Symbols

${\displaystyle r}$	cylindrical radius
${\displaystyle \theta }$	angle of curve from beginning of spiral to a particular point on the spiral
${\displaystyle \operatorname {sn} (s,k)}$ ${\displaystyle \operatorname {cn} (s,k)}$	basic Jacobi Elliptic Function[2]
${\displaystyle \vartheta _{i}(s)}$	Jacobi Theta Functions (where ${\displaystyle i}$ the kind of Theta Functions show)[3]
${\displaystyle k}$	elliptic modulus (any positive real constant)[4]

Representation via equations

The Seiffert's spherical spiral can be expressed in cylindrical coordinates as

${\displaystyle r=\operatorname {sn} (s,k),\,\theta =k\cdot s{\text{ and }}z=\operatorname {cn} (s,k)}$

or expressed as Jacobi theta functions

${\displaystyle r={\frac {\vartheta _{3}(0)\cdot \vartheta _{1}(s\cdot \vartheta _{3}^{-2}(0))}{\vartheta _{2}(0)\cdot \vartheta _{4}(s\cdot \vartheta _{3}^{-2}(0))}},\,\theta ={\frac {\vartheta _{2}^{2}(q)}{\vartheta _{3}^{2}(q)}}\cdot s{\text{ and }}z={\frac {\vartheta _{4}(0)\cdot \vartheta _{3}(s\cdot \vartheta _{3}^{-2}(0))}{\vartheta _{3}(0)\cdot \vartheta _{4}(s\cdot \vartheta _{3}^{-2}(0))}}}$.^[5]