# BRANCHES OF GEOMETRY

It is a classic branch of mathematics that involves the study of sizes and shapes in various dimensions. Over the years, the subject has been advancing, and it serves a crucial role in solving problems with figures on two or three dimensions. It also offers solutions to complex issues like the study of differentials and gravitational fields. In this article, you will study the numerous branches of geometry.

## Euclidean Geometry as one of the Branches of Geometry

It is the most commonly known type of geometry. Well known to many as classical geometry, it is a mathematical system that is described in the form of elements. The element is considered to be the epitome of mathematics, and it consists of assumptions and tells how many small sets spontaneously appeal to the axioms. On the other hand, Euclidean geometry deduces the other diverse theorems of the subject.

Due to this, mathematicians refer to it as an axiomatic system of mathematics. The elements include subfields such as solid geometry of three dimensions, plane geometry to mention a few, thus, as a result, led to the rise of concepts in number theory and algebra. Euclidean geometry posits the sum of triangle angles summing up to 180 degrees while quadrilateral angles add up to 360 degrees.

## Non-Euclidean Geometry

It is an extension of Euclid's principle of geometry. This branch of mathematics focuses on Euclid's principles on three-dimension objects. This kind of geometry shows how theorem like the sum of triangle angles differs in three-dimensional spaces. It results from affine planes linked with planar algebras which give rise to kinematic geometries due to the relaxation of the metric geometries.

It is further categorized into two subfields, hyperbolic geometry, and elliptic geometry. With hyperbolic geometry, it has the final postulates altered. The mathematical subfield contains an infinite number of parallel lines that go to line 'l' via point 'p.' The angles of the triangles sum up to less than 180 degrees.

These type of triangles are either singly asymptotic, double asymptotic or triply asymptotic. Elliptic geometry has no infinite number of parallel lines. The sum of the triangle angles totals to more than 180 degrees. These types of triangles can well be classified as obtuse angles that sum up to 270 degrees. This type of geometry also deals with spherical objects.

## Riemannian Geometry as one of the Branches of Geometry

This is a branch of differential geometry that involves the study of smooth manifolds, with an inner product on the tangent space that varies from each point and contains the Riemannian metric. The field is based on the line element, length of curves, volume, surface area, and angles.

The development of the riemannian geometry resulted in the synthesis of different results concerning the geometry of surfaces and the techniques that can be used to study the differentiable manifolds of higher dimensions.

## Differential Geometry as one of the Branches of Geometry

This is a branch of geometry that studies planes, surfaces and lines using the principles of integral and differential calculus. The scientific field uses the techniques of integral calculus, differential calculus, linear and multi-linear algebra to study geometric problems.

The application of this discipline ranges from engineering problems to calculation of gravitational fields. It is also involved in the geometric structures on differential manifolds. One can deal with things that can be defined and it is beneficial to scholars as it facilitates them to articulate the concept of the curvature notion.

## Analytic Geometry as one of the Branches of Geometry

This is a classical subfield of geometry that studies geometric figures and constructions with the help of the coordinate system. It is well known as Cartesian geometry or coordinate geometry. Analytic geometry is the part of the foundation of modern geometry, and it is applied in curves and lines to illustrate the set of coordinates.

Sequentially, it often uses the Cartesian coordinate system and the parametric systems to manipulate straight lines, planes, and squares in both two and three dimensions where possible. The discipline defines and illustrates geometric shapes in the form of numerical figures.

Moreover, it provides vital information from the representations and definition of its shapes. It is widely applied in subjects such as engineering and physics. In classical mathematics, it involves differential, computational, algebraic and discrete geometry.

## Algebraic Geometry

It is a discipline of mathematics that deals with the study of zeros of multivariate polynomials. The modern algebraic geometry is based on the application of abstract algebraic methods that are crucial in solving geometric problems on the sets. It specializes in studying geometries that come from algebra, asserting that algebra is a ring of polynomial also called as an algebraic variety.

This is the manifestation of solutions of systems of the polynomial equations. Some of the algebraic variety includes plane algebraic curves like parabolas, circles, lines, ellipses to mention a few. It holds a central position in modern mathematics and contains multiple conceptual connections with diverse fields like topology, number theory, and complex analysis.

Ultimately, it depends on what you mean by 'branches,' but it's enough to say that geometry is rich and robust with lots of exciting subfields. All in all, you can seek for geometry answers online by contacting us today for prompt help.

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