orthogonal coordinate system

A 3D orthogonal coordinate system with its origin at O.

In an orthogonal coordinate system the lengths of the basis vectors are known as scale factors.

One- and two-dimensional wave equations in orthogonal coordinate systems.

In 3D orthogonal coordinate systems are 3: Cartesian, cylindrical, and spherical.

Elliptic coordinates, an orthogonal coordinate system based on families of ellipses and hyperbolas.

Bipolar coordinates are a two-dimensional orthogonal coordinate system.

If the curvilinear coordinates form an orthogonal coordinate system, the element of arc length ds is expressed as:

If the basis vectors are orthogonal at every point, the coordinate system is an orthogonal coordinate system.

However, there are other orthogonal coordinate systems in three dimensions that cannot be obtained by projecting or rotating a two-dimensional system, such as the ellipsoidal coordinates.

Parabolic coordinates are a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal parabolas.

Conventional solutions for the equations of equilibrium based on the well-known Vlasov thin-walled beam theory uncouple the equations by adopting orthogonal coordinate systems.

A simple method for generating orthogonal coordinates systems in two dimensions is by a conformal mapping of a standard two-dimensional grid of Cartesian coordinates (x, y).

Bispherical coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional bipolar coordinate system about the axis that connects the two foci.

Elliptic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional elliptic coordinate system in the perpendicular -direction.

Similar to the related ellipsoidal coordinates, the paraboloidal coordinate system has orthogonal quadratic coordinate surfaces that are not produced by rotating or projecting any two-dimensional orthogonal coordinate system.

Laplace's equation is separable in 13 orthogonal coordinate systems, and the Helmholtz equation is separable in 11 orthogonal coordinate systems.

CCA defines coordinate systems that optimally describe the cross-covariance between two datasets while PCA defines a new orthogonal coordinate system that optimally describes variance in a single dataset.

This right-handed orthogonal coordinate system is named in honor of the German scientist Dr. Walter Lode because of his seminal paper written in 1926 describing the effect of the middle principal stress on metal plasticity.

The true utility of this formulation is seen when the flow is two dimensional in nature and the equation is written in a general orthogonal coordinate system, in other words a system where the basis vectors are orthogonal.

While spherical polar coordinates are one orthogonal coordinate system for expressing vectors and tensors using polar and azimuthal angles and radial distance, the spherical basis are constructed from the standard basis and use complex numbers.

But for the higher order terms (the two coming from the divergence of the deviatoric stress that distinguish Navier-Stokes equations from Euler equations) some tensor calculus is required for deducing an expression in non-cartesian orthogonal coordinate systems.

Expressing the Navier-Stokes vector equation in Cartesian coordinates is quite straightforward and not much influenced by the number of dimensions of the euclidean space employed, and this is the case also for the first-order terms (like the variation and convection ones) also in non-cartesian orthogonal coordinate systems.

Here σ is the surface tension, n, t and s are unit vectors in a local orthogonal coordinate system (n,t,s) at the free surface (n is outward normal to the free surface while the other two lie in the tangential plane and are mutually orthogonal).