In classical mathematics, analytic geometry, also known as coordinate geometry, or Cartesian geometry, is the study of geometry using a coordinate system and the principles of algebra and analysis.

Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and squares, often in two and sometimes in three dimensions.

Illustration of a Cartesian coordinate plane.

The most common coordinate system to use is the Cartesian coordinate system, where each point has an x-coordinate representing its horizontal position, and a y-coordinate representing its vertical position.

Expanded in Cartesian coordinates ( see: Del in cylindrical and spherical coordinates for spherical and cylindrical coordinate representations ), ∇× F is, for F composed of F < sub > y </ sub >, F < sub > z </ sub >:

The two integers a and b are coprime if and only if the point with coordinates ( a, b ) in a Cartesian coordinate system is " visible " from the origin ( 0, 0 ), in the sense that there is no point with integer coordinates between the origin and ( a, b ).

Thus, u < sub > ρ </ sub > and u < sub > θ </ sub > form a local Cartesian coordinate system attached to the particle, and tied to the path traveled by the particle.

To introduce the unit vectors of the local coordinate system, one approach is to begin in Cartesian coordinates and describe the local coordinates in terms of these Cartesian coordinates.

# REDIRECT Cartesian coordinate system

Illustration of a Cartesian coordinate plane.

A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length.

Cartesian coordinate system with a circle of radius 2 centered at the origin marked in red.

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.

The development of the Cartesian coordinate system would play an intrinsic role in the development of the calculus by Isaac Newton and Gottfried Wilhelm Leibniz.

Choosing a Cartesian coordinate system for a one-dimensional space — that is, for a straight line — means choosing a point O of the line ( the origin ), a unit of length, and an orientation for the line.

The modern Cartesian coordinate system in two dimensions ( also called a rectangular coordinate system ) is defined by an ordered pair of perpendicular lines ( axes ), a single unit of length for both axes, and an orientation for each axis.

A three dimensional Cartesian coordinate system, with origin O and axis lines X, Y and Z, oriented as shown by the arrows.

Choosing a Cartesian coordinate system for a three-dimensional space means choosing an ordered triplet of lines ( axes ), any two of them being perpendicular ; a single unit of length for all three axes ; and an orientation for each axis.

A line with a chosen Cartesian system is called a number line.

The Coordinate system # Coordinate surface | coordinate surfaces of the Cartesian coordinates ( x, y, z ).

The four quadrants of a Cartesian coordinate system.

The axes of a two-dimensional Cartesian system divide the plane into four infinite regions, called quadrants, each bounded by two half-axes.

A forerunner of the modern ideas of cyberspace is the Cartesian notion that people might be deceived by an evil demon that feeds them a false reality.

In modern mathematics, it is more common to define Euclidean space using Cartesian coordinates and the ideas of analytic geometry.

On several occasions he mentions that in order to keep his judgment as unprepossessed as might be with any of the modern theories of philosophy, until he was " provided of experiments " to help him judge of them, he refrained from any study of the Atomical and the Cartesian systems, and even of the Novum Organum itself, though he admits to " transiently consulting " them about a few particulars.

However, virtually all modern philosophers have noted that not only is Cartesian Dualism untenable since the interactionism problem breaks the causal closure of the physical, but the cogito itself is logically fallacious.

Contemporary philosophers dispute that Socrates ’ s conceptions of knowing truth, and of ethical conduct, can be equated with modern, post – Cartesian conceptions of knowledge and of rational intellectualism.

Instead, JPL integrates the equations of motion in Cartesian coordinates ( x, y, z ), and adjusts the initial conditions in order to fit modern, highly accurate measurements of planetary positions.

Foucault's use of parrhesia, he tells us, is troubled by our modern day Cartesian model of evidential necessity.

The Cartesian theater is a derisive term coined by philosopher Daniel Dennett to pointedly refer to a defining aspect of what he calls Cartesian materialism, which he considers to be the often unacknowledged remnants of Cartesian dualism in modern materialistic theories of the mind.

In the C-plane we construct a set of rectangular Cartesian coordinates u, V with the origin at Q and such that both C and Af have finite slope at Q.

* The Cartesian coordinate system ( also called the " rectangular coordinate system "), which, for two-and three-dimensional spaces, uses two and three numbers ( respectively ) representing distances from the origin in three mutually perpendicular directions.

Expressed in terms of components with respect to a rectangular Cartesian coordinate system, the governing equations of linear elasticity are:

In quantum mechanics, idealized situations occur in rectangular Cartesian coordinates, such as the potential well, particle in a box, quantum harmonic oscillator, and more realistic approximations in spherical polar coordinates such as electrons in atoms and molecules.

One would describe the motion of a particle in a rectangular box in Cartesian coordinates, whereas one would prefer spherical coordinates for a particle in a sphere.

where h < sub > 1 </ sub >, h < sub > 2 </ sub >, h < sub > 3 </ sub > are scale factors ( also called Lamé coefficients after Gabriel Lamé ) that account for the deformation from the rectangular Cartesian coordinates to the curvilinear system ( see below ), h < sub > 1 </ sub > q < sub > 1 </ sub >, h < sub > 2 </ sub > q < sub > 2 </ sub >, h < sub > 3 </ sub > q < sub > 3 </ sub > are the coordinates of this position vector, and b < sub > 1 </ sub >, b < sub > 2 </ sub >, b < sub > 3 </ sub > are the curvilinear basis.

Especially when using rectangular grid finite-difference models ( e. g. MODFLOW, made by the USGS ), we deal with Cartesian coordinates.

Kane Yee's seminal 1966 paper proposed spatially staggering the vector components of the E-field and H-field about rectangular unit cells of a Cartesian computational grid so that each E-field vector component is located midway between a pair of H-field vector components, and conversely.

The futuristic building looks like a strict rectangular Cartesian coated stainless steel and glass, with a base of 183 to 100 metres.

Platform heading consists of orienting the sensitive axis of the accelerometers parallel to the desired coordinate system of the navigator.

For the case of a purely inertial autonavigator consisting of three restrained gyros, a coordinate system is used where the sensitive axis of the X accelerometer is parallel to the east-west direction at the base point, and the Y accelerometer sensitive axis is parallel to the north-south direction at the base point.

Ataxia is a non-specific clinical manifestation implying dysfunction of the parts of the nervous system that coordinate movement, such as the cerebellum.

In geography, the location of any point on the Earth can be identified using a geographic coordinate system.

In astronomy, a given point on the celestial sphere ( that is, the apparent position of an astronomical object ) can be identified using any of several astronomical coordinate systems, where the references vary according to the particular system.

If we use a cylindrical coordinate system with basis vectors, then the gradient of and the divergence of are given by

However, although Apollonius came close to developing analytic geometry, he did not manage to do so since he did not take into account negative magnitudes and in every case the coordinate system was superimposed upon a given curve a posteriori instead of a priori.

In analytic geometry, the plane is given a coordinate system, by which every point has a pair of real number coordinates.

The black grid shows lines of latitude and longitude in an Altair-centric coordinate system.

This enables a convenient choice of a coordinate system to be made, called comoving coordinates.

In this coordinate system the grid expands along with the Universe, and objects that are moving only due to the expansion of the Universe remain at fixed points on the grid.

The vector A can be written using any set of basis vectors and corresponding coordinate system.

* Conical coordinates, a three-dimensional orthogonal coordinate system

Continuum mechanics deals with physical properties of solids and fluids which are independent of any particular coordinate system in which they are observed.

These physical properties are then represented by tensors, which are mathematical objects that have the required property of being independent of coordinate system.

Thus, the sum of all applied forces and torques ( with respect to the origin of the coordinate system ) in the body can be given by

In a general coordinate system, the curl is given by

This formula shows how to calculate the curl of F in any coordinate system, and how to extend the curl to any oriented three dimensional Riemannian manifold.