209 Parameterization
A function that labels points of a curve by a real parameter that is used to write coordinates along the curve.
definition [d] (Parameterization = Parametrization = Parametric Representation) A continuous vector-valued function that traces a curve \(C\) as the parameter varies over an interval \(I\):
- \(\mathbf{r}(t) = \langle f(t),\, g(t),\, h(t) \rangle\) , \(t \in I\) .
where
- \(\mathbf{r}(t)\) is the position vector of a point on \(C\).
- \(C\) is the curve being traced.
- \(f, g, h\) are continuous real-valued functions on \(I\).
- \(t\) is the parameter.
- \(I\) is the parameter interval.
Note:
- a given curve admits many different parameterizations; speed and direction may differ.
definition [d] (Parameterization = Parametrization = Parametric Form) A description of a curve by expressing its coordinates as continuous functions of a single parameter \(t \in I\):
- \(x = f(t)\), \(y = g(t)\), \(z = h(t)\) .
where
- \(f, g, h\) are continuous real-valued functions on \(I\).
- \(t\) is the parameter.
- \(I\) is the parameter interval.
- \(x, y, z\) are coordinates of a point on the curve.
Note:
- as \(t\) increases, the point \((f(t), g(t), h(t))\) moves along the curve with a definite orientation.