,
,
)
direct orthonormal frame, with axes Ox, Oy and Oz.
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SURFACES
See the notations below.
Surfaces starting with
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ASYMPTOTIC PLANE OF A GENERATRIX OF A RULED SURFACE
ATTRACTION (SURFACE OF EQUAL/)
ATTRACTION (SOLID OF MAXIMAL/)
CENTRAL POINT OF A GENERATRIX OF A RULED SURFACE
CHARACTERISTIC OF A MANIFOLD, OF A SURFACE (EULER/)
CONE OR CONICAL SURFACE
CURVATURE (SURFACE OF REVOLUTION WITH CONSTANT GAUSSIAN/)
CUSPIDAL EDGE OF A DEVELOPABLE RULED SURFACE
CYLINDER
OF REVOLUTION
DARBOUX CYCLIDE
DEVELOPABLE SURFACE (INVOLUTE OF A/)
DYCK'S SURFACE
EDGE
(CUSPIDAL/)
ELASTICITY SURFACE (FRESNEL'S)
ENVELOPE OF A FAMILY OF SURFACES
EULER CHARACTERISTIC
OF A MANIFOLD, OF A SURFACE
FLIPPABLE SURFACE
GYROID
HANDLES (SURFACE WITH n/)
HYPERBOLOID
OF ONE
SHEET (H1)
OF TWO
SHEETS (H2)
HYPERSPHERE (3-DIMENSIONAL/, n DIMENSIONAL/)
HYPERTORURS
INDICATRIX
(DUPIN/)
INVERSE OF A SURFACE WITH RESPECT TO A SPHERE
INVOLUTE OF A DEVELOPABLE SURFACE
ISOMETRIC TO ANOTHER SURFACE (SURFACE/)
LINE
CURVATURE/,
ASYMPTOTIC/,
GEODESIC/
LINE (TOPOGRAPHIC/):
LEVEL/,
SLOPE/,
THALWEG/,
APEX/,
LIOUVILLE SURFACE
MANIFOLD (TOPOLOGICAL/, DIFFERENTIAL/,
ALGEBRAIC/)
MERIDIAN OF A SURFACE OF REVOLUTION
MÖBIUS STRIP, OR BAND, OR RING
MORIN'S SURFACE
NADIRASHVILI
SURFACE
OBLATE (ELLIPSOID OF REVOLUTION/)
OVOID
PAPER LANTERN (SCHWARZ/)
PARABOLIC (POINT / D'UNE SURFACE)
PARALLEL TO ANOTHER SURFACE (SURFACE/)
POLAR DEVELOPABLE OF A SKEW CURVE
POLAR OF A SURFACE, OF A CURVE, WITH RESPECT TO A SPHERE (RECIPROCAL/)
PRESSURE (TOWER WITH CONSTANT/)
PROLATE ELLIPSOID OF REVOLUTION
RIEMANN FINITE MINIMAL SURFACE
SCREW WITH SQUARE THREAD (SURFACE OF THE)
SCREW WITH TRIANGULAR THREAD (SURFACE OF THE/)
SLOPE LINE, OR OF GREATEST SLOPE
STRAIGHT DIRECTRIX (SURFACE WITH A/)
STRICTION LINE OF A NON DEVELOPABLE RULED SURFACE
SUM OF TWO SURFACES (CONNECTED/)
SYMMETRY (SURFACE WITH ROTATIONAL/)
SYSTEM (TRIPLE ORTHOGONAL)
TAKAGI (MOUNT/)
TRICKLE OF WATER (SURFACE OF THE/)
TRIPLE ORTHOGONAL SYSTEM OF SURFACES
TUBE or TUBULAR SURFACE
UMBILIC
(S) surface currently under study.
M: current point of the surface.
(O, ,,)
direct orthonormal frame, with axes Ox, Oy and Oz.
(
):
Cartesian coordinates of M.
(
):
cylindrical coordinates of M; 
.
(r, q, l) or (r, q, j): spherical coordinates of M (q is the longitude, l is the latitude and j the colatitude).
Generalization to toroidal coordinates (r,
r,
q,l):
u, v: parameters.
Cartesian equation, parametrization: characterization in terms of x, y et z.
Cylindrical equation, parametrization: characterization in terms of r, q and z.
Spherical equation, parametrization: characterization in terms of r, q and l.
, 
, 
,
,
.
,
, 
:
coefficients of the first fundamental quadratic form (s =
curvilinear abscissa of a curve traced on the surface):
.
:
Surface element.
:
normal vector.
, 
, 
:
coefficients of the second fundamental quadratic form:
R1 and R2: principal (i.e. extremal) radii of curvature at M.
and 
: principal
curvatures at M.
: Gaussian (or total) curvature at M.
: mean curvature at M.
.
Let be (C) a curve drawn on (S) passing through
M
, of Frenet trihedron 
.
The Darboux-Ribaucour (or geodesic) trihedron
is 
where 
.
,
angle of the rotation passing from Frenet to Darboux, angle between the
osculating plane and the tangent plane to the surface:
.
Frenet formulas : 
;
Darboux formulas : 
,
: normal radius of curvature (radius of curvature of the normal section
of the surface, tangent to the curve), 
: normal curvature.
: geodesic
radius of curvature, 
: geodesic curvature ; it is the curvature of the two asymptotic
lines passing through M. (So 
).
:
geodesic torsion radius ; 
: geodesic torsion ; it is the torsion of asymptotic lines, and geodesics
passing through M (as well as pseudo-geodesics).
Let 
the angle be between the first principal direction and the tangent to (C);
we have the
Euler formula : 
and the Bonnet formula : 
.
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