![]() Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. This enlargement of the scope of geometry led to a change of meaning of the word "space", which originally referred to the three-dimensional space of the physical world and its model provided by Euclidean geometry presently a geometric space, or simply a space is a mathematical structure on which some geometry is defined. Since then, the scope of geometry has been greatly expanded, and the field has been split in many subfields that depend on the underlying methods- differential geometry, algebraic geometry, computational geometry, algebraic topology, discrete geometry (also known as combinatorial geometry), etc.-or on the properties of Euclidean spaces that are disregarded- projective geometry that consider only alignment of points but not distance and parallelism, affine geometry that omits the concept of angle and distance, finite geometry that omits continuity, and others. The geometry that underlies general relativity is a famous application of non-Euclidean geometry. ![]() Later in the 19th century, it appeared that geometries without the parallel postulate ( non-Euclidean geometries) can be developed without introducing any contradiction. This implies that surfaces can be studied intrinsically, that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. One of the oldest such discoveries is Carl Friedrich Gauss' Theorema Egregium code: lat promoted to code: la ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. ĭuring the 19th century several discoveries enlarged dramatically the scope of geometry. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. A mathematician who works in the field of geometry is called a geometer. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. This process is called geometry instantiation.Geometry (from Ancient Greek γεωμετρία ( geōmetría) 'land measurement' from γῆ ( gê) 'earth, land', and μέτρον ( métron) 'a measure') is, with arithmetic, one of the oldest branches of mathematics. In the cases where a format or transformation can no longer be treated as a geometry instance, it is instantiated so that it no longer refers to the shared geometry definition, but instead becomes an independent copy of the original geometry definition, adjusted with the size and placement for that particular geometry instance. ![]() Some formats or transformations do not or cannot support geometry instances. For example, if a user wishes to change the park bench from red to blue, a modification can be made to the geometry definition only, and it will subsequently be reflected in all geometry instances. With a single geometry definition, it is also possible to easily modify only the geometry definition, and subsequently update all instances of that definition without needing to modify every instance. This is especially true if the geometry instance is very large or detailed. If a single geometry definition is used, with many geometry instances, the overall file size, processing required, and effort to display the geometry is greatly reduced than if many copies of the same geometry are used. For instance, in a visualization of a city, many copies of the same park bench may be used in different locations. Geometry instances are useful for complex geometries where many copies of the same object are required. Geometry definitions may contain any type of geometry, including instances of other geometry definitions. A geometry instance consists of a geometry definition reference, a placement location, and a 3D affine matrix. ![]() A geometry definition is a shared geometry that can have several geometry instances in the same or different features.
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