Roto-translation (rigid transformation)

The rigid transformation (or roto-translation motion) is the composition between reflection, translation, and rotation, and therefore it is an isometry, that is, a geometric transformation that leaves the distances unchanged. In other words, we can think of roto-translation as a rigid movement in which a geometric figure first rotates and then translates. The rototranslation motion of a rigid body […]


Rotation is defined as rigid movement having as fixed points a point called center (in two dimensions) or a straight line called axis (in three dimensions) of rotation. This movement shifts all points around the center, or axis, by a fixed angle. In other words a rotation is the movement of a body following a circular […]


In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure or space by the same distance in a given direction. It can also be interpreted as the addition of a constant vector at each point, or as a displacement of the origin of the coordinate system. In Physics, translation is defined […]


Euclidean geometry. Non-Euclidean geometry.

Coordinate systems

A coordinate system is a system that uses one or more numbers, called coordinates, to uniquely determine the position of a point or other geometric elements on 1D, 2D, and 3D dimensions. Each of these numbers indicates the distance between the point and some fixed reference point, called the origin. The first number, known as the \(x\) value, indicates how far left […]


Vectors are indicated in the scientific literature with a letter, generally lowercase, with an arrow above it: \(\vec{v}\). Vectors are essential to physics and engineering. Many fundamental physical quantities are vectors, including displacement, velocity, force, and electric and magnetic vector fields. In this context, the following fundamental entities are assigned: free vectors: characterized by magnitude, direction, […]


In geometry, the volume is defined as the measure of space occupied by a three-dimensional body. The volume of a three-dimensional body is a numerical characteristic of the body; in the simplest case, when the body can be decomposed into a finite set of unit cubes (i.e., cubes with edges of unit length), it is equal to […]


In order to describe the motion of an object, you must first be able to describe its position (where it is at any particular time). More precisely, you need to specify its position relative to a convenient reference frame. So the position of a point \(P\) can be described by a pair or a set of coordinates, […]


The displacement of an object is defined as the vector distance from some initial point to a final point. This change in position is known as displacement. The word “displacement“ implies that an object has moved, or has been displaced. It is therefore distinctly different from the distance traveled except in the case of straight-line motion in […]


It is defined monomial as a literal algebraic expression, consisting of a numerical part (coefficient) and a literal part among which only multiplication and exponentiation operations appear; for example: \[\dfrac{1}{2}x;\;7x^2y;\;-9x^n\] The monomial degree is defined as the sum of all the exponents of the literal part. Monomes that have the same literal part (with identical exponent) are […]

Statistical hypothesis testing

Hypothesis testing is a branch of statistics that tries to evaluate the reliability of a hypothesis through the results of an experiment or a series of observations. The hypotheses considered can be the most varied and in the most disparate fields; that is, one can try to ascertain by experiment the validity of a natural law […]


Exponentiation is one of the mathematical operations that replace multiple multiplications between equal numbers or variables, simplifying both writing and processing. If the exponent is greater than 1, the power is the product of as many factors as are indicated by the number of the exponent, all equal to the base. From this statement it is […]


In mathematics, an isometry (from the Greek ἴσος, isos, which means equal | called also congruence, or congruent transformation) is a notion that generalizes that of rigid movement of an object or a geometric figure. Formally, it is a function between two metric spaces that preserves distances. Examples of isometries are translations, rotations, and reflections in the plane or […]


The inequations, unlike the equations, are inequalities between monomials, or polynomials, for which we seek the solution of one or more literal variables, called unknowns (as for the equations). Some examples of inequations are: \(a<b\) \(x+y+z\leq 1\) \(n>1\) \(x\neq 0\) Intervals In mathematics, an interval is defined as the set of all the elements of an ordered set which are preceded […]


The length, quantitatively and objectively, identifies a material body according to a single main or prevalent dimension of the body itself. The measure of length leads to the knowledge of the geometry of bodies, that is to their “dimensions.“ Many length measurements are also based on many (indirect) measurements of other physical quantities. Length measurements can be obtained […]


Natural number (counting number). Whole number. Integer. Rational numbers. Real numbers. Complex number. Parity (odd and even numbers).


Notable products between binomials Notable products are used in algebra for the literal calculation of the product between binomials. They are said to be notable because the product of some particular polynomials always reaches the same result. For this reason, it is possible to avoid, for these particular polynomials, the carrying out of all the calculation steps […]


Polynomial is defined as the algebraic sum of monomials. The monomials that make up a polynomial are called polynomial terms. For an expression to be a polynomial term, any variables in the expression must have whole-number powers (or else the “understood“ power of 1, as in \(x^1\), which is normally written as \(x\)).


Sequence. Arithmetic progression. Harmonic series. Geometric series.


Spacetime is any mathematical model that fuses the three dimensions of space (length, width, and depth) and the one dimension of time into a single four-dimensional manifold. It represents the “stage“ in which physical phenomena take place. Spacetime is a physical concept that combines our classic traditionally distinct notions of space and time into a single homogeneous entity. […]

Euclid of Alexandria

Euclid of Alexandria (Ancient Greek: Εὐκλείδης – Eukleídēs, lived c. 300 BCE, Alexandria, Egypt) systematized ancient Greek and Near Eastern mathematics and geometry. He wrote The Elements, the most widely used mathematics and geometry textbook in history. Older books sometimes confuse him with Euclid of Megara. Modern economics has been called “a series of footnotes to Adam Smith,” […]