199 Curve
A continuous mapping from an interval of real numbers into a space of points that is used to represent a path between points.
definition [d] (Curve = Parametric Curve) The set of all points \((x,y,z)\) given by continuous parametric equations on an interval \(I\):
- \(x = f(t)\), \(y = g(t)\), \(z = h(t)\) , \(t \in I\) .
where
- \(f, g, h\) are continuous real-valued functions on \(I\).
- \(t\) is the parameter.
- \(I\) is an interval.
- \(x, y, z\) are the coordinates of a point on the curve.
Note:
- equivalently, \(\mathbf{r}(t) = \langle f(t),\, g(t),\, h(t) \rangle\) traces the curve.
definition [d] (Curve = Path) The image of a continuous map \(\boldsymbol{\gamma}: [a,b] \rightarrow \mathbb{R}^{n}\):
- \(C = \{\, \boldsymbol{\gamma}(t) \mid t \in [a,b] \,\}\) .
where
- \(\boldsymbol{\gamma}\) is a continuous map on \([a,b]\).
- \(C\) is the image set of points traced by the map.
- \([a,b]\) is the parameter interval.
- \(\mathbb{R}^{n}\) is \(n\)-dimensional Euclidean space.
Note:
- the map is also written as a continuous vector function \(\mathbf{r}(t)\).
- the map itself is often called a path.