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where r is a three-dimensional vector. Parametric surfaces A ... Such a parametric equation is called a parametric form of the solution of the system. [10]
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.
Suppose that F is a static vector field, that is, a vector-valued function with Cartesian coordinates (F 1,F 2,...,F n), and that x(t) is a parametric curve with Cartesian coordinates (x 1 (t),x 2 (t),...,x n (t)). Then x(t) is an integral curve of F if it is a solution of the autonomous system of ordinary differential equations,
The second fundamental form of a general parametric surface S is defined as follows. Let r = r(u 1,u 2) be a regular parametrization of a surface in R 3, where r is a smooth vector-valued function of two variables. It is common to denote the partial derivatives of r with respect to u α by r α, α = 1, 2.
For a plane whose equation is given in parametric form (,) = + +, where is a point on the plane and , are non-parallel vectors pointing along the plane, a normal to the plane is a vector normal to both and , which can be found as the cross product =.
For a parametric curve this is an easy task: One just computes the points of a sequence of parametric values. For an implicit curve one has to solve two subproblems: determination of a first curve point to a given starting point in the vicinity of the curve,
In differential geometry, the first fundamental form is the inner product on the tangent space of a surface in three-dimensional Euclidean space which is induced canonically from the dot product of R 3. It permits the calculation of curvature and metric properties of a surface such as length and area in a manner consistent with the ambient space.
Parametric design is a design method in which features, such as building elements and engineering components, are shaped based on algorithmic processes rather than direct manipulation. In this approach, parameters and rules establish the relationship between design intent and design response.