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

Crusades

The Crusades were a series of religious wars initiated, supported, and sometimes directed by the Christian Latin Church in the medieval period. The best known of these military expeditions are those to the Holy Land in the period between 1095 and 1291 that were intended to conquer Jerusalem and its surrounding areas from Muslim rule. Beginning with the First Crusade, which resulted in the conquest of Jerusalem in 1099, dozens of military campaigns were organized, providing a focal point of European history for centuries. Crusading declined rapidly after the 15th century. In 1095, Pope Urban II proclaimed the first expedition at the Council of Clermont. He encouraged military support for Byzantine emperor Alexios I Komnenos and called for an armed pilgrimage to Jerusalem. Across all social strata in western Europe there was an enthusiastic response. Participants came from all over Europe and had a variety of motivations, including religious salvation, satisfying feudal obligations,...

Upclose and Personal: Satan

Satan, also known as the Devil, is an entity in the Abrahamic religions that seduces humans into sin or falsehood. In Christianity and Islam, he is usually seen as either a fallen angel or a genie, who used to possess great piety and beauty, but rebelled against God, who nevertheless allows him temporary power over the fallen world and a host of demons. In Judaism, Satan is typically regarded as a metaphor for the yetzer hara or "evil inclination", or as an agent subservient to God. A figure known as "the satan" first appears in the Tanakh as a heavenly prosecutor, a member of the sons of God subordinate to Yahweh, who prosecutes the nation of Judah in the heavenly court and tests the loyalty of Yahweh's followers by forcing them to suffer. During the intertestamental period, possibly due to influence from the Zoroastrian figure of Angra Mainyu, the satan developed into a malevolent entity with abhorrent qualities in dualistic opposition to God. In the apocrypha...

Narrative Conflict

In works of narrative, conflict is the challenge main characters need to solve to achieve their goals. Traditionally, conflict is a major literary element of narrative or dramatic structure that creates challenges in a story by adding uncertainty as to whether the goal will be achieved. A narrative is not limited to a single conflict. While conflicts may not always resolve in narrative, the resolution of a conflict creates closure or fulfillment, which may or may not occur at a story's end. Conflict in literature refers to the different drives of the characters or forces involved. Conflict may be internal or external--that is, it can occur within a character's mind or between a character and exterior forces (or point/s of view). Conflict is most visible between two or more characters, usually between a protagonist and antagonist/villain/enemy, but can occur in many different forms. A character may as easily find himself or herself in conflict with a natural force, such as an an...

Search This Blog