems emerging from the Industrial Revolution. Problems included being able to convert straight line motion to circular motion. However, towards the end of the 19th century the subject went to sleep. This was probably because without new ideas and approaches to the problems that were of engineering and mathematical interest, research had been carried as far as the engineering and mathematics of the times made possible.
In 1978 the field of computational geometry was born. Computational geometry concerns itself with finding algorithms to carry out geometric processes and studying the computational complexity of geometric phenomena. Computational geometry encouraged the drawing together of a variety of strands of mathematical ideas from disparate areas including: geometry, computer science, graph theory, combinatorics, and rigidity theory. Some of these subjects had old roots and others were much newer. In particular, a variety of problems involving linkages were reexamined anew (hence this column's title). Our goal here is to look at some problems involving linkages and related ideas from a recent perspective. As usual, we will see the value of careful definitions to capture subtly different phenomena and the value of the mathematical point of view.
Progress in mathematical understanding lays the foundation for progress in technological innovation. In the 19th century robots were a dream. Now mathematicians assist engineers in designing algorithms to get a robot arm from one position to another in as fast a way as possible while avoiding objects in the work space of the arm. Mathematicians are also working with biologists to understand rigidity issues associated with biological molecules.
York College (CUNY)
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