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In the case of two parameters, the point describes a surface, called a parametric surface. In all cases, the equations are collectively called a parametric representation, [2] or parametric system, [3] or parameterization (also spelled parametrization, parametrisation) of the object. [1] [4] [5]
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] "
Given a line and any point A on it, we may consider A as decomposing this line into two parts. Each such part is called a ray and the point A is called its initial point. It is also known as half-line (sometimes, a half-axis if it plays a distinct role, e.g., as part of a coordinate axis). It is a one-dimensional half-space. The point A is ...
Parametric representation is a very general way to specify a surface, as well as implicit representation. Surfaces that occur in two of the main theorems of vector calculus , Stokes' theorem and the divergence theorem , are frequently given in a parametric form.
Any straight line (,) with fixed parameter = is called a generator. The vectors () describe the directions of the generators. The curve () is called the directrix of the representation. The directrix may collapse to a point (in case of a cone, see example below).
Given: two planes : =, =,,, linearly independent, i.e. the planes are not parallel.. Wanted: A parametric representation = + of the intersection line.. The direction of the line one gets from the crossproduct of the normal vectors: =.
Let γ(t) = (x(t), y(t)) be a proper parametric representation of a twice differentiable plane curve. Here proper means that on the domain of definition of the parametrization, the derivative dγ / dt is defined, differentiable and nowhere equal to the zero vector.
The intersection of a line and a plane in general position in three dimensions is a point. Commonly a line in space is represented parametrically ((), (), ()) and a plane by an equation + + =. Inserting the parameter representation into the equation yields the linear equation