Parametric curve

From Computer Graphics

A parametric curve γ(t) in Rⁿ is one which is defined by an n-dimensional function of one real parameter t.

[edit] Definition

Let n be a natural number, r a natural number or ∞, I be a non-empty interval of real numbers and t in I. A vector valued function

of class C^r (i.e. γ is r times continuously differentiable is called a parametric curve of class C^r or a C^r parametrization of the curve γ. t is called the parameter of the curve γ. γ(I) is called the image of the curve.

It is important to distinguish between a curve γ and the image of a curve γ(I) because a given image can be described by several different C^r curves.

One may think of the parameter t as representing time and the curve γ(t) as the trajectory of a moving particle in space.

If I is a closed interval [a, b] we call γ(a) the starting point and γ(b) the endpoint of the curve γ.

If γ(a) = γ(b) we say γ is closed or a loop. Furthermore we call γ a closed C^r-curve if γ^(k)(a) = γ^(k)(b) for all k ≤ r.

If γ:(a,b) → Rⁿ is injective, we call the curve simple.

If γ is a parametric curve which can be locally described as a power series, we call the curve analytic or of class .

We write -γ to say the curve is traversed in opposite direction.

[edit] Example

Some parametrizatios of a circle