GEOMETRY IN MATHEMATICS; ITS TYPES & BASICS

What is Geometry?

Geometry is derived from two different words "Geo" which means "Earth" and "metry" which means "measurement". Geometry thus means the measurement of the earth. 

Similarly, the study of things' sizes, shapes, positions, angles, and dimensions is what is called geometry. Geometry is a subfield of Mathematics.

We learn about various angles, transformations, and similarities in the figures in geometry. Points, lines, angles, and planes are major components of the fundamentals of geometry. 

Concentrating on calculation gives understudies numerous fundamental abilities and assists them with building their sensible reasoning abilities, insightful thinking, logical thinking, and critical thinking abilities. thereby contributing to their complete growth.

Which are the Common Types of Geometry?

Geometry can take many different forms, but the most common ones are listed below.

• Algebraic Geometry
• Complex Geometry
• Computational Geometry
• Convex Geometry
• Discrete Geometry
• Euclidean Geometry
• Non-Euclidean Geometry
• Topology

Algebraic Geometry

The Cartesian geometry of coordinates gave rise to the field of algebraic geometry. Projective geometry, birational geometry, algebraic varieties, and commutative algebra, among other topics, were created and studied alongside it during its periodic growth periods. 

It had undergone significant foundational development from the late 1950s to the middle of the 1970s, largely as a result of the work of Jean-Pierre Serre and Alexander Grothendieck. Schemes were developed as a result, and topological approaches, such as various cohomology theories, received more attention. 

Complex Geometry

The study of the nature of geometric structures modeled on or arising from the complex plane is known as complex geometry. String theory and mirror symmetry are two areas in which complex geometry has been found useful. It is a branch of differential geometry that sits at the intersection of algebraic geometry, complex variable analysis, and differential geometry. 

Holomorphic vector bundles and coherent sheaves over complex spaces, complex algebraic varieties, and complex analytic varieties are the primary subjects of complex geometry research.

Computational Geometry

Algorithms and their implementations for manipulating geometrical objects are the focus of computational geometry. The traveling salesman problem, minimum spanning trees, hidden-line removal, and linear programming are all historically significant issues. Even though it is a relatively new subfield of geometry, it has numerous uses in computer vision, image processing, computer-aided design, medical imaging, and other fields.

Convex Geometry

Convex geometry employs real analysis and discrete mathematics to investigate convex shapes in Euclidean space and their more abstract counterparts. It has important applications in number theory and close connections to convex analysis, optimization, and functional analysis.

Discrete Geometry

Discrete geometry is a subject that has close associations with curved calculation. Questions regarding the relative positions of straightforward geometric elements like points, lines, and circles are the primary focus of this research. The investigation of sphere packings, triangulations, and the Kneser-Poulsen conjecture are just a few examples. Combinatorics shares numerous principles and methods.

Euclidean Geometry

Classical geometry refers to euclidean geometry. It is utilized in numerous scientific and technical fields, including mechanics, astronomy, crystallography, engineering, architecture, geodesy, aerodynamics, and navigation, as it models the physical world's space. The majority of nations include the study of Euclidean concepts like points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, and analytic geometry in the mandatory educational curriculum. 

Topology

The study of the properties of continuous mappings falls under the umbrella of topology, which can be thought of as an extension of Euclidean geometry. In practice, topology frequently entails dealing with space's large-scale connectedness and compactness properties. 

In a technical sense, topology, which saw significant growth during the 20th century, is a subfield of transformation geometry in which transformations are homeomorphisms. The expression "topology is rubber-sheet geometry" has frequently been used to describe this. Geometric topology, differential topology, algebraic topology, and general topology are all subfields of topology. 

What are the Main Concepts/Terms of Geometry?

Following are the main terms or concepts which help us to understand geometry easily. 

• Angles
• Axioms
• Curves
• Dimension
• Length, Area, & Volume
• Lines
• Manifolds
• Metrics and measures
• Planes
• Points
• Surfaces
• Symmetry

Angles

In modern terms, an angle is a figure formed by two rays sharing a common endpoint, called the vertex of the angle. Euclid defined a plane angle as the inclination to each other, in a plane, of two lines that meet and do not lie straight with respect to each other. Angles that are acute (a), obtuse (b), and straight (c). 

The oblique angles include the acute and obtuse angles. Angles are used to study polygons and triangles in Euclidean geometry, as well as forming an object of study in and of themselves. 

The study of angles in a unit circle or a triangle is the foundation of trigonometry. The derivative can be used to calculate the angles between plane, space, or surface curves in differential geometry and calculus.

Axioms

One of the most influential books ever written, Euclid's Elements, took an abstract approach to geometry. A number of axioms, or postulates, that expressed the fundamental or obvious properties of points, lines, and planes were introduced by Euclid. Using mathematical reasoning, he continued to rigorously deduce additional properties. 

The rigorous nature of Euclid's approach to geometry has earned it the moniker "axiomatic" or "synthetic" geometry.

Curve

A curve or bend is a 1-layered object that might be straight (like a line) or not; Plane curves are the names given to curves in two dimensions, while space curves are the names given to curves in three dimensions. A function from a real number interval to another space defines a curve in topology. 

The same definition is used in differential geometry, but the defining function must be differentiable. Algebraic geometry studies algebraic curves, which are defined as one-dimensional algebraic varieties.

Dimension

Mathematicians and physicists have utilized higher dimensions for nearly two centuries in places where traditional geometry permitted dimensions 1 (a line), 2 (a plane), and 3 (our actual world, which is conceived of as three-dimensional space). 

The configuration space of a physical system, which has a dimension equal to the system's degrees of freedom, is one example of a mathematical application for higher dimensions. Five coordinates, for instance, can be used to describe a screw's configuration. 

The concept of dimension has been extended from natural numbers to infinite dimensions (such as Hilbert spaces) and positive real numbers (in fractal geometry) in topology as a whole. The dimension of an algebraic variety has been given a number of seemingly different definitions in algebraic geometry, all of which are equivalent in the majority of cases.

Length, Area, & Volume

The size or extent of an object in one dimension, two dimensions, or three dimensions is referred to as its length, area, and volume, respectively. In Euclidean math and scientific math, the length of a line section can frequently be determined by the Pythagorean hypothesis. 

Area and volume can be described and calculated in terms of lengths in a plane or three-dimensional space, or they can be defined as fundamental quantities distinct from length. 

Numerous explicit area and volume formulas for various geometric objects have been discovered by mathematicians. Area and volume can be defined using integrals, such as the Riemann or Lebesgue integral, in calculus.

Line

A line, according to Euclid, "lies equally with respect to the points on itself" and has "breadthless length." The idea of a line is closely related to how geometry is described in contemporary mathematics because of the large number of geometries. 

In a more abstract setting, such as incidence geometry, a line may be an independent object distinct from the set of points that lie on it. For instance, in analytic geometry, a line in the plane is frequently defined as the set of points whose coordinates satisfy a given linear equation.

On the other hand, in incidence geometry, a line may be an independent object. A geodesic is an extension of the concept of a line to curved spaces in differential geometry.

Manifold

The terms "curve" and "surface" can be applied to a manifold. A manifold is a topological space with a neighborhood for each point that is homomorphic to Euclidean space in topology. 

A differential manifold is a space where each neighborhood is diffeomorphic to Euclidean space in differential geometry. Physics makes extensive use of manifolds, such as in string theory and general relativity.

Metrics and measures

Metrics are a natural extension of the concept of length or distance. The distance between two points in the Euclidean plane is measured by the Euclidean metric, while the distance between two points in the hyperbolic plane is measured by the hyperbolic metric. 

The Lorentz metric of special relativity and the semi-Riemannian metrics of general relativity are two additional significant examples of metrics. 

Measure theory, which studies methods of assigning a size or measure to sets and where the measures follow rules similar to those of classical area and volume, extends the concepts of length, area, and volume in a different direction.

Plane

A plane is a two-dimensional, flat, and infinitely long surface in Euclidean geometry; The definitions for the various other kinds of geometries are just broadening that. Numerous fields of geometry make use of planes. 

Planes, for instance, can be studied as a topological surface without taking into account angles or distances; It can be studied as an affine space, but distances and collinearity cannot be studied there; Using methods of complex analysis, it can be studied as the complex plane; and so forth.

Points

Most people think that the building blocks for geometry are points. They can be defined in synthetic geometry or by the properties that they must have, as in Euclid's definition of "that which has no part." They are typically referred to as components of a set known as space, which is itself axiomatically defined in modern mathematics. 

Every geometric shape is defined by these contemporary definitions as a collection of points; This is not the case in synthetic geometry, where a line is seen as a fundamental object independent of the set of points it passes through. However, there are some contemporary geometries in which points are either not primitive objects at all, or even not present at all.

Surface

A two-dimensional object like a paraboloid or sphere is called a surface. In differential geometry and topology, surfaces are described by diffeomorphism- or homeomorphism-assembled two-dimensional "patches" (or neighborhoods). Polynomial equations describe surfaces in algebraic geometry.

Symmetry

Congruences and rigid motions in classical Euclidean geometry represent symmetry, whereas collineations, geometric transformations that transform straight lines into straight lines, play an analogous role in projective geometry. In projective geometry, among other fields, the principle of duality is a different kind of symmetry. 

This meta-peculiarity can generally be portrayed as follows: The result of any theorem is the same for any exchange point with a plane, join with meet, lies with contains, and so on. A vector space and its dual space share a type of duality that is very similar to it and related to it.

20 Common Geometric Formulas 

1. Area = ½ ab
2. Area of a Circle = A = π×r2 
3. Area of a Rectangle = Length × Breadth.
4. Area of a Square = Side2
5. Area of a Triangle = ½ × base × height.
6. Area of a Trapezoid = ½ × (base1+base2) ( b a s e 1 + b a s e 2 ) × height.
7. Circumference of a Circle = 2πr
8. Curved Surface Area of a Cone =  πrl
9. Curved Surface Area of a Cylinder  = 2πrh
10. Diagonal, d = √(l² + w²)
11. Perimeter = a + b + √(a² + b²)
12. Perimeter of a Square = 4(Side)
13. Perimeter of a Rectangle = 2(Length + Breadth)
14. Pythagoras Theorem: a² + b² = c² (a = altitude of a triangle, b = base of a triangle, c = hypotenuse of a triangle)
15. Surface Area of a Sphere = S = 4πr2
16. Total surface area of a cone = πr(r+l) = πr[r+√(h2+r2)]
17. Total Surface Area of a Cylinder = 2πr(r + h)
18. Volume of a Cone = V = ⅓×πr2h
19. Volume of a Cylinder = V = πr2h
20. Volume of a Sphere = V = 4/3×πr3

Also Read 

300 Basic Maths Solved MCQs