Geometry is the study of shapes and sizes in various dimensions. Most of the foundation of geometry was written in Euclid's "Elements," one of the oldest mathematical texts. Geometry has progressed since the ancient times, however. Modern geometry problems involve not only figures on two or three dimensions, but also more complex problems like the study of differentials and gravitational fields.
Euclidean, or classical, geometry is the most commonly known geometry, and is the geometry taught most often in schools, especially at the lower levels. Euclid described this form of geometry in detail in "Elements," which is considered one of the cornerstones of mathematics. The impact of "Elements" was so big that no other kind of geometry was used for almost 2,000 years.
Non-Euclidean geometry is essentially an extension of Euclid's principles of geometry to three dimensional objects. Non-Euclidean geometry, also called hyperbolic or elliptic geometry, includes spherical geometry, elliptic geometry and more. This branch of geometry shows how familiar theorems, such as the sum of the angles of a triangle, are very different in a three-dimensional space.
Analytic geometry is the study of geometric figures and constructions using a coordinate system. Lines and curves are represented as set of coordinates, related by a rule of correspondence which usually is a function or a relation. The most used coordinate systems are the Cartesian, polar and parametric systems.
Differential geometry studies planes, lines and surfaces in a three-dimensional space using the principles of integral and differential calculus. This branch of geometry focuses on a variety of problems, such as contact surfaces, geodesics (the shortest path between two points on the surface of a sphere), complex manifolds and many more. The application of this branch of geometry ranges from engineering problems to the calculation of gravitational fields.