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TRIPLE ORTHOGONAL SYSTEM SURFACES
A triple orthogonal system of surfaces consists in three
given families with one parameter such that at any point common to three
representatives of each family, the three tangent planes of the surfaces
are 2 by 2 orthogonal. This notion generalizes to 3D the notion of double
orthogonal systems of curves.
According to the Dupin theorem, two surfaces taken
among two of the families of the system intersect along curvature
lines.
If the three families are given in a parametric
form: ,
fixed u, variable v,w for the first family, fixed v, variable u,w for the second one, fixed w, variable u,v for the third one, then the three families form a triple orthogonal system iff (in other words, the columns of the Jacobian matrix of are orthogonal to one another). 
A first series of examples is provided by a given double
orthogonal system
translated perpendicularly to its plane (the three orthogonal families
are composed of two families of cylinders based on the initial curves and
the family of planes orthogonal to the direction of translation); the triple
orthogonal system is parametrized by .
Some examples (first family, fixed u, in red,
second one, fixed v, in blue, third one, fixed w, in green):
double orthogonal system  associated cylindrical triple system 

equations of the three families  name of the coordinate system (u,v,w) 

planes 


Cartesian coordinates 

cylinders of revolution, planes 


cylindrical coordinates 

parabolic cylinders, planes 


parabolic cylindrical coordinates 

elliptic cylinders, hyperbolic cylinders, planes 


elliptic cylindrical coordinates 
A second series is provided by a given double orthogonal
system
rotated around an axis of its plane (or rather an axis of symmetry to avoid
conical points); the orthogonal families are composed of two families of
surfaces of revolution based on the initial curves and the family of planes
passing by the axis of rotation; if the rotation is around Oz, then
the triple orthogonal system is parametrized by .
Some examples:
double orthogonal system  associated triple system of revolution  parametrization  equations of the three families  name of the coordinate system (u,v,w) 

spheres, cones, planes 


spherical coordinates 

prolate ellipsoids, twosheeted hyperboloids, planes 


prolate ellipsoidal coordinates 

oblate ellipsoids, onesheeted hyperboloids, planes 


oblate ellipsoidal coordinates 

ring tori, spheres, planes 

toroidal coordinates  

spheres, spindle tori, planes 

bispherical coordinates 
A third series is obtained by the images by an inversion
of a triple orthogonal system, the resulting system still being a triple
orthogonal system (since the map is conformal).
initial orthogonal system  2D equivalent  triple system  parametrization  equations of the three families  name of the coordinate system (u,v,w) 





trispherical coordinates 
system associated to the cylindrical coordinates 

horn tori, planes, spheres 

inverse cylindrical coordinates 
Fourth series: systems of homofocal quadrics.
triple system  parametrization  intervals of definition  equations of the three families  name of the coordinate system (u,v,w) 


if :


homofocal ellipsoidal coordinates;
see also quadric. 


if :


homofocal paraboloidal coordinates;
see also hyperbolic paraboloid 
Notice the double orthogonal system of biquadratics on the sphere (remember that on the sphere, any line is a curvature line). 
if :


conical coordinates 
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© Robert FERRÉOL
2017