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Along with arithmetic, geometry is also an ancient field of mathematics. A figure’s property of space about its shape, size, and distance is studied under geometry. Euclid geometry dominated the time till the 19th century but, currently, there are vast developments with computational, differential, and other areas. Geometry aims to develop a model for the physical world. Geometry has its applications in multiple fields of science and architecture, and arts. For instance, the techniques of algebraic geometry are feasible in solving the Wiles’ solution, which remained unsolved for years.
It is the geometry of axiomatic systems in which the derivation of theorems happens from the simple axioms and their small numbers. The name ‘geometry’ brings the images of squares, triangles, points, and angles but, Euclidean geometry is the name for plane geometry. The relationship and properties of multiple things are studied under Euclidean geometry. Non-Euclidean geometry differs from Euclidean geometry in terms of parallel lines. In Euclid Geometry, there is only one that passes through the given points, and it is in the same plane without the possibility of intersection. The ancient mathematician Euclid has his name on 13 books comprising the geometrical and mathematical work, which are elements. The Elements are divided into postulates, axioms, proofs, and propositions about mathematics and geometry.
In differential geometry, integral and differential calculus is used to study the geometric properties. It deals with the properties of surfaces and curves concerning mathematical analysis and differential calculus. Mathematical analysis of surfaces and curves is the reason for the development of differential geometry. Some of the questions do not have answers in calculus. To answer those questions, differential geometry comes into play. The geometric properties are the ones that completely depend on the geometry of the shape and not on how the object is observed in space. Congruence does not affect the nature of these properties. Their derivation is possible from the geometry of an object. Differential Geometry is the most-chosen area of research and development by the Ph.D. or postgraduate students. We have Ph.D. veterans who have been researching geometry and are capable of providing an ideal dissertation.
The geometric properties of the answers to the polynomial equations are studied under algebraic geometry. It includes the solutions, which are over three-dimensions. Analytic geometry gave birth to algebraic geometry. A birational transformation uses the map present in the dual directions to match the points on the two curves with the coordinates and their rational functions. The genus and other intrinsic properties of the curves are preserved under birational transformation. It helps the geometers by removing any singularities for the classification and simplification of the curves. This topic gives the scope of development in the dissertation. Hence, it becomes the ideal geometry type for writing a dissertation.
It is a mathematical field involving the designing and implementation of algorithms for solving geometric problems. Computational geometry is used for referring to pattern recognition and for defining modeling algorithms to manipulate the surfaces and curves. This field of geometry has its name in solving some interesting problems. In the 1980s, this geometry contributed to designing the divide and conquer model. The researchers can develop frameworks for the best geometric algorithms. Though computational geometry cannot handle any continuous process, the scope of development and research is high for this discipline. Hence, it becomes the choice of Ph.D. students for writing their dissertations.
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