Length of Element of Arc in Orthogonal Curvilinear Coordinates

Theorem
Let $\tuple {q_1, q_2, q_3}$ denote a set of orthogonal curvilinear coordinates.

Let the relation between those orthogonal curvilinear coordinates and Cartesian coordinates be expressed as:

where $\tuple {x, y, z}$ denotes the Cartesian coordinates.

Let $S$ be an infinitesimal arc.

Let $\d s$ be the length of $S$

Then:

where:


 * $\d q_i$ is the projection of $S$ onto the unit normal to the curvilinear coordinate surface determined by $q_i$, for $i \in \set {1, 3}$


 * ${h_i}^2 = \paren {\dfrac {\partial x} {\partial q_i} }^2 + \paren {\dfrac {\partial y} {\partial q_i} }^2 + \paren {\dfrac {\partial z} {\partial q_i} }^2$

Proof
By definition of the metric of $\tuple {q_1, q_2, q_3}$:

From Value of Curvilinear Coordinate Metric:
 * $\forall i, j \in \set {1, 2, 3}: {h_{i j} }^2 = \dfrac {\partial x} {\partial q_i} \dfrac {\partial x} {\partial q_j} + \dfrac {\partial y} {\partial q_i} \dfrac {\partial y} {\partial q_j} + \dfrac {\partial z} {\partial q_i} \dfrac {\partial z} {\partial q_j}$

But we have that $\tuple {q_1, q_2, q_3}$ is orthogonal.

From Definition 1 of orthogonal curvilinear coordinates:


 * $\dfrac {\partial x} {\partial q_i} \dfrac {\partial x} {\partial q_j} + \dfrac {\partial y} {\partial q_i} \dfrac {\partial y} {\partial q_j} + \dfrac {\partial z} {\partial q_i} \dfrac {\partial z} {\partial q_j} = 0$

wherever $i \ne j$.

Hence we have:

Elements $h_{i i}$ are those elements of the metric which do not vanish when $\tuple {q_1, q_2, q_3}$ is orthogonal.

To streamline notation, we rename them $h_1$, $h_2$ and $h_3$.

Hence:

where:


 * ${h_i}^2 = \paren {\dfrac {\partial x} {\partial q_i} }^2 + \paren {\dfrac {\partial y} {\partial q_i} }^2 + \paren {\dfrac {\partial z} {\partial q_i} }^2$

Also see

 * Definition:Scale Factor for Orthogonal Curvilinear Coordinates