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CONE OF REVOLUTION
- give honor where honor is due: the planar sections or conics: ellipses, parabolas and hyperbolas.They can be developped into the polygasteroids of index n >1. Cutting by the plane gives an ellipse, a parabola, or a hyperbola, depending on whether is greater, equal, or less than ; if the cone rolls over a plane, it is developped into the polygasteroid with polar equation: .ENNEPER SURFACE
It can be geometrically defined as the envelope of the mediatrix planes of two points located on two homofocal parabolas (i.e. parabolas the planes of which are perpendicular and such that the vertex of one passes by the focus of the other one; compare to the definition of the symmetric parabolic Dupin cyclide).. As is the configuration of homofocal parabolas, the Enneper surface is invariant CATENOID - MATHCURVE.COM The catenoid is the surface of revolution generated by the rotation of a catenary around its base. Since the mean curvature is zero at all points, it is a minimal surface; for that matter, it is the only minimal surface of revolution.It is also the only minimal surface with a circle as a geodesic.. We get the parametrization by taking and in the Weierstrass parametrization of a minimal surface.CUBIC SURFACE
A property that has really excited mathematicians is that any smooth complex surface of degree 3 contains exactly 27 straight lines (whereas any smooth surface of degree 2 is a ruled surface, and a smooth projective surface of degree greater than or equal to 4 may not contain any lines).. In real projective geometry, this Salmon-Cayley theorem becomes: any projective smooth surface of degree 3 DOUBLE SIX - MATHCURVE.COM There exists a unique smooth cubic surface (S) containing the 12 lines of the double six.. For the double six represented above, where the vertices of the chosen cube are etc, and the line passes by , the equation of the cubic surface is: .. The 27 lines of this cubic are the 12 lines of the double six (in red and blue opposite) plus the 15 lines (in yellow) defined above. TREFOIL - MATHCURVE.COM The trefoil knot is the only knot for which the crossing number is minimal, namely 3; there are in fact two of them, they are enantiomorphic (images of one another by reflection).. It is an overhand knot: the ends of which were connected.. The trefoil knot is also the torus knot of type (3,2) (3 coils around the torus, on two turns), as well as that of type (2,3): SPHEROID - MATHCURVE.COM The oblate ellipsoid of revolution is the surface of revolution obtained by rotating the ellipse around its minor axis, having the shape of a pebble or a flying saucer, or also a go stone. NEILOID - MATHCURVE.COM It is used in dendrometry to model the base of a tree trunk.: United States Forest Service (USDA FS)STRICTION LINE
Parametrization: , for the ruled surface represented by ; If we normalize (), the formula simplifies to and then M 1 describes the striction line iff .. The distribution parameter: can be geometrically interpreted by the formula , where is the angle between the tangent plane at C and the tangent plane at M; in an equivalent fashion, the surface generated by the normals along the generatrix isBICYLINDRIQUE
Dans le cas de 2 cylindres orthogonaux de rayons a et b, et d'axes distants de 2c: Système d’équations cartésiennes :. Biquadratique. Paramétrisation cartésienne : . Equation cartésienne de la projection sur xOy : (voir la courbe d'Alain). Pour a £ b et c = 0 : - Aire de la portion de cylindre délimitée par chaque composante, donnée par une intégrale elliptique de deuxièmeCONE OF REVOLUTION
- give honor where honor is due: the planar sections or conics: ellipses, parabolas and hyperbolas.They can be developped into the polygasteroids of index n >1. Cutting by the plane gives an ellipse, a parabola, or a hyperbola, depending on whether is greater, equal, or less than ; if the cone rolls over a plane, it is developped into the polygasteroid with polar equation: .ENNEPER SURFACE
It can be geometrically defined as the envelope of the mediatrix planes of two points located on two homofocal parabolas (i.e. parabolas the planes of which are perpendicular and such that the vertex of one passes by the focus of the other one; compare to the definition of the symmetric parabolic Dupin cyclide).. As is the configuration of homofocal parabolas, the Enneper surface is invariant CATENOID - MATHCURVE.COM The catenoid is the surface of revolution generated by the rotation of a catenary around its base. Since the mean curvature is zero at all points, it is a minimal surface; for that matter, it is the only minimal surface of revolution.It is also the only minimal surface with a circle as a geodesic.. We get the parametrization by taking and in the Weierstrass parametrization of a minimal surface.CUBIC SURFACE
A property that has really excited mathematicians is that any smooth complex surface of degree 3 contains exactly 27 straight lines (whereas any smooth surface of degree 2 is a ruled surface, and a smooth projective surface of degree greater than or equal to 4 may not contain any lines).. In real projective geometry, this Salmon-Cayley theorem becomes: any projective smooth surface of degree 3 DOUBLE SIX - MATHCURVE.COM There exists a unique smooth cubic surface (S) containing the 12 lines of the double six.. For the double six represented above, where the vertices of the chosen cube are etc, and the line passes by , the equation of the cubic surface is: .. The 27 lines of this cubic are the 12 lines of the double six (in red and blue opposite) plus the 15 lines (in yellow) defined above. TREFOIL - MATHCURVE.COM The trefoil knot is the only knot for which the crossing number is minimal, namely 3; there are in fact two of them, they are enantiomorphic (images of one another by reflection).. It is an overhand knot: the ends of which were connected.. The trefoil knot is also the torus knot of type (3,2) (3 coils around the torus, on two turns), as well as that of type (2,3): SPHEROID - MATHCURVE.COM The oblate ellipsoid of revolution is the surface of revolution obtained by rotating the ellipse around its minor axis, having the shape of a pebble or a flying saucer, or also a go stone. NEILOID - MATHCURVE.COM It is used in dendrometry to model the base of a tree trunk.: United States Forest Service (USDA FS)STRICTION LINE
Parametrization: , for the ruled surface represented by ; If we normalize (), the formula simplifies to and then M 1 describes the striction line iff .. The distribution parameter: can be geometrically interpreted by the formula , where is the angle between the tangent plane at C and the tangent plane at M; in an equivalent fashion, the surface generated by the normals along the generatrix isBICYLINDRIQUE
Dans le cas de 2 cylindres orthogonaux de rayons a et b, et d'axes distants de 2c: Système d’équations cartésiennes :. Biquadratique. Paramétrisation cartésienne : . Equation cartésienne de la projection sur xOy : (voir la courbe d'Alain). Pour a £ b et c = 0 : - Aire de la portion de cylindre délimitée par chaque composante, donnée par une intégrale elliptique de deuxième MATHCURVE.COMTRANSLATE THIS PAGE page d'accueil de l'encyclopédie des formes remarquables ;home page of the encyclopaedia of curves, surfaces, fractals and polyhedrons TREFOIL - MATHCURVE.COM The trefoil knot is the only knot for which the crossing number is minimal, namely 3; there are in fact two of them, they are enantiomorphic (images of one another by reflection).. It is an overhand knot: the ends of which were connected.. The trefoil knot is also the torus knot of type (3,2) (3 coils around the torus, on two turns), as well as that of type (2,3):CUBIC SURFACE
A property that has really excited mathematicians is that any smooth complex surface of degree 3 contains exactly 27 straight lines (whereas any smooth surface of degree 2 is a ruled surface, and a smooth projective surface of degree greater than or equal to 4 may not contain any lines).. In real projective geometry, this Salmon-Cayley theorem becomes: any projective smooth surface of degree 3 SPHEROID - MATHCURVE.COM The oblate ellipsoid of revolution is the surface of revolution obtained by rotating the ellipse around its minor axis, having the shape of a pebble or a flying saucer, or also a go stone.MÖBIUS STRIP
Here is already a Möbius strip with one half-twist for which the ratio length/width is equal to 3Ö3 (slightly greater in fact for the sake of clarity). The view from above is a regular hexagon. The initial pattern is indicated on the right (the 3 folds are dotted).SCHWARZ SURFACE
The Schwarz "D" (for diamond) minimal surface is the triply periodic minimal surface the fundamental patch of which is the solution of the Plateau problem for a contour that is one of the 6 skew quadrilaterals made from a skew hexagon inscribed in the edges of a cube, as represented opposite.. The Schwarz "P" (for primitive) minimal surface is the surface built similarly to the previous oneBICYLINDRIQUE
Dans le cas de 2 cylindres orthogonaux de rayons a et b, et d'axes distants de 2c: Système d’équations cartésiennes :. Biquadratique. Paramétrisation cartésienne : . Equation cartésienne de la projection sur xOy : (voir la courbe d'Alain). Pour a £ b et c = 0 : - Aire de la portion de cylindre délimitée par chaque composante, donnée par une intégrale elliptique de deuxièmeMONKEY SADDLE
Surface shaped like a saddle that allows to place a monkey's legs but also its tail (the monkey is not the mount, but the rider!). By the point O (which is a planar point of the surface) there pass 3 real lines of the surface, forming between one another angles of 120° (in red above): the point O is an Eckardt point of the surface; by the point at infinity of Oz, which is a singular point ofSTRICTION LINE
Parametrization: , for the ruled surface represented by ; If we normalize (), the formula simplifies to and then M 1 describes the striction line iff .. The distribution parameter: can be geometrically interpreted by the formula , where is the angle between the tangent plane at C and the tangent plane at M; in an equivalent fashion, the surface generated by the normals along the generatrix isVERONESE SURFACE
The Veronese surface is the image of the quotient of the 2-dimensional sphere by the antipodal relation (in other words, the real projective plane), by the map: .. Since this function is injective, the Veronese surface is a surface (i.e. a 2-manifold) without singular points embedded in (since it is included in the hyperplane of ) and homeomorphic to the real projective plane. MATHCURVE.COMTRANSLATE THIS PAGEBIBLIOGRAPHIECOURBES 3DFRACTALSPOLYÈDRESCOURBE DE LA CRÊPECOURBE SOLÉNOÏDALE page d'accueil de l'encyclopédie des formes remarquables ;home page of the encyclopaedia of curves, surfaces, fractals and polyhedrons CATENOID - MATHCURVE.COM The catenoid is the surface of revolution generated by the rotation of a catenary around its base. Since the mean curvature is zero at all points, it is a minimal surface; for that matter, it is the only minimal surface of revolution.It is also the only minimal surface with a circle as a geodesic.. We get the parametrization by taking and in the Weierstrass parametrization of a minimal surface.CONE OF REVOLUTION
The cone of revolution is the surface generated by the revolution of a line secant to an axis, around this axis; it is a special case of elliptic cone. Writing the colatitude, spherical equation: , the axis being Oz and the half-angle at the vertex . Cylindrical equation: . Cartesian equation: . SPHEROID - MATHCURVE.COM Volume : . The oblate ellipsoid of revolution is the surface of revolution obtained by rotating the ellipse around its minor axis, having the shape of a pebble or a flying saucer. Opposite, views of a closed geodesic of the oblate ellipsoid, whose top view forms a crossed octahedron. Differential system whose solutions give thesegeodesics:
MÖBIUS STRIP
A Möbius strip is a surface obtained by sewing together two sides of a rectangular strip with a half-twist, or any topologically equivalent surface. Maple program giving an animation of the opposite construction. the boundary of the strip with 2p + 1 half-twists is a toroidal knot of order (2p + 1, 2). OLOID - MATHCURVE.COM The oloid is the convex hull of two orthogonal circles each passing by the center of the other. Its surface is a part of the developable surface supported on the two circles (proof: H. POTTMANN, J. WALLNER : Computational Line Geometry, Springer-Verlag Telos (2001) p. 405)N -HOLED TORUS
n-HOLED TORUS. The notion of n-holed torus, or n-torus, or n-uple torus, or sphere with n handles, refers to any topological space homeomorphic to the connected sum of the simple torus n times with itself: ; by convention, we set . Any orientable connected compact surface without boundary is homeomorphic to an n-torus. The Euler characteristic of the n-torus is equal to 2–2n.VIVIANI CURVE
The Viviani curve is the intersection between a sphere with radius R (here, ) and a cylinder of revolution with diameter R a generatrix of which passes by the center of the sphere (here, ); it is therefore a special case of hippopede, a curve that is at the same time spherical and cylindrical, as well as a special case of conical rose.. Therefore, we get a Viviani curve by sticking the tip ofCONICAL HELIX
The conical helix can be defined as a helix traced on a cone of revolution (i.e. a curve forming a constant angle with respect to the axis of the cone), or a rhumb line of this cone (i.e. a curve forming a constant angle with the meridians); it is not a geodesic of the cone. In concrete terms, we get a conical helix when we trace a path with constant slope on a cone placed vertically. EULER CHARACTERISTIC EULER CHARACTERISTIC OF A SURFACE CHROMATIC NUMBER OF A SURFACE. A cell decomposition of a finite type manifold of dimension n (i.e. a topological space locally homeomorphic to a closed ball of and that is the finite union of parts homeomorphic to such balls) is a finite partition of this surface is subsets homeomorphic to for p £ n. If we write c p the number of subsets homeomorphic to , it MATHCURVE.COMTRANSLATE THIS PAGEBIBLIOGRAPHIECOURBES 3DFRACTALSPOLYÈDRESCOURBE DE LA CRÊPECOURBE SOLÉNOÏDALE page d'accueil de l'encyclopédie des formes remarquables ;home page of the encyclopaedia of curves, surfaces, fractals and polyhedrons CATENOID - MATHCURVE.COM The catenoid is the surface of revolution generated by the rotation of a catenary around its base. Since the mean curvature is zero at all points, it is a minimal surface; for that matter, it is the only minimal surface of revolution.It is also the only minimal surface with a circle as a geodesic.. We get the parametrization by taking and in the Weierstrass parametrization of a minimal surface.CONE OF REVOLUTION
The cone of revolution is the surface generated by the revolution of a line secant to an axis, around this axis; it is a special case of elliptic cone. Writing the colatitude, spherical equation: , the axis being Oz and the half-angle at the vertex . Cylindrical equation: . Cartesian equation: . SPHEROID - MATHCURVE.COM Volume : . The oblate ellipsoid of revolution is the surface of revolution obtained by rotating the ellipse around its minor axis, having the shape of a pebble or a flying saucer. Opposite, views of a closed geodesic of the oblate ellipsoid, whose top view forms a crossed octahedron. Differential system whose solutions give thesegeodesics:
MÖBIUS STRIP
A Möbius strip is a surface obtained by sewing together two sides of a rectangular strip with a half-twist, or any topologically equivalent surface. Maple program giving an animation of the opposite construction. the boundary of the strip with 2p + 1 half-twists is a toroidal knot of order (2p + 1, 2). OLOID - MATHCURVE.COM The oloid is the convex hull of two orthogonal circles each passing by the center of the other. Its surface is a part of the developable surface supported on the two circles (proof: H. POTTMANN, J. WALLNER : Computational Line Geometry, Springer-Verlag Telos (2001) p. 405)N -HOLED TORUS
n-HOLED TORUS. The notion of n-holed torus, or n-torus, or n-uple torus, or sphere with n handles, refers to any topological space homeomorphic to the connected sum of the simple torus n times with itself: ; by convention, we set . Any orientable connected compact surface without boundary is homeomorphic to an n-torus. The Euler characteristic of the n-torus is equal to 2–2n.VIVIANI CURVE
The Viviani curve is the intersection between a sphere with radius R (here, ) and a cylinder of revolution with diameter R a generatrix of which passes by the center of the sphere (here, ); it is therefore a special case of hippopede, a curve that is at the same time spherical and cylindrical, as well as a special case of conical rose.. Therefore, we get a Viviani curve by sticking the tip ofCONICAL HELIX
The conical helix can be defined as a helix traced on a cone of revolution (i.e. a curve forming a constant angle with respect to the axis of the cone), or a rhumb line of this cone (i.e. a curve forming a constant angle with the meridians); it is not a geodesic of the cone. In concrete terms, we get a conical helix when we trace a path with constant slope on a cone placed vertically. EULER CHARACTERISTIC EULER CHARACTERISTIC OF A SURFACE CHROMATIC NUMBER OF A SURFACE. A cell decomposition of a finite type manifold of dimension n (i.e. a topological space locally homeomorphic to a closed ball of and that is the finite union of parts homeomorphic to such balls) is a finite partition of this surface is subsets homeomorphic to for p £ n. If we write c p the number of subsets homeomorphic to , itCONE OF REVOLUTION
The cone of revolution is the surface generated by the revolution of a line secant to an axis, around this axis; it is a special case of elliptic cone. Writing the colatitude, spherical equation: , the axis being Oz and the half-angle at the vertex . Cylindrical equation: . Cartesian equation: .ENNEPER SURFACE
It can be geometrically defined as the envelope of the mediatrix planes of two points located on two homofocal parabolas (i.e. parabolas the planes of which are perpendicular and such that the vertex of one passes by the focus of the other one; compare to the definition of the symmetric parabolic Dupin cyclide).. As is the configuration of homofocal parabolas, the Enneper surface is invariantSCHWARZ SURFACE
The Schwarz "D" (for diamond) minimal surface is the triply periodic minimal surface the fundamental patch of which is the solution of the Plateau problem for a contour that is one of the 6 skew quadrilaterals made from a skew hexagon inscribed in the edges of a cube, as represented opposite.. The Schwarz "P" (for primitive) minimal surface is the surface built similarly to the previous one DOUBLE SIX - MATHCURVE.COM There exists a unique smooth cubic surface (S) containing the 12 lines of the double six.. For the double six represented above, where the vertices of the chosen cube are etc, and the line passes by , the equation of the cubic surface is: .. The 27 lines of this cubic are the 12 lines of the double six (in red and blue opposite) plus the 15 lines (in yellow) defined above.FENÊTRE DE VIVIANI
La courbe de Viviani est l’intersection d'une sphère de rayon R (ici, ) et d'un cylindre de révolution de diamètre R dont une génératrice passe par le centre de la sphère (ici, ) ; c’est donc un cas particulier d’hippopède, une courbe à la fois sphérique et cylindrique, ainsi qu'un cas particulier de rosace conique.. On obtient donc une fenêtre de Viviani en plantant la pointeENSEMBLE DE JULIA
À toute fonction f entière du plan complexe dans lui-même et tout point de départ est associée la suite des itérés successifs de par f.. Le plan est alors partagé entre l'ensemble des point pour lesquels la suite est bornée (les prisonniers de f), et les autres (les fugitifs).. L'ensemble de Julia associé à f est alors la frontière commune de ces deux ensembles ; c'est, autrementTÉTRAÈDRE
TÉTRAÈDRE. Tetrahedron, Tetraeder. Du grec "tetra" quatre et "edros" siège, base. Un tetraèdre est un polyèdre à 4 faces (ou 4 sommets), nombre minimal possible ; il n'en existe qu'un seul type, équivalent au tétraèdre régulier dont voici la carte de visite : Famille. polyèdres réguliers.KLEIN BOTTLE
The Möbius strip is a one-sided surface (with one face), therefore is not orientable, of genus 2, zero Euler characteristic, and chromatic number equal to 6 like the torus:(and not 7 like the Heawood formula would indicate) : In this 6 country map traced on the Klein bottle, each country meets the 5 others; 6 is the maximum number possible, and any map can be colored with no more than 6 colors.DODÉCAÈDRE
Un dodécaèdre est un polyèdre à 12 faces. Le plus célèbre est le dodécaèdre régulier, ou dodécaèdre pentagonal (20 sommets), dont on trouvera ci-dessous la carte de visite ; mais il y a aussi le dodécaèdre rhombique (14 sommets), le triaki-tétraèdre (8 sommets) et le dodécadeltaèdre (8 sommets).STRICTION LINE
Parametrization: , for the ruled surface represented by ; If we normalize (), the formula simplifies to and then M 1 describes the striction line iff .. The distribution parameter: can be geometrically interpreted by the formula , where is the angle between the tangent plane at C and the tangent plane at M; in an equivalent fashion, the surface generated by the normals along the generatrix is MATHCURVE.COMBIBLIOGRAPHIECOURBES 3DFRACTALSPOLYÈDRESCOURBE DE LA CRÊPECOURBE SOLÉNOÏDALE page d'accueil de l'encyclopédie des formes remarquables ;home page of the encyclopaedia of curves, surfaces, fractals and polyhedronsCONE OF REVOLUTION
The cone of revolution is the surface generated by the revolution of a line secant to an axis, around this axis; it is a special case of elliptic cone. Writing the colatitude, spherical equation: , the axis being Oz and the half-angle at the vertex . Cylindrical equation: . Cartesian equation: .GUIMARD'S SURFACE
Hector Guimard's surface is the ruled surface union of the lines indicated above, when the point M has a linear sinusoidal movement and the point N a circular sinusoidal movement with two arches, and is therefore describing the pancake curve.. View showing the various segment lines The back of the circular part of this canopy of the Parisian underground, due to Hector Guimard, is aGAUDI'S SURFACE
Gaudi's surface is the right conoid the directrix of which is a sinusoid located in a plane parallel to the axis (here, the directrix plane is yOz and the axis y = z = 0 ). We named it this way because it was used by Gaudi for the roof of the "escoles" of the Sagrada familia DOUBLE SIX - MATHCURVE.COM There exists a unique smooth cubic surface (S) containing the 12 lines of the double six.. For the double six represented above, where the vertices of the chosen cube are etc, and the line passes by , the equation of the cubic surface is: .. The 27 lines of this cubic are the 12 lines of the double six (in red and blue opposite) plus the 15 lines (in yellow) defined above.ENNEPER SURFACE
It can be geometrically defined as the envelope of the mediatrix planes of two points located on two homofocal parabolas (i.e. parabolas the planes of which are perpendicular and such that the vertex of one passes by the focus of the other one; compare to the definition of the symmetric parabolic Dupin cyclide).. As is the configuration of homofocal parabolas, the Enneper surface is invariant SURFACE OF CONSTANT WIDTH Given a compact convex K of the space, define its width in the direction of the line D as the length of the orthogonal projection of K on D, length which also is the smallest width of a strip orthogonal to D and containing K, or, which amounts to the same thing, the distance between two supporting planes of the convex K that are orthogonal to D.. The convex K is then said to be "of constantN -HOLED TORUS
n-HOLED TORUS. The notion of n-holed torus, or n-torus, or n-uple torus, or sphere with n handles, refers to any topological space homeomorphic to the connected sum of the simple torus n times with itself: ; by convention, we set . Any orientable connected compact surface without boundary is homeomorphic to an n-torus. The Euler characteristic of the n-torus is equal to 2–2n.STRICTION LINE
Parametrization: , for the ruled surface represented by ; If we normalize (), the formula simplifies to and then M 1 describes the striction line iff .. The distribution parameter: can be geometrically interpreted by the formula , where is the angle between the tangent plane at C and the tangent plane at M; in an equivalent fashion, the surface generated by the normals along the generatrix is SPIRALE CONIQUE DE PAPPUS La spirale conique de Pappus est la trajectoire d'un point se déplaçant uniformément sur une droite passant par un point O, cette droite tournant uniformément autour d'un axe Oz en conservant un angle a avec Oz. Elle est donc intersection du cône de révolution (C) : avec l’hélicoïde droit: . Si l’on développe le cône (C) sur un plan, le point M devenant le point de coordonnées MATHCURVE.COMBIBLIOGRAPHIECOURBES 3DFRACTALSPOLYÈDRESCOURBE DE LA CRÊPECOURBE SOLÉNOÏDALE page d'accueil de l'encyclopédie des formes remarquables ;home page of the encyclopaedia of curves, surfaces, fractals and polyhedronsCONE OF REVOLUTION
The cone of revolution is the surface generated by the revolution of a line secant to an axis, around this axis; it is a special case of elliptic cone. Writing the colatitude, spherical equation: , the axis being Oz and the half-angle at the vertex . Cylindrical equation: . Cartesian equation: .GUIMARD'S SURFACE
Hector Guimard's surface is the ruled surface union of the lines indicated above, when the point M has a linear sinusoidal movement and the point N a circular sinusoidal movement with two arches, and is therefore describing the pancake curve.. View showing the various segment lines The back of the circular part of this canopy of the Parisian underground, due to Hector Guimard, is aGAUDI'S SURFACE
Gaudi's surface is the right conoid the directrix of which is a sinusoid located in a plane parallel to the axis (here, the directrix plane is yOz and the axis y = z = 0 ). We named it this way because it was used by Gaudi for the roof of the "escoles" of the Sagrada familia DOUBLE SIX - MATHCURVE.COM There exists a unique smooth cubic surface (S) containing the 12 lines of the double six.. For the double six represented above, where the vertices of the chosen cube are etc, and the line passes by , the equation of the cubic surface is: .. The 27 lines of this cubic are the 12 lines of the double six (in red and blue opposite) plus the 15 lines (in yellow) defined above.ENNEPER SURFACE
It can be geometrically defined as the envelope of the mediatrix planes of two points located on two homofocal parabolas (i.e. parabolas the planes of which are perpendicular and such that the vertex of one passes by the focus of the other one; compare to the definition of the symmetric parabolic Dupin cyclide).. As is the configuration of homofocal parabolas, the Enneper surface is invariant SURFACE OF CONSTANT WIDTH Given a compact convex K of the space, define its width in the direction of the line D as the length of the orthogonal projection of K on D, length which also is the smallest width of a strip orthogonal to D and containing K, or, which amounts to the same thing, the distance between two supporting planes of the convex K that are orthogonal to D.. The convex K is then said to be "of constantN -HOLED TORUS
n-HOLED TORUS. The notion of n-holed torus, or n-torus, or n-uple torus, or sphere with n handles, refers to any topological space homeomorphic to the connected sum of the simple torus n times with itself: ; by convention, we set . Any orientable connected compact surface without boundary is homeomorphic to an n-torus. The Euler characteristic of the n-torus is equal to 2–2n.STRICTION LINE
Parametrization: , for the ruled surface represented by ; If we normalize (), the formula simplifies to and then M 1 describes the striction line iff .. The distribution parameter: can be geometrically interpreted by the formula , where is the angle between the tangent plane at C and the tangent plane at M; in an equivalent fashion, the surface generated by the normals along the generatrix is SPIRALE CONIQUE DE PAPPUS La spirale conique de Pappus est la trajectoire d'un point se déplaçant uniformément sur une droite passant par un point O, cette droite tournant uniformément autour d'un axe Oz en conservant un angle a avec Oz. Elle est donc intersection du cône de révolution (C) : avec l’hélicoïde droit: . Si l’on développe le cône (C) sur un plan, le point M devenant le point de coordonnéesENNEPER SURFACE
It can be geometrically defined as the envelope of the mediatrix planes of two points located on two homofocal parabolas (i.e. parabolas the planes of which are perpendicular and such that the vertex of one passes by the focus of the other one; compare to the definition of the symmetric parabolic Dupin cyclide).. As is the configuration of homofocal parabolas, the Enneper surface is invariantASTROIDAL ELLIPSOID
The astroidal ellipsoid is the surface with the above equation. Its name comes from the fact that its sections by planes parallel to the axes are astroids.. It is the envelope of the planes intersecting the three axes at the three vertices of a triangle for which the distance between the center of gravity and O is constant, equal to a /3.. It has the same vertices and same symmetries as the INDEX COURBES SPATIALES ALGEBRAIC CURVE (3D/) ANAMORPHOSE. ARCHYTAS (CURVE) ASYMPTOTIC LINE OF A SURFACE. ASYMPTOTIC LINE OF THE TORUS. BALL (SEAM LINE OF THE TENNIS/) BASIN (3D/) BERTRAND CURVE. BEZIER CURVE (3D/) OLOID - MATHCURVE.COM The oloid is the convex hull of two orthogonal circles each passing by the center of the other. Its surface is a part of the developable surface supported on the two circles (proof: H. POTTMANN, J. WALLNER : Computational Line Geometry, Springer-Verlag Telos (2001) p. 405)MÖBIUS STRIP
A Möbius strip is a surface obtained by sewing together two sides of a rectangular strip with a half-twist, or any topologically equivalent surface. Maple program giving an animation of the opposite construction. the boundary of the strip with 2p + 1 half-twists is a toroidal knot of order (2p + 1, 2). TREFOIL - MATHCURVE.COM The trefoil knot is the only knot for which the crossing number is minimal, namely 3; there are in fact two of them, they are enantiomorphic (images of one another by reflection).. It is an overhand knot: the ends of which were connected.. The trefoil knot is also the torus knot of type (3,2) (3 coils around the torus, on two turns), as well as that of type (2,3):VERONESE SURFACE
The Veronese surface is the image of the quotient of the 2-dimensional sphere by the antipodal relation (in other words, the real projective plane), by the map: .. Since this function is injective, the Veronese surface is a surface (i.e. a 2-manifold) without singular points embedded in (since it is included in the hyperplane of ) and homeomorphic to the real projective plane.KLEIN BOTTLE
The Möbius strip is a one-sided surface (with one face), therefore is not orientable, of genus 2, zero Euler characteristic, and chromatic number equal to 6 like the torus:(and not 7 like the Heawood formula would indicate) : In this 6 country map traced on the Klein bottle, each country meets the 5 others; 6 is the maximum number possible, and any map can be colored with no more than 6 colors.WHITNEY'S UMBRELLA
Whitney's umbrella is a right conoid with directrix a parabola with axis parallel to its axis; it is therefore a parabolic conoid. It is a surface that crosses itself along a half-line (here, Oz) the ends of which are cuspidal points or Whitney singular points (one of them atinfinity).
EULER CHARACTERISTIC EULER CHARACTERISTIC OF A SURFACE CHROMATIC NUMBER OF A SURFACE. A cell decomposition of a finite type manifold of dimension n (i.e. a topological space locally homeomorphic to a closed ball of and that is the finite union of parts homeomorphic to such balls) is a finite partition of this surface is subsets homeomorphic to for p £ n. If we write c p the number of subsets homeomorphic to , it MATHCURVE.COMBIBLIOGRAPHIECOURBES 3DFRACTALSPOLYÈDRESCOURBE DE LA CRÊPECOURBE SOLÉNOÏDALE page d'accueil de l'encyclopédie des formes remarquables ;home page of the encyclopaedia of curves, surfaces, fractals and polyhedronsCONE OF REVOLUTION
- give honor where honor is due: the planar sections or conics: ellipses, parabolas and hyperbolas.They can be developped into the polygasteroids of index n >1. Cutting by the plane gives an ellipse, a parabola, or a hyperbola, depending on whether is greater, equal, or less than ; if the cone rolls over a plane, it is developped into the polygasteroid with polar equation: .GUIMARD'S SURFACE
Hector Guimard's surface is the ruled surface union of the lines indicated above, when the point M has a linear sinusoidal movement and the point N a circular sinusoidal movement with two arches, and is therefore describing the pancake curve.. View showing the various segment lines The back of the circular part of this canopy of the Parisian underground, due to Hector Guimard, is aGAUDI'S SURFACE
Gaudi's surface is the right conoid the directrix of which is a sinusoid located in a plane parallel to the axis (here, the directrix plane is yOz and the axis y = z = 0).. We named it this way because it was used by Gaudi for the roof of the "escoles" of the Sagrada familiain Barcelona.
ENNEPER SURFACE
It can be geometrically defined as the envelope of the mediatrix planes of two points located on two homofocal parabolas (i.e. parabolas the planes of which are perpendicular and such that the vertex of one passes by the focus of the other one; compare to the definition of the symmetric parabolic Dupin cyclide).. As is the configuration of homofocal parabolas, the Enneper surface is invariant DOUBLE SIX - MATHCURVE.COM There exists a unique smooth cubic surface (S) containing the 12 lines of the double six.. For the double six represented above, where the vertices of the chosen cube are etc, and the line passes by , the equation of the cubic surface is: .. The 27 lines of this cubic are the 12 lines of the double six (in red and blue opposite) plus the 15 lines (in yellow) defined above. SURFACE OF CONSTANT WIDTH Given a compact convex K of the space, define its width in the direction of the line D as the length of the orthogonal projection of K on D, length which also is the smallest width of a strip orthogonal to D and containing K, or, which amounts to the same thing, the distance between two supporting planes of the convex K that are orthogonal to D.. The convex K is then said to be "of constantN -HOLED TORUS
n-HOLED TORUS. The notion of n-holed torus, or n-torus, or n-uple torus, or sphere with n handles, refers to any topological space homeomorphic to the connected sum of the simple torus n times with itself: ; by convention, we set . Any orientable connected compact surface without boundary is homeomorphic to an n-torus. The Euler characteristic of the n-torus is equal to 2–2n.MÖBIUS STRIP
Here is already a Möbius strip with one half-twist for which the ratio length/width is equal to 3Ö3 (slightly greater in fact for the sake of clarity). The view from above is a regular hexagon. The initial pattern is indicated on the right (the 3 folds are dotted). SPIRALE CONIQUE DE PAPPUS La spirale conique de Pappus est la trajectoire d'un point se déplaçant uniformément sur une droite passant par un point O, cette droite tournant uniformément autour d'un axe Oz en conservant un angle a avec Oz. Elle est donc intersection du cône de révolution (C) : avec l’hélicoïde droit: . Si l’on développe le cône (C) sur un plan, le point M devenant le point de coordonnées MATHCURVE.COMBIBLIOGRAPHIECOURBES 3DFRACTALSPOLYÈDRESCOURBE DE LA CRÊPECOURBE SOLÉNOÏDALE page d'accueil de l'encyclopédie des formes remarquables ;home page of the encyclopaedia of curves, surfaces, fractals and polyhedronsCONE OF REVOLUTION
- give honor where honor is due: the planar sections or conics: ellipses, parabolas and hyperbolas.They can be developped into the polygasteroids of index n >1. Cutting by the plane gives an ellipse, a parabola, or a hyperbola, depending on whether is greater, equal, or less than ; if the cone rolls over a plane, it is developped into the polygasteroid with polar equation: .GUIMARD'S SURFACE
Hector Guimard's surface is the ruled surface union of the lines indicated above, when the point M has a linear sinusoidal movement and the point N a circular sinusoidal movement with two arches, and is therefore describing the pancake curve.. View showing the various segment lines The back of the circular part of this canopy of the Parisian underground, due to Hector Guimard, is aGAUDI'S SURFACE
Gaudi's surface is the right conoid the directrix of which is a sinusoid located in a plane parallel to the axis (here, the directrix plane is yOz and the axis y = z = 0).. We named it this way because it was used by Gaudi for the roof of the "escoles" of the Sagrada familiain Barcelona.
ENNEPER SURFACE
It can be geometrically defined as the envelope of the mediatrix planes of two points located on two homofocal parabolas (i.e. parabolas the planes of which are perpendicular and such that the vertex of one passes by the focus of the other one; compare to the definition of the symmetric parabolic Dupin cyclide).. As is the configuration of homofocal parabolas, the Enneper surface is invariant DOUBLE SIX - MATHCURVE.COM There exists a unique smooth cubic surface (S) containing the 12 lines of the double six.. For the double six represented above, where the vertices of the chosen cube are etc, and the line passes by , the equation of the cubic surface is: .. The 27 lines of this cubic are the 12 lines of the double six (in red and blue opposite) plus the 15 lines (in yellow) defined above. SURFACE OF CONSTANT WIDTH Given a compact convex K of the space, define its width in the direction of the line D as the length of the orthogonal projection of K on D, length which also is the smallest width of a strip orthogonal to D and containing K, or, which amounts to the same thing, the distance between two supporting planes of the convex K that are orthogonal to D.. The convex K is then said to be "of constantN -HOLED TORUS
n-HOLED TORUS. The notion of n-holed torus, or n-torus, or n-uple torus, or sphere with n handles, refers to any topological space homeomorphic to the connected sum of the simple torus n times with itself: ; by convention, we set . Any orientable connected compact surface without boundary is homeomorphic to an n-torus. The Euler characteristic of the n-torus is equal to 2–2n.MÖBIUS STRIP
Here is already a Möbius strip with one half-twist for which the ratio length/width is equal to 3Ö3 (slightly greater in fact for the sake of clarity). The view from above is a regular hexagon. The initial pattern is indicated on the right (the 3 folds are dotted). SPIRALE CONIQUE DE PAPPUS La spirale conique de Pappus est la trajectoire d'un point se déplaçant uniformément sur une droite passant par un point O, cette droite tournant uniformément autour d'un axe Oz en conservant un angle a avec Oz. Elle est donc intersection du cône de révolution (C) : avec l’hélicoïde droit: . Si l’on développe le cône (C) sur un plan, le point M devenant le point de coordonnéesENNEPER SURFACE
It can be geometrically defined as the envelope of the mediatrix planes of two points located on two homofocal parabolas (i.e. parabolas the planes of which are perpendicular and such that the vertex of one passes by the focus of the other one; compare to the definition of the symmetric parabolic Dupin cyclide).. As is the configuration of homofocal parabolas, the Enneper surface is invariantASTROIDAL ELLIPSOID
The astroidal ellipsoid is the surface with the above equation. Its name comes from the fact that its sections by planes parallel to the axes are astroids.. It is the envelope of the planes intersecting the three axes at the three vertices of a triangle for which the distance between the center of gravity and O is constant, equal to a /3.. It has the same vertices and same symmetries as the INDEX COURBES SPATIALES ALGEBRAIC CURVE (3D/) ANAMORPHOSE. ARCHYTAS (CURVE) ASYMPTOTIC LINE OF A SURFACE. ASYMPTOTIC LINE OF THE TORUS. BALL (SEAM LINE OF THE TENNIS/) BASIN (3D/) BERTRAND CURVE. BEZIER CURVE (3D/) OLOID - MATHCURVE.COM The oloid is the convex hull of two orthogonal circles each passing by the center of the other. Its surface is a part of the developable surface supported on the two circles (proof: H. POTTMANN, J. WALLNER : Computational Line Geometry, Springer-Verlag Telos (2001) p. 405)MÖBIUS STRIP
A Möbius strip is a surface obtained by sewing together two sides of a rectangular strip with a half-twist, or any topologically equivalent surface. Maple program giving an animation of the opposite construction. the boundary of the strip with 2p + 1 half-twists is a toroidal knot of order (2p + 1, 2). TREFOIL - MATHCURVE.COM The trefoil knot is the only knot for which the crossing number is minimal, namely 3; there are in fact two of them, they are enantiomorphic (images of one another by reflection).. It is an overhand knot: the ends of which were connected.. The trefoil knot is also the torus knot of type (3,2) (3 coils around the torus, on two turns), as well as that of type (2,3):VERONESE SURFACE
The Veronese surface is the image of the quotient of the 2-dimensional sphere by the antipodal relation (in other words, the real projective plane), by the map: .. Since this function is injective, the Veronese surface is a surface (i.e. a 2-manifold) without singular points embedded in (since it is included in the hyperplane of ) and homeomorphic to the real projective plane.KLEIN BOTTLE
The Möbius strip is a one-sided surface (with one face), therefore is not orientable, of genus 2, zero Euler characteristic, and chromatic number equal to 6 like the torus:(and not 7 like the Heawood formula would indicate) : In this 6 country map traced on the Klein bottle, each country meets the 5 others; 6 is the maximum number possible, and any map can be colored with no more than 6 colors.WHITNEY'S UMBRELLA
Whitney's umbrella is a right conoid with directrix a parabola with axis parallel to its axis; it is therefore a parabolic conoid. It is a surface that crosses itself along a half-line (here, Oz) the ends of which are cuspidal points or Whitney singular points (one of them atinfinity).
EULER CHARACTERISTIC EULER CHARACTERISTIC OF A SURFACE CHROMATIC NUMBER OF A SURFACE. A cell decomposition of a finite type manifold of dimension n (i.e. a topological space locally homeomorphic to a closed ball of and that is the finite union of parts homeomorphic to such balls) is a finite partition of this surface is subsets homeomorphic to for p £ n. If we write c p the number of subsets homeomorphic to , it P age perso de l'auteurbibliographie
courbes 2D
courbes 3D
surfaces
fractals
polyèdres
ENCYCLOPÉDIE DES FORMES MATHÉMATIQUES REMARQUABLES Prix Anatole Decerf 2008 Sauvez le Palais de la Découverte ! Nouveautés au 01 / 12 / 2020 : Ellipsoïde de révolution, avec une belle géodésique Surfaces à géodésiques fermées qui ne sont pas des sphères "Polyèdres" à base de cylindres Courbe à accélération angulaire constante Courbe barycentrique Surfaces à lignes de courbure plane Développantes obliques Trochoïdes à base quelconque Inverses des tractoires : les courbes équitangentielles Un exemple pour voir : Ce site sur les formes mathématiques est issu d'un petit résumé commencé en 1993, destiné à des élèves de math sup désireux d’avoir un récapitulatif des courbes classiques dont quelques exemples parsemaient mon cours. Ce résumé a commencé à prendre du volume, et j’ai pensé que bien que le sujet ne soit pas particulièrement à la mode, il pourrait intéresser, d'où ce site. Les courbes et les surfaces m'ont fait découvrir pas mal de trésors connus de nos prédécesseurs et que nous avons tendance à oublier. J'ai même eu l'impression que les anciens avaient une vue beaucoup plus abstraite et générale de ces objets que celle que nous avonsaujourd'hui.
Une note plus contemporaine est donnée par les fractals qui auraient certainement enthousiasmé les anciens. Je remercie dans cette entreprise : - Alain ESCULIER, qui me donne de judicieux conseils pour la réalisation des animations, et réalise toutes les figures de qualité utilisant le logiciel povray. - Pierre LAMBERT, ancien professeur au lycée de Montmorency, qui profite de sa retraite pour mettre au clair toute cette vieille géométrie qu'il est encore l'un des rares à maîtriser parfaitement (mais aussi les maths "de maintenant"). Je lui dois pas mal de propriétés présentées ici, mais il est en train de rédiger des notes plus complètes que j'espère qu'il pourra publier. - Jacques MANDONNET, mon professeur en terminale puis ex-collègue en prépa, qui a fait beaucoup de figures et animations et m'a donné de nombreux conseils en informatique. - Jacques BOUTELOUP,
qui m'a bien mis au point sur la notion de courbe algébrique. - François RIDEAU, qui m'a prêté de nombreux livres rares et anciens qu'il conserve amoureusement, dont le Gomes Teixeira. - Robert MARCH, enseignant à l’École d’Architecture Paris-Val-de-Seine, qui a réalisé certaines des illustrations et m'a donné de très bonnes idées. - Mon cousin par alliance Christoph SOLAND, et Jean-Claude CHASTANG, tous deux fans d'ovales de Descartes, l'un en Suisse, l'autre aux États-Unis. Et je remercie WORD, NETSCAPE COMPOSER, MAPLE, et ILLUSTRATOR, qui m'ont permis de réaliser quelque chose que je n'aurais même pasimaginé en 93.
MODE D'EMPLOI
Les fiches sont regroupées en 5 thèmes : les courbes 2D, les courbes 3D (ce qui fait plus "in" que courbe plane et courbe gauche...), les surfaces, les fractals et les polyèdres. Chaque fiche commence par l'intitulé de la forme que l'on vadécrire.
Certaines formes qui m'ont paru devoir figurer dans ce répertoire n'avaient pas de nom reconnu, du moins en français. J'ai donc usé d'un droit de baptême, et si vous n'êtes pas d'accord, faites le moisavoir !
Il s'agit par exemple de : AFC , boiteœufs , bouche
, courbe de la crêpe , courbe de filature , courbe du nageur, poisson
, serpentine droite
, courbe
solénoïdale , talus, torpille
.
L’intitulé est toujours écrit au singulier, mais dans le corps de l’article, doit-on parler de la forme au singulier ou au pluriel ? Quand ses divers représentants sont tous semblables (dans un sens que nous allons essayer de définir), on a tendance de parler de cette forme au singulier : la parabole, l’ellipse, l’ellipsoïde etc. Dans le cas contraire, on en parle plutôt au pluriel : les coniques, les ovales de Descartes. Quel sens donner alors au mot semblable ? Pas son sens mathématique, puisque, si les paraboles sont toutes semblables entre elles au sens mathématique, ce n’est pas le cas des ellipses. J'ai donc étendu le sens de semblable à " image par une transformation affine ". Avec cette définition, j'ai été obligé de parler _des_ strophoïdes obliques par exemple, bien qu’elles aient toutes un peu la même tête, mais je dis _la_ strophoïde droite... J'ai eu aussi quelques problèmes avec le genre des noms de courbes et surfaces en " oïde " (du grec eidos : apparence), qui change suivant les auteurs. Je n'ai pas fait d’exception à la règle générale qui semble se dégager : féminin pour une courbe (une deltoïde, une ovoïde), masculin pour une surface (un caténoïde (bien que le Larousse le donne féminin), un onduloïde, un ovoïde). Sont indiqués ensuite les mathématiciens qui ont étudié l’objet, avec la date correspondante. N’étant pas spécialiste d’histoire des mathématiques, je me suis contenté de transcrire les informations des diverses sources, en particulier de ; signalez-moi les erreurs, svp. Suit une étymologie succincte de l’intitulé. J'ai indiqué l’origine de la plupart de ses composants sauf celles qui semblent évidentes. Je n'ai par exemple pas dit que le mot cercle vient de circulum qui signifie cercle, ni répété qu’oïde signifie : enforme de.
Dans la suite, plus mathématique, j'ai essayé de concilier la rigueur moderne, avec l’élégance et la concision des expressions de la géométrie traditionnelle. Par exemple, je parle du " lieu _du_ point _M_ astreint à telle condition" et non pas de " l’ensemble _des_ points _M_ ... ". Et je parle de _l_’équation cartésienne (ou polaire) bien qu’il n’y en ait pas qu’une. D'autre part, le cadre géométrique est la géométrie affine réelle. Mais lorsqu'on parle de point à l'infini, ou de point complexe, on considére le prolongement dans le complété projectif réel ou complexe. Cependant, distinguer les courbes de , de, de et de conduit à des lourdeurs inacceptables. En encadré j'indique donc tout d’abord la carte d’identité de la forme, ou, pour être plus exact, d’un modèle de la forme. La définition générale qui suit donnera précisément l’ensemble de tous les objets qui ont pour nom celui de l’intitulé, mais dans la carte d’identité, n’est décrit qu’un sous-ensemble représentatif, associé au repère considéré ; autrement dit, ce que j'annonce comme équation cartésienne n'est en général que l'équation cartésienne _réduite_. Par exemple pour les cercles, je prends les cercles de centre _O_ et de rayon _R_. Tous les objets de l’ensemble doivent être semblables (au sens mathématique) l’un des objets du sous-ensemble. Mais je ne suis pas allé jusqu’à mettre dans la carte d’identité du cercle, uniquement le cercle de centre O et de rayon 1, car j'ai veillé à écrire des équations homogènes par rapport aux longueurs. Dans les équations apparaissent donc en général un facteur d’échelle, presque toujours appelé _a_, puis des constantes appelées en général _b_, _c_ si ce sont des longueurs. Elles sont supposées non nulles et très souvent positives. La première équation est celle qui est la plus classique, et non forcément l’équation cartésienne. Suivent d’autres équations, et, dans le cas des courbes, un calcul de l’abscisse curviligne et du rayon de courbure, éventuellement des calculs de longueurs etd’aires.
Après l’encadré carte d’identité, vient la définition la plus usitée de l’objet ; elle est énoncée de façon générale, mais le lien est fait avec le cas particulier (qui représente en fait la généralité) de l’encadré. Lorsque j'emploie l’expression " ici ", je me réfère à ce cas particulier. Suivent alors d’autres caractérisations qui constituent autant de propriétés de l’objet, et parfois d’autres propriétés si elles sont classiques. Lorsque l’objet a été défini moins pour lui-même que pour illustrer une propriété, j'allège la partie équations et définitions pour insister sur cette propriété. J'ai enfin essayé de répertorier l'appartenance de la forme aux diverses familles définies dans d'autres fiches, et parfois découvert ainsi des liens inhabituels : les besaces dans les courbes de Lissajous, les coniques dans les anticaustiques , les courbes de Booth dans les courbes de Wattpar exemple !
Bonne lecture et vos commentaires, encouragements et critiques sontles bienvenus !
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