Surfaces

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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

H

M

P

S

T

ALGEBRAIC (SURFACE/)

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

BOUR SURFACE

BOTTLE (KLEIN/)

BOY'S SURFACE

CANAL SURFACE

CARTAN'S UMBRELLA

CASSINI SURFACE

CASSINIAN SURFACE

CATALAN SURFACE

CATALAN'S MINIMAL SURFACE

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

CONOID (PARABOLIC/)

CONSTANT SLOPE (SURFACE OF/)

COONS PATCH

COSTA'S SURFACE

CROSSED TROUGH

CURVATURE (SURFACE OF REVOLUTION WITH CONSTANT GAUSSIAN/)

CUBIC SURFACE

CUBIC (RULED/)

CURVATURE LINE

CUSPIDAL EDGE OF A DEVELOPABLE RULED SURFACE

CYCLIDE (DUPIN/)

CYCLOTOMIC SURFACE

CYLINDER (ELLIPTIC/)

CYLINDER (HYPERBOLIC/)

CYLINDER (PARABOLIC/)

CYLINDER OF REVOLUTION

DARBOUX CYCLIDE

DARBOUX SURFACE

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 OF REVOLUTION

ELLIPTIC (CONE/)

ELLIPTIC CYLINDER

ELLIPTIC PARABOLOID

ELLIPTIC POINT OF A SURFACE

ENNEPER SURFACE

ENRIQUES SURFACE

ENVELOPE OF A FAMILY OF SURFACES

EULER CHARACTERISTIC OF A MANIFOLD, OF A SURFACE

FLIPPABLE SURFACE

FLOWER (JEENER'S/)

FLYING SAUCER

FORTUNATUS (GOLD SACK OF/)

FRESNEL'S ELASTICITY SURFACE

FRESNEL'S WAVE SURFACE

GABRIEL'S HORN

GAUDI'S SURFACE

GENUS OF A SURFACE

GEODESIC OF A SURFACE

GERGONNE SURFACE

GOURSAT SURFACE

GREAT CIRCLE

GUIMARD'S SURFACE (HECTOR/)

GUTHRIE'S SOLID

GYROID

HANDLES (SURFACE WITH n/)

HELICO-CONIC SURFACE

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/)

INVERSE OF A SURFACE WITH RESPECT TO A SPHERE

INVOLUTE OF A DEVELOPABLE SURFACE

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

MÖBIUS SHORTS

MÖBIUS STRIP, OR BAND, OR RING

MÖBIUS SURFACE

MOLDING SURFACE

MONGE SURFACE

MONGE SPHERE

MORIN'S SURFACE

NADIRASHVILI SURFACE

NEOVIUS SURFACE

NORMAL SURFACE

OBLATE (ELLIPSOID OF REVOLUTION/)

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/)

PEDAL SURFACE

PILLOW (BOULIGAND'S/)

PINCH POINT

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/)

PROJECTIVE PLANE

PROLATE ELLIPSOID OF REVOLUTION

PRUFER SURFACE

PSEUDOSPHERE

QUADRIC

QUARTIC SURFACE

RATIONAL SURFACE

REVOLUTION (SURFACE OF/)

REVOLUTION OF THE SINUSOID

RICHMOND SURFACE

RIEMANN'S MINIMAL SURFACE

RIEMANN FINITE MINIMAL SURFACE

RIGHT CONOID

RIGHT HELICOID

RING (MÖBIUS/)

ROMAN SURFACE

ROSILLO SURFACE

RULED SURFACE

S²

SACK OF FORTUNATUS (GOLD/)

SADDLE (HORSE/)

SADDLE (MONKEY/)

SCHERK SURFACE

SCHWARZ MINIMAL SURFACES

SCREW (SAINT GILLES/)

SCREW WITH SQUARE THREAD (SURFACE OF THE)

SCREW WITH TRIANGULAR THREAD (SURFACE OF THE/)

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 (n-DIMENSIONAL/)

SPHERIFORM SURFACE

STEINER SURFACE

STRAIGHT DIRECTRIX (SURFACE WITH A/)

STRICTION LINE OF A NON DEVELOPABLE RULED SURFACE

STRIP (MÖBIUS/)

SUM OF TWO SURFACES (CONNECTED/)

SYMMETRY (SURFACE WITH ROTATIONAL/)

SYSTEM (TRIPLE ORTHOGONAL)

TAKAGI (MOUNT/)

TANNERY'S PEAR

TETRAHEDRAL SURFACE (KUMMER)

TIGHT SURFACE

THALWEG LINE

TITEICA SURFACE

T_n

TOPOGRAPHIC LINE

TORUS (GEOMETRIC NOTION)

TORU (TOPOLOGICAL TORUS)

TORUS (n-DIMENSIONAL/)

TORUS WITH n HANDLES

TORUS (CLIFFORD'S/)

TORUS (KLEIN/)

TORUS (SINE/)

TOWER WITH CONSTANT PRESSURE

TRACTROID (SECOND/)

TRANSCENDENTAL SURFACE

TRANSLATION SURFACE

TRICKLE OF WATER (SURFACE OF THE/)

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.
: 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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