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SURFACES

Translated from French by David Michel
VERSION FRANCAISE
See the notations below.

Surfaces starting with

A
B
C
D
E
F G
H
I  J  K  L
M
N O
P
Q R
S
T
U  V  W  X  Y  Z

ALGEBRAIC (SURFACE/)

ALYSSEID

ANALLAGMATIC (SURFACE/)

APPLICABLE (SURFACE/)

APSIS (SURFACE/)

ASTROIDAL (ELLIPSOID/)

ASYMPTOTIC (LINE/)

ASYMPTOTIC PLANE OF A GENERATRIX OF A RULED SURFACE

ATTRACTION (SURFACE OF EQUAL/)

ATTRACTION (SOLID OF MAXIMAL/)

AXOID

BALL (RUGBY/)

BAND (MÖBIUS/)

BARTH SEXTIC

BELTRAMI SURFACE

BEZIER (SURFACE/)

BISPHERICAL ALGEBRAIC SURFACE

BOHEMIAN DOME

BOX (EGG/)

BOUR SURFACE

BOTTLE (KLEIN/)

BOY'S SURFACE

CANAL SURFACE

CARTAN'S UMBRELLA

CASSINI SURFACE

CASSINIAN SURFACE

CATALAN SURFACE

CATALAN'S MINIMAL SURFACE

CATENOID

CATENOID (SKEW/)

CAUSTIC SURFACE

CAYLEY SURFACE

CAYLEY CUBIC RULED SURFACE

CENTRAL POINT OF A GENERATRIX OF A RULED SURFACE

CHARACTERISTIC OF A MANIFOLD, OF A SURFACE (EULER/)

CIRCLED SURFACE

CHROMATIC NUMBER OF A SURFACE

CIRCULAR CONE, CYLINDER

CLEBSCH CUBIC SURFACE

CLIFFORD'S TORUS

COIL

CONE OR CONICAL SURFACE

CONE (ELLIPTIC/)

CONE OF REVOLUTION

CONE (SINUSOIDAL/)

CONICAL POINT

CONICAL WEDGE

CONOCUNEUS

CONOID

CONOID (PARABOLIC/)

CONSTANT SLOPE (SURFACE OF/)

COONS PATCH

CORNUCOPIA

COSTA'S SURFACE

CURVATURE (SURFACE OF REVOLUTION WITH CONSTANT TOTAL/)

CROSS-CAP

CUBIC SURFACE

CUBIC (RULED/)

CURVATURE LINE

CUSHION (BOULIGAND'S/)

CUSPIDAL EDGE OF A DEVELOPABLE RULED SURFACE

CYCLIDE

CYCLIDE (DUPIN/)

CYCLOTOMIC SURFACE

CYLINDER

CYLINDER (ELLIPTIC/)

CYLINDER (HYPERBOLIC/)

CYLINDER (PARABOLIC/)

CYLINDER OF REVOLUTION

DARBOUX CYCLIDE

DARBOUX SURFACE

DEFERENT

DELAUNAY SURFACE

DEVELOPABLE SURFACE

DEVELOPABLE (TANGENT/)

DEVELOPABLE SURFACE (INVOLUTE OF A/)

DINI'S SURFACE

DOME (BOHEMIAN/)

DOUBLE SIX (SCHÄFLI'S/)

DRESSING OF THE SPHERE

DROP OF WATER (HANGING)

DUPIN CYCLIDE

DUPIN INDICATRIX

DYCK'S SURFACE

EDGE (CUSPIDAL/)

EIGHT (THICKENED FIGURE-/)

ELASTICITY SURFACE (FRESNEL'S)

ELLIPSOID

ELLIPTIC (CONE/)

ELLIPTIC CYLINDER

ELLIPTIC PARABOLOID

ELLIPTIC POINT OF A SURFACE

EMBANKMENT

ENNEPER SURFACE

ENRIQUES SURFACE

ENVELOPE OF A FAMILY OF SURFACES

EULER CHARACTERISTIC OF A MANIFOLD, OF A SURFACE

FLOWER (JEENER'S/)

FLYING SAUCER

FOCAL

FORTUNATUS (GOLD SACK OF/)

FRESNEL'S ELASTICITY SURFACE

FRESNEL'S WAVE SURFACE

FUNNEL

GABRIEL'S HORN

GAUDI'S SURFACE

GENUS OF A SURFACE

GEOID

GEODESIC OF A SURFACE

GERGONNE SURFACE

GÖMBÖC

GOURSAT SURFACE

GREAT CIRCLE

GUIMARD'S SURFACE (HECTOR/)

GUTHRIE'S SOLID

GYROID

HACHETTE'S CONE

HANDLES (SURFACE WITH n/)

HELICO-CONIC SURFACE

HELICOID

HELICOID (CIRCLED/)

HELICOID (DEVELOPABLE/)

HELICOID (RIGHT/)

HELICOID (MINIMAL/)

HELICOID (RULED/)

HENNEBERG'S SURFACE

HORN (GABRIEL'S)

HYPERBOLIC (PARABOLOID/)

HYPERBOLIC POINT OF A SURFACE

HYPERBOLOID
    OF ONE SHEET (H1)
    OF TWO SHEETS (H2)

HYPERSPHERE (3-DIMENSIONAL/, n DIMENSIONAL/)

HYPERTORURS

INDICATRIX (DUPIN/)

INNER TUBE

INVERSE OF A SURFACE WITH RESPECT TO A SPHERE

INVOLUTE OF A DEVELOPABLE SURFACE

ISOHYPSE

ISOMETRIC TO ANOTHER SURFACE (SURFACE/)

JEENER'S FLOWER

KLEIN'S CONE

KLEIN BOTTLE

KUEN'S SURFACE

KUMMER SURFACE

LAMÉ SURFACE

LEVEL CURVE OR LINE

LINE TRACED ON A SURFACE

LINE
    CURVATURE/, ASYMPTOTIC/, GEODESIC/

LINE (TOPOGRAPHIC/):
    LEVEL/, SLOPE/, THALWEG/, APEX/,

LINE (FLOW/)

LIOUVILLE SURFACE

MANIFOLD (TOPOLOGICAL/, DIFFERENTIAL/, ALGEBRAIC/)

MANIFOLD (3-DIMENSIONAL)

MERIDIAN OF A SURFACE OF REVOLUTION

MEUSNIER HELICOID

MILK CARTON

MINIMAL SURFACE

MITRE

MÖBIUS SHORTS

MÖBIUS STRIP, OR BAND, OR RING

MÖBIUS SURFACE

MOLDING SURFACE

MONGE SURFACE

MONGE SPHERE

MORIN'S SURFACE

NADIRASHVILI SURFACE

NEOVIUS SURFACE

NEILOID

N-NOID

NODOID

NORMAL SURFACE

OBLATE (ELLIPSOID OF REVOLUTION/)

OLOID

ONDULOID

ONE-SIDED SURFACE

ORIENTABLE SURFACE

ORTHOBICYCLE

ORTHOPTIC SURFACE

OUTLINE (VISIBLE/)

OVOID

PAPER LANTERN (SCHWARZ/)

PATCH (COONS/)

PARABOLIC (POINT / D'UNE SURFACE)

PARABOLOID ELLIPTIC

PARABOLOID HYPERBOLIC

PARABOLOID DE REVOLUTION

PARALLEL TO ANOTHER SURFACE (SURFACE/)

PEAR (TANNERY'S/)

PEBBLE

PEDAL SURFACE

PINCH POINT

PLANE

PLANE (REAL PROJECTIVE/)

PLANAR POINT OF A SURFACE

PLÜCKER'S CONOID

POLAR DEVELOPABLE OF A SKEW CURVE

POLAR OF A SURFACE, OF A CURVE, WITH RESPECT TO A SPHERE (RECIPROCAL/)

PRESSURE (TOWER WITH CONSTANT/)

PRETZEL

PROJECTIVE PLANE

PROLATE ELLIPSOID OF REVOLUTION

PRUFER SURFACE

PSEUDOSPHERE

QUADRIC

QUARTIC SURFACE

RATIONAL SURFACE

FLIPPABLE SURFACE

REVOLUTION (SURFACE OF/)

REVOLUTION OF THE SINUSOID

RICHMOND SURFACE

RIDGE LINE

RIEMANN'S MINIMAL SURFACE

RIEMANN FINITE MINIMAL SURFACE

RIGHT CONOID

RIGHT HELICOID

RING (MÖBIUS/)

ROTOID

ROMAN SURFACE

ROSILLO SURFACE

RUGBY BALL

RULED SURFACE

S2

S3

Sn

SACK OF FORTUNATUS (GOLD/)

SAUSAGE

SADDLE (HORSE/)

SADDLE (MONKEY/)

SANDPILE

SCHERK SURFACE

SCHWARZ MINIMAL SURFACES

SCREW (SAINT GILLES/)

SCREW WITH SQUARE THREAD (SURFACE OF THE)

SCREW WITH TRIANGULAR THREAD (SURFACE OF THE/)

SEA-SHELL

SEIFERT SURFACE

SEXTIC (BARTH/)

SHORTS (MÖBIUS/)

SIEVERT'S SURFACE

SINGULARITIES

SINE SURFACE

SLOPE LINE, OR OF GREATEST SLOPE

SLOPE (SURFACE OF CONSTANT/)

SMOOTH SURFACE

SPACE (3-DIMENSIONAL)

SPHERE

SPHERE (n-DIMENSIONAL/)

SPHERIFORM SURFACE

SPHEROID

STEINER SURFACE

STRICTION LINE OF A NON DEVELOPABLE RULED SURFACE

STRIP (MÖBIUS/)

SUM OF TWO SURFACES (CONNECTED/)

SURFACE

SYMMETRY (SURFACE WITH ROTATIONAL/)

SYSTEM (TRIPLE ORTHOGONAL)

TAKAGI (MOUNT/)

TANNERY'S PEAR

TETRAHEDRAL SURFACE (KUMMER)

THALWEG LINE

TITEICA SURFACE

Tn

Tn

TOPOGRAPHIC LINE

TORTI

TORUS (GEOMETRIC NOTION)

TORU (TOPOLOGICAL TORUS)

TORUS (n-DIMENSIONAL/)

TORUS WITH n HANDLES

TORUS (CLIFFORD'S/)

TORUS (KLEIN/)

TORUS (SINE/)

TOWER WITH CONSTANT PRESSURE

TRANSCENDENTAL SURFACE

TRANSLATION SURFACE

TRICKLE OF WATER (SURFACE OF THE/)

TRINOID

TRIPLE ORTHOGONAL SYSTEM OF SURFACES

TUBE or TUBULAR SURFACE

UMBILIC

UMBRELLA (CARTAN'S/)

UMBRELLA (WHITNEY'S)

VERONESE SURFACE

WATER DIVIDE

WAVE SURFACE (FRESNEL'S)

WEINGARTEN SURFACE

WENTE TORUS

WIDTH (SURFACE OF CONSTANT/)

WILLMORE SURFACE, TORUS

WHITNEY'S UMBRELLA

ZINDLER'S CONOID

NOTATIONS

(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.

: total (or Gaussian) curvature at M.

: mean curvature at M.

: normal curvature at M of a curve traced on the surface, equal to the curvature in the direction of the tangent to the curve at M (j is the angle between the tangent and the first principal direction)

: geodesic curvature at M of a curve traced on the surface, equal to the norm of the projection of on the tangent plane of the surface ( is the curvature of the curve and  is the normal vector).

: geodesic torsion at M of a curve traced on the surface; it is the torsion of the geodesic passing by M in the same direction as the curve.
 
 
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