Cartesian Coordinate System

A Cartesian coordinate system in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the same unit of length. Each reference line is called a coordinate axis or just axis (plural axes) of the system, and the point where they meet is its origin, at ordered pair (0 , 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.

One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines). In general, n Cartesian coordinates (an element of real n-space) specify the point in an n-dimensional Euclidean space for any dimensional n. These coordinates are real, up to sign, to distances from the point to n mutually perpendicular hyperplanes.

The invention of Cartesian coordinates in the 17th centuty by René Descartes (Latinized name: Cartesius) revolutionized mathematics by providing the first systemic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes (such as curves) can be described by Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, may be described as the set of all points whose coordinates x and y satisfy the equation x^2 + y^2 = 4.

Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory, and more. A familiar example is the concept of the graph of a function. Cartesian coordinates are also essential tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering, and more. They are the most common coordinate system used in computer graphics, computer-aided geometric design, and other geometry-related data processing.

x-------x

This post is a sponsorship of Louis Vuitton shoes designed by Virgil Ablou.

Comments

How to Create a Richly Imagined World

For someone who likes fantasy and sci-fi fiction, most of the time, a lot of people ask me about how to create a richly imagined world. Fantasy and sci-fi elements rest heavily on how an author weave the setting and the world in which the heroes dwell in, and it helps to make the novel to be imagined vividly in the readers' minds. A convincing world should be relatable, something that we can associate ourselves with. For us to be associated with a world an author created in his mind, and wrote on the pages of a book, this world has to be close to the real thing. It has to be systematic, real and alive, and very convincing. A real world has certain elements, and an author must consider them in writing a vividly imagined world: Cartography - a fantasy or sci-fi world depend heavily on geography and maps, especially if the plot requires war and the belligerents occupy so much space in the plot. A convincing world has the world separated in territories, and every part of the...

The Roman Empire

The Roman Empire was the post-Rupublican period of ancient Rome. As a polity it included large territorial holdings around the Mediterranean Sea in Europe, Northern Africa, and Western Asia ruled by emperors. From the accession of Caesar Augustus to the military anarchy of the third century, it was a principate with Italy as metropole of the provinces and the city of Rome as sole capital (27 BC - 286 AD). After the military crisis, the empire was ruled by military emperors who shared rule over the Western Roman Empire (based in Milan and later in Ravenna) and over the Eastern Roman Empire (also known as the Byzantine Empire; centered on Nicomedia and Antioch, later based in Constantinopole). Rome remained the nominal capital of both parts until 476 AD, when the imperial insignia were sent to Constantinopole, following the capture of Ravenna by the barbarians of Odoacer and the subsequent deposition of Romulus Augustulus. The fall of the Western Roman Empire to Germanic Kings, alon...

Theodicy

Theodicy means vindication of God. It is to answer the question why a good God permits the manifestation of evil, thus resolving the issue of the problem of evil. Some theodicies also address the evidential problem of evil by attempting "to make the existence of an all-knowing, all-powerful, and all-good or omnibenevolent God consistent with the existence of evil or suffering in this world." Unlike a defense, which tries to demonstrate that God's existence is logically possible in the light of evil, a theodicy attempts to provide a framework where God's existence is also plausible. The German philosopher and mathematician Gottfried Leibniz coined the term "theodicy" in 1710 in his work Théodecée, through various responses to the problem of evil that had been previously proposed. The British philosopher John Hick traced the history of moral theodicy in his 1966 work, Evil and the Love of God, identifying three major traditions: the Plotinian theodicy, named a...

Search This Blog