A triply orthogonal system consists of three smooth one-parameter families of surfaces in a region of three-dimensional Euclidean space. Exactly one surface from each family passes through each point of the region, and their normal vectors are pairwise orthogonal (Darboux 1898, pp. 1-2).
Locally the three families are level surfaces of smooth functions 
, 
, and 
with nonzero gradients.
The orthogonality conditions are
 |
The gradients are linearly independent, so 
, 
, and 
define a local orthogonal
coordinate system. For cylindrical coordinates 
away from the axis, fixing 
, 
, and 
gives coaxial circular cylinders,
radial half-planes, and horizontal planes,
respectively.
By Dupin's theorem, two surfaces from different families intersect along a common line of curvature (Darboux 1898, p. 9). The Darboux-Halphen system arose from a problem of finding a third family of quadrics orthogonal to two prescribed families of surfaces (Cruz Morales et al. 2018).
