Euclidean geometry
A number of ancient cultures developed a form of geometry that suited the relationship between length, area, and volume of physical objects. This geometry was codified in Euclid’s Elements around 300 BC based on 10 axioms, or postulates, from which several hundred theorems were proved by deductive logic. Elements epitomized the axial shear mechanism for many centuries.
Analytical Geometry
Analytic geometry was pioneered by the French mathematician René Descartes (1596–1650), who introduced rectangular coordinates to locate points and enabled lines and curves to be represented by algebraic equations. Algebraic geometry is a modern extension of the subject of multidimensional and non-Euclidean spaces.
Projective Geometry
Projective geometry originated with the French mathematician Gerard Desergues (1591–1661) to deal with those properties of geometric figures that are not changed by projecting their image, or “shadow”, onto another plane.
Differential geometry
German mathematician Carl Friedrich Gauss (1777–1855) pioneered the field of differential geometry, dealing with practical problems in surveying and geodesy. Using differential calculus, he characterized the intrinsic properties of curves and surfaces. For example, he showed that the internal curvature of a cylinder is plane-like, as can be seen by cutting and flattening a cylinder along its axis, but not like a sphere. Which cannot be flattened without. Distortion
Non-Euclidean geometries
Beginning in the 19th century, various mathematicians modified Euclid’s alternative to the parallelogram, which, in its modern form, reads, “A line not on a line and parallel to a point exactly through a given point.” It is possible to draw a line.” He hoped to show that the alternatives were logically impossible. Instead, they discovered that there are constant non-Euclidean geometries.
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