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Classical Geometry: Euclidean, Transformational, Inversive, and Projective

Reed Mitchell

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Classical geometry has its origins in ancient Greece and Egypt, where it was developed as a means of solving practical problems in land measurement, architecture, and astronomy. Over the centuries, geometry has evolved into a vast and complex subject with applications in a wide range of fields, including mathematics, physics, engineering, and computer science.

This article provides a brief overview of the four main branches of classical geometry:

Classical Geometry: Euclidean, Transformational, Inversive, and Projective

by J. E. Lewis

4.6 out of 5

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* Euclidean geometry * Transformational geometry * Inversive geometry * Projective geometry

Each branch of geometry has its own unique set of axioms and theorems, and they can be used to solve different types of problems.

Euclidean Geometry

Euclidean geometry is the most familiar type of geometry. It is based on the axioms of Euclid, which were first set forth in his Elements around 300 BC. Euclidean geometry is concerned with the properties of figures in two- and three-dimensional space. The basic figures of Euclidean geometry are points, lines, planes, and circles.

Euclidean geometry has many applications in everyday life. For example, it is used in architecture to design buildings, in engineering to design bridges and other structures, and in surveying to measure land.

Transformational Geometry

Transformational geometry is concerned with the study of transformations, which are mappings from one set of points to another set of points. The most common types of transformations are translations, rotations, reflections, and dilations.

Transformational geometry has many applications in computer graphics, animation, and robotics. For example, it is used to create realistic animations of moving objects, and to design robots that can move and interact with their environment.

Inversive Geometry

Inversive geometry is concerned with the study of inversions, which are mappings from one set of points to another set of points that preserve angles. The most common type of inversion is the inversion in a circle.

Inversive geometry has many applications in optics, crystallography, and computer vision. For example, it is used to design lenses and mirrors, to study the structure of crystals, and to develop algorithms for image processing.

Projective Geometry

Projective geometry is concerned with the study of projective transformations, which are mappings from one set of points to another set of points that preserve straight lines. The most common type of projective transformation is the perspective projection.

Projective geometry has many applications in computer graphics, architecture, and engineering. For example, it is used to create realistic drawings and renderings, to design buildings and bridges, and to study the flow of fluids.

Classical geometry is a vast and complex subject with a wide range of applications. The four main branches of classical geometry are Euclidean geometry, transformational geometry, inversive geometry, and projective geometry. Each branch of geometry has its own unique set of axioms and theorems, and they can be used to solve different types of problems.

This article has provided a brief overview of the four main branches of classical geometry. For more information, please consult the references below.

References

* Coxeter, H. S. M. (1969). to geometry. New York: John Wiley & Sons. * Euclid. (1956). The elements. Translated by T. L. Heath. New York: Dover Publications. * Hilbert, D. (1902). Grundlagen der geometrie. Leipzig: B. G. Teubner. * Kline, M. (1972). Mathematical thought from ancient to modern times. New York: Oxford University Press.

Classical Geometry: Euclidean, Transformational, Inversive, and Projective

by J. E. Lewis

4.6 out of 5