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The theory of curves is much simpler and narrower in scope than the theory of surfaces and its higher-dimensional generalizations because a regular curve in a Euclidean space has no intrinsic geometry. Any regular curve may be parametrized by the arc length (the natural parametrization).
Parametrization, also spelled parameterization, parametrisation or parameterisation, is the process of defining or choosing parameters. Parametrization may refer more specifically to: Parametrization (geometry), the process of finding parametric equations of a curve, surface, etc. Parametrization by arc length, a natural parametrization of a curve
This is a natural assumption in Euclidean geometry, because the arclength is a Euclidean invariant of the curve. In the terminology of physics, the arclength parametrization is a natural choice of gauge. However, it may be awkward to work with in practice. A number of other equivalent expressions are available.
In mathematics, and more specifically in geometry, parametrization (or parameterization; also parameterisation, parametrisation) is the process of finding parametric equations of a curve, a surface, or, more generally, a manifold or a variety, defined by an implicit equation. The inverse process is called implicitization. [1] "
In the case of a single parameter, parametric equations are commonly used to express the trajectory of a moving point, in which case, the parameter is often, but not necessarily, time, and the point describes a curve, called a parametric curve. In the case of two parameters, the point describes a surface, called a parametric surface.
This method is based on the observation that the "ideal" parameterization of a curve is arclength. Pseudo-arclength is an approximation of the arclength in the tangent space of the curve. The resulting modified natural continuation method makes a step in pseudo-arclength (rather than ). The iterative solver is required to find a point at the ...
The parametrization is regular for the given values of the parameters if the vectors , are linearly independent. The tangent plane at a regular point is the affine plane in R 3 spanned by these vectors and passing through the point r ( u , v ) on the surface determined by the parameters.
A space curve is a curve for which is at least three-dimensional; a skew curve is a space curve which lies in no plane. These definitions of plane, space and skew curves apply also to real algebraic curves , although the above definition of a curve does not apply (a real algebraic curve may be disconnected ).