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Parametric curve

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

Definition Edit

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

\mathbf{\gamma}:I \to {\mathbb R}^n

of class Cr (i.e. γ is r times continuously differentiable is called a parametric curve of class Cr or a Cr 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 Cr 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 Cr-curve if γ(k)(a) = γ(k)(b) for all kr.

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 C^\omega.

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

Example Edit

Some parametrizatios of a circle

\gamma_1(t) = \langle r \cos(t), r \sin(t)\rangle
\gamma_2(t) = \langle r \frac{1-t^2}{1+t^2}, r \frac{2 t}{1+t^2}\rangle

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