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This article provides an introduction to analytic geometry and its application in Euclidean and non-Euclidean spaces. The article begins by discussing the basics of analytic geometry, such as how it is used in the study of space, shapes, and equations for transformation. It then discusses applications such as pattern recognition, computer graphics, and computer vision. The final section considers what is new in this area of mathematics as it pertains to computational complexity theory. This introduction provides an overview of analytic geometry and its application to various fields today. It reviews what can be considered to be the foundation of analytic geometry. This includes such topics as axioms such as the axiom of choice (AC). The article discusses numerous applications of analytic geometry including pattern recognition, computer graphics, and computer vision. It also covers the work of Riemannian geometry and mentions its usage in physics to explain relativistic effects. The final section focuses on what is new in this area of mathematics as it pertains to computational complexity theory. This article reviews key concepts and techniques used together with an explanation on how they can be applied in practice. It provides a framework for the study and understanding of modern analytic geometry and its many diverse application areas. The article starts by giving an introduction to the language of analysis. It then defines what is meant in the context of mathematics when referring to "steady change", or in other words when discussing calculus. It also details how the concept of differentiation is applied when analyzing functions. The section ends by introducing the fundamental theorem of calculus, which can be used in analyzing differentiable functions. The article proceeds to detail how analytic functions are originally derived using these fundamental concepts. The final section gives an overview on what is newer in this field especially for computational complexity theory . A review of modern aspects on analytic geometry and its applications to various fields today. It highlights what is new in this area of mathematics especially for computational complexity theory. It also discusses the application of Riemannian geometry to relativistic physics. This review of analytic geometry focuses on the application of the subject to pattern recognition, computer graphics, and computer vision. It discusses that although this area is new in the context of mathematics it is not new in that sense that it is an established area, with an origin dating back over 100 years ago. Bibliography Category:Geometry stubs Category:Analytic geometry stubs Category:Mathematics stubs https://books.google. com/books?id=yACACwAAQBAJ&pg=PA136&lpg=PA136&dq=analytic+geometry+%22modern+applications%22&source=bl&ots=yjGjZZR4Bq&sig=-EbvhcJO0f6gN3eUxX-LQyY2Yg8Iw#v=onepage&q=analytic%20geometry%20%22modern%20applications%22&f=false https://books.google.com/books?id=Y7_CwAAQBAJ&l... cfa1e77820