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

Mariology

Mariology is the theological study of Mary, the mother of Jesus. Mariology seeks to relate doctrine or dogma about Mary to other doctrines of the faith, such as those concerning Jesus and notions about redemption, intercession, and grace. Mariology aims to place the role of the historic Mary in the context of scripture, tradition and the teachings of the Church on Mary. In terms of social history, Mariology may be broadly defined as the study of devotion to and thinking about Mary throughout the history of Christianity. There exist a variety of Christian (and non-Christian) views about Mary as a figure ranging from the focus on the veneration of the Blessed Virgin Mary in Roman Catholic Mariology to criticisms of "mariolatry" as a form of idolatry. The latter would include certain Protestant objections to Marian devotion. There are also more distinctive approaches to the role of Mary in Lutheran Marian theology and Anglican Marian theology. As a field of theology, the most ...

Stoichiometry

Stoichiometry refers to the relationship between the quantities of reactants and products before, during, and following chemical reactions. Stoichiometry was founded on the law of conservation of mass where the total mass of the reactants equals the total mass of the products, leading to the insight that the relations among qualities of reactants and products typically form a ratio of positive integers. This means that if the amounts of separate reactants are known, then the amount of the product can be calculated. Conversely, if one reactant has a known quantity and the quantity of the products can be empirically determined, then the amount of other reactants can also be calculated. This is illustrated in this example, where the balanced equation is: CH4 + 2O4 -----> CO2 + 2H2O Here, one molecule of methane reacts with two molecules of oxygen gas to yield one molecule of carbon dioxide and two molecules of water. This particular chemical...

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...

Search This Blog