The so-called abscissa axis represents the horizontal coordinate axis. Surfaces: r - spheres, θ - cones, φ - half-planes Lines: r - straight beams, θ - vertical semicircles, φ - horizontal circles Axes: r - straight beams, θ - tangents to vertical semicircles, φ - tangents to horizontal circlesįor now, consider 3-D space.It is known as abscissa (word derived from Latin abscissa, 'cut') to a horizontal direction coordinate appearing on a rectangular Cartesian plane and that is expressed as the distance that exists between a point and the vertical axis. 2 - Coordinate surfaces, coordinate lines, and coordinate axes of spherical coordinates. Orthogonal curvilinear coordinates in 3 dimensions Coordinates, basis, and vectors įig. 10 Fictitious forces in general curvilinear coordinates.9 Vector and tensor calculus in three-dimensional curvilinear coordinates.8.1 The metric tensor in orthogonal curvilinear coordinates.7 Vector and tensor algebra in three-dimensional curvilinear coordinates.4.2 Constructing a covariant basis in three dimensions.

4.1 Constructing a covariant basis in one dimension.1 Orthogonal curvilinear coordinates in 3 dimensions.Spherical coordinates are the most common curvilinear coordinate systems and are used in Earth sciences, cartography, quantum mechanics, relativity, and engineering. While one might describe the motion of a particle in a rectangular box using Cartesian coordinates, the motion in a sphere is easier with spherical coordinates. Equations with boundary conditions that follow coordinate surfaces for a particular curvilinear coordinate system may be easier to solve in that system. The motion of particles under the influence of central forces is usually easier to solve in spherical coordinates than in Cartesian coordinates this is true of many physical problems with spherical symmetry defined in R 3. Such expressions then become valid for any curvilinear coordinate system.Ī curvilinear coordinate system may be simpler to use than the Cartesian coordinate system for some applications. Mathematical expressions involving these quantities in vector calculus and tensor analysis (such as the gradient, divergence, curl, and Laplacian) can be transformed from one coordinate system to another, according to transformation rules for scalars, vectors, and tensors. The formalism of curvilinear coordinates provides a unified and general description of the standard coordinate systems.Ĭurvilinear coordinates are often used to define the location or distribution of physical quantities which may be, for example, scalars, vectors, or tensors. In the same space, the coordinate surface r = 1 in spherical coordinates is the surface of a unit sphere, which is curved. A Cartesian coordinate surface in this space is a coordinate plane for example z = 0 defines the x- y plane.

Well-known examples of curvilinear coordinate systems in three-dimensional Euclidean space ( R 3) are cylindrical and spherical coordinates. The name curvilinear coordinates, coined by the French mathematician Lamé, derives from the fact that the coordinate surfaces of the curvilinear systems are curved. This means that one can convert a point given in a Cartesian coordinate system to its curvilinear coordinates and back. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally invertible (a one-to-one map) at each point. In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. Curvilinear (top), affine (right), and Cartesian (left) coordinates in two-dimensional space

Definition abscissa