The " figure 8 " immersion ( Klein bagel ) of the Klein bottle has a particularly simple parameterization.

The fact that cloud processes are not perfectly parameterized is due in part to a lack of perfect understanding of them, but parameterization itself is a fundamental and useful modeling technique, not a crutch ..

The database of compounds used for parameterization, i. e., the resulting set of parameters and functions is called the force field, is crucial to the success of molecular mechanics calculations.

This is the natural parameterization and has the property that

With the above parameterization the cross section is a 2: 1 Lissajous curve.

However, in metric geometry the geodesic considered is often equipped with natural parameterization, i. e. in the above identity v = 1 and

Over local fields this is expected to give a parameterization of L-packets of admissible irreducible representations of a reductive group over the local field.

Lindzen's work on ozone photochemistry has been important in studies that look at the effects that anthropogenic ozone depletion will have on climate .< ref > See for instance the widely cited study The Lindzen and Blake formalism is used in the parameterization of radiative-photochemical damping ( see Appendix A ).</ ref >

In the same way the hyperspherical space of 3D rotations can be parameterized by three angles ( Euler angles ), but any such parameterization is degenerate at some points on the hypersphere, leading to the problem of gimbal lock.

Suppose now that a different parameterization is selected, by allowing u and v to depend on another pair of variables u ′ and v ′.

which defines a rational parameterization of the ellipse and hence shows the ellipse is a rational curve.

For example, the projective parameterization of above ellipse is

But as is compact, given a continuous affine parameterization of the geodesic generators, there exists a lower bound to the absolute value of the expansion parameter.

Third-level parametric continuity ( C2 ), however, differs from curvature continuity in that its parameterization is also continuous.

Another parameterization, with being the angle between OA and the x-axis, increasing anti-clockwise is

For example if we use three angles ( Euler angles ), such parameterization is degenerate at some points on the hypersphere, leading to the problem of gimbal lock.

Slightly different techniques apply depending on whether one is interested in the submanifold along with its parameterization, or the submanifold up to reparameterization.

Parametrization ( or parameterization ; also parameterisation, parametrisation in British English ) is the process of deciding and defining the parameters necessary for a complete or relevant specification of a model or geometric object.

A parameterization with such a transformation rule is called flat.

A flat parameterization for is an exponential or parameterization, because the parameters are in the exponent of.

The parameterization above implies that the curve is rational which implies it has genus zero.

Let γ ( s ) be a plane curve, parameterized by its arclength s. The unit tangent vector to the curve is, by virtue of the arclength parameterization,

In computability theory the s < sub > mn </ sub > theorem, ( also called the translation lemma, parameter theorem, or parameterization theorem ) is a basic result about programming languages ( and, more generally, Gödel numberings of the computable functions ) ( Soare 1987, Rogers 1967 ).

Under some circumstances, and in some technical contexts, transcluded sections of text may not require strict adherence to the " context neutrality " principle, because the transcluded sections are capable of parameterization.

In this parameterization, u follows the length of the bottle's body while v goes around its circumference.

The alternative name arises from the parameterization of the equation of the ellipse describing a meridian section.

A likelihood function arises from a conditional probability distribution considered as a function of its distributional parameterization argument, conditioned on the data argument.

Cedar's main additions were garbage collection, dynamic types, a limited form of type parameterization, and special syntax to identify the " type-safe " parts of a multi-module software package.

Using four properties which are history, features, parameterization, and high level constraints.

It consists of steel strips which represent the image of a polar coordinate grid under a parameterization given by Robert Bryant and Rob Kusner.

The parameterization of these very coarse-grained models must be done empirically, by matching the behavior of the model to appropriate experimental data or all-atom simulations.

Such a rational parameterization may be considered in the projective space by equating the first projective coordinates to the numerators of the parameterization and the last one to the common denominator.

( This gives motivation to the use of the symbol " sigma " in the above parameterization of the Rayleigh density.

The original Chou-Fasman parameters, determined from the small sample of structures solved in the mid-1970s, produce poor results compared to modern methods, though the parameterization has been updated since it was first published.

* It facilitates parameterization.

In geometry, Boy's surface is an immersion of the real projective plane in 3-dimensional space found by Werner Boy in 1901 ( he discovered it on assignment from David Hilbert to prove that the projective plane could not be immersed in 3-space ).

Photon mapping is another method that uses both light-based and eye-based ray tracing ; in an initial pass, energetic photons are traced along rays from the light source so as to compute an estimate of radiant flux as a function of 3-dimensional space ( the eponymous photon map itself ).

The theorem says that the Gaussian curvature of a surface can be determined entirely by measuring angles, distances and their rates on the surface itself, without further reference to the particular way in which the surface is embedded in the ambient 3-dimensional Euclidean space.

The kingdom itself takes the form of a large table containing a 3-dimensional map where the player clicks where to attack next from the highlighted regions-this is quite similar to Dungeon Keeper's world map with mainly graphical improvements.

For example, the vertices, edges and faces of a cube have dimension 0, 1, and 2 respectively ; the cube itself is 3-dimensional.

The group PSL ( 2, C ) of 2 by 2 complex matrices of determinant 1 modulo its center has several natural representations: as conformal transformations of the Riemann sphere, and as orientation-preserving isometries of 3-dimensional hyperbolic space H < sup > 3 </ sup >, and as orientation preserving conformal maps of the open unit ball B < sup > 3 </ sup > in R < sup > 3 </ sup > to itself.

With time and space considered to be outside the realm of physics itself, under classical mechanics a 3-dimensional coordinate system is enough to describe the world.