What is Euclidean Geometry?
An isometry, of the Euclidean space, is said to be a mapping that preserves the Euclidean distance and is denoted by the letter d between points.
This topic focuses on the rigid motions (isometries) of Euclidean space, euclidean architecture, as well as Euclidean geometry, which illustrates how congruency theorems of triangles can be extended to other geometric objects.
Euclidean Geometry is known to emphasize that an arbitrary isometry of Euclidean space can be uniquely expressed as an orthogonal transformation followed by a translation.
On this page, we are going to prove an analogue for curves of the various criteria for congruence of triangles in plane geometry; more specifically, it showed that a necessary and sufficient condition for two curves in R3 to be congruent is that they have the same curvature and torsion (and speed), and the unit-speed curve for a position in R3 is determined by its curvature as well as by its torsion. Furthermore, the sufficiency proof of Euclidean geometry shows how to find the required isometry explicitly.
What is Euclidean Space?
Euclidean space definition and Euclidean space linear algebra:
Euclidean space can be defined as a finite-dimensional vector space over the reals R, with an inner product.
As it is taught in schools all over the world, two-dimensional geometry, as well as three-dimensional geometry, was first described by Euclid more than two thousand years ago. And it is still useful for dealing with physical space even though modern physics has shown that geometry in the universe is far more complicated.
This is commonly known as Euclidean space is based on a few fundamental concepts, the notions point, straight line, plane, and how they are related.
Two points determine a straight line or we can say two points determine a line segment, and a line and a point determine a line through that point as well as parallel to the given line. A line, as well as a point (not on that line), determines a plane, and a plane and a point (not on that plane) "generate" 3-space.
Euclidean Space can be defined as the set of all n-tuples of real numbers, formally
E\[_{n}\] = {[x\[_{1}\], x\[_{2}\], x\[_{3}\], x\[_{4}\]......x\[_{n}\]]|x\[_{i}\] ∈ R, i = 1, 2, 3….., n} with a number is known as distance assigned to every pair of its elements.
Formally, if X = [x\[_{1}\], x\[_{2}\], x\[_{3}\], x\[_{4}\]......x\[_{n}\]], Y = [y\[_{1}\], y\[_{2}\], y\[_{3}\],......x\[_{n}\]] we define
ρ(X, Y) = \[\sqrt{(x_{1} - y_{1})^{2} + (x_{2} - y_{2})^{2} + ….. + (x_{n} - y_{n})^{2}}\]
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Properties of Vector Operations in Euclidean Space
Properties of Vector Operations in Euclidean Space, the various Euclidean spaces share properties that will be of significance in our study of linear algebra. Many of these properties are listed in the following theorem: If u, v, and w are vectors in n dimensional Euclidean space, and k and m are scalars (real numbers), then:
(a) u + v equals v + u
(b) u + (v + w) equals (u + v) + w
(c) u + 0 equals u
(d) u + (−u) equals 0
(e) k(u + v) equals ku + kv
(f) (k + m)u equals ku + mu
(g) (km) u equals k(mu)
(h) 1u equals u
Vectors in Euclidean Space
In vector also known as multivariable calculus, we will basically deal with functions of two variables or three variables (usually x,y or x,y,z, respectively). The graph of a function of two variables says z equals f(x,y), lies in Euclidean space, which in the Cartesian coordinate system consists of all ordered triples of real numbers say (a,b,c). Since Euclidean space is known to be three-dimensional, we can denote it by R3. The graph of f consists of the points (x,y,z) equals (x,y,f(x,y))
Vector - Since we have already discussed what vectors are, we can also perform some of the usual algebraic operations on them (For example - addition, subtraction, multiplication, etc). Before doing that, let’s discuss what is the notion of a scalar. Let’s know why was the term scalar invented? It was invented in the first space to convey the sense of something that could be represented by a point on a scale/ ruler. The word vector comes from Latin, where the word means "carrier''. A few examples of scalar quantities are mass, electric charge, as well as speed (not velocity).
Cross Product - We will define a product of two vectors that does result in another vector. This product is known as the cross product, and it is only defined for vectors in R3.
FAQs on Euclidean Space
1. What is a Euclidean space in simple terms?
In simple terms, a Euclidean space is the familiar space we experience every day, like a flat sheet of paper (2D) or the world around us (3D). It's a mathematical space where we can measure distances and angles using the standard rules of geometry, such as the Pythagorean theorem. All points, lines, and planes behave exactly as you'd expect them to in high school geometry.
2. Can you give a simple example of a Euclidean space?
The most common example is the Cartesian coordinate system. A flat piece of graph paper with x and y axes is a perfect 2D Euclidean space. Similarly, the 3D coordinate system with x, y, and z axes used to locate points in space is a 3D Euclidean space. In these spaces, you can define any point with coordinates like (x, y) or (x, y, z).
3. Why is this concept named after Euclid?
It is named after the ancient Greek mathematician Euclid, who lived around 300 BC. In his famous book, "Elements," he systematically laid out the fundamental principles, or axioms, that describe the properties of flat space. Because his work is the foundation for this type of geometry, the space it describes is called Euclidean.
4. Can a Euclidean space have more than three dimensions?
Yes, absolutely. While we can only visualise three dimensions, mathematicians and scientists often work with n-dimensional Euclidean spaces. These are abstract concepts where a point can be described by four, five, or even hundreds of coordinates. These higher-dimensional spaces are essential tools in fields like data science, theoretical physics, and advanced engineering.
5. How is a Euclidean space different from a Cartesian space?
This is a subtle but important distinction. A Euclidean space is the space itself, defined by its properties of distance and angles, without any reference points. A Cartesian space is a Euclidean space that has been equipped with a coordinate system (like x, y, and z axes and an origin). In other words, a Cartesian system is a way to *map and describe* a Euclidean space.
6. What is the main difference between a Euclidean space and a vector space?
The key difference is the concept of a special point, or origin. A vector space must have a zero vector, which acts as the origin. All other elements (vectors) are defined in relation to this origin. A Euclidean space, on the other hand, is a space of points where no single point is more special than another; it has no fixed origin. It's a space of locations, while a vector space is a space of displacements.
7. Why is understanding Euclidean space important for subjects like physics or computer graphics?
Euclidean space is the mathematical foundation for describing the physical world and creating virtual ones.
- In Physics, it's used to model everything from the path of a thrown ball to the forces acting on a bridge.
- In Computer Graphics, every 3D model in a video game or animated film is built using the rules of 3D Euclidean geometry to calculate positions, perspectives, and movement.
8. If our world is mostly Euclidean, what would a non-Euclidean space be like?
A non-Euclidean space is one where the standard rules of geometry don't apply, usually because the space is curved. For example:
- On the surface of a sphere (like Earth), the shortest distance between two cities is a curved arc, and the angles of a large triangle add up to more than 180 degrees.
- In a hyperbolic space (shaped like a saddle), a triangle's angles add up to less than 180 degrees.
