📂Geometry

Parametric Curves on a Simple Surface

Definition^{1 2}

Let $\mathbf{x} : U \to \R^{3}$ be called a simple surface. Let the coordinates of $U$ be called $(u, v)$. For any point $(u_{0}, v_{0})$, the following curve is called the $u-$parameter curve at $v = v_{0}$ of $\mathbf{x}$.

$$ u \mapsto \mathbf{x}(u, v_{0}) $$

The following curve is called the $v-$parameter curve at $u = u_{0}$ of $\mathbf{x}$.

$$ v \mapsto \mathbf{x}(u_{0}, v) $$

The velocity vectors $\dfrac{\partial \mathbf{x}}{\partial u} = \mathbf{x}_{u}=\mathbf{x}_{1}$, $\dfrac{\partial \mathbf{x}}{\partial v} = \mathbf{x}_{v}=\mathbf{x}_{2}$ of the two parameter curves at point $(u_{0}, v_{0})$ are called the partial velocity vectors of $\mathbf{x}$ at $(u_{0}, v_{0})$.

Explanation

The coordinates of $U$ are often also written as $(u^{1}, u^{2})$, so the above-mentioned curves are also called the $u^{1}$-curve and the $u^{2}$-curve, respectively.

According to the definition, it can be understood that the surface $\mathbf{x}$ is covered by a family of such parameter curves.

The grid formed by these two parameter curves is called a curvilinear coordinate system, which includes the spherical and cylindrical coordinate systems.