In mathematics, familiarity does not breed contempt, it generates underappreciation for the power and majesty of the mathematical operations and representations. This is especially the case for the marriage between complex numbers and geometry. Whether you use the familiar z = a + bi or the often more powerful polar representation, geometric operations such as translations, rotations and even more complex manipulations can be expressed using arithmetic and fairly simple functional transformations. The ease with which you can shift between representations gives you the best of both worlds.
In this book, Roland Deaux gives some powerful reminders of how effective a representation complex numbers are when you want to create, alter, describe and explain geometric figures. The opening chapter is an introduction to what complex numbers are, how to perform arithmetic on them, the fundamental transformations and anharmonic ratios. In chapter two, the representations of fundamental geometric shapes such as the line, conic sections, cycloids, and unicursal curves in polar coordinates are described. The topics of the third and final chapter are circular transformations, similitude groups, homographies, Mobius inversions, antigraphy and symmetries.
It does all mathematicians an enormous amount of good to stop on occasion and ponder the fundamentals of what they are doing. By pausing on occasion and looking back and downward through the foundations of how things are done, you have a better lens on your vision looking forward as you work to make progress in mathematics.
Charles Ashbacher splits his time between consulting with industry in projects involving math and computers, teaching college classes and co-editing The Journal of Recreational Mathematics. In his spare time, he reads about these things and helps his daughter in her lawn care business.
|I. Geometric Representation of Complex Numbers|
|1. Fundamental Operations|
|2. Fundamental Transformations|
|3. Anharmonic Ratio|
|II. Elements of Analytic Geometry in Complex Numbers|
|2. Straight Line|
|3. The Circle|
|4. The Ellipse|
|5. Cycloidal Curves|
|6. Unicursal Curves|
|8. Unicursal Bicircular Quartics and Unicursal Circular Cubics|
|III. Circular Transformations|
|1. General Properties of the Homography|
|2. The Similitude Group|
|3. Non-similitude Homography|
|4. Möbius Involution|
|5. Permutable Homographies|
|7. Product of Symmetries|