Parametric Curve

Related Concepts

Algebraic Curve; Parametric Surface; Splines

Definition

A parametric curve S in 2-dimensional Euclidean space has the following form:

$$\begin{array}{lll} S (\cdot): \mathbb{R} & \rightarrow & \mathbb{R}^2 \nonumber\\ t & \mapsto & (x(t), y(t)), t\in[a, b] \end{array}$$

(1)

where t is the parameter and varies in the domain [a, b]. In practice, the domain [a, b] is often normalized as a specific region, such as [0, 1]. And x(t), y(t) are real-valued functions continuously mapping to a 2D point on a curve.

Similarly, a parametric curve S in 3-dimensional space has the following form:

$$\begin{array}{lll} S (\cdot): \mathbb{R} & \rightarrow & \mathbb{R}^3 \nonumber\\ t &\mapsto & (x(t), y(t), z(t)), t\in[a, b] \end{array}$$

(2)

For example, an ellipse can be represented in a parametric form as: x = a cos t, y = b sin t, with t ∈ [0, 2π), in contrast with its implicit representation: $\frac{x^2}{a^2}+\frac{y^2}{b^2}-1=0$.

Background

Since parametric functions are easy to...