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Generalized Helicoid
In geometry, a generalized helicoid is a surface in Euclidean space generated by rotating and simultaneously displacing a curve, the ''profile curve'', along a line, its ''axis''. Any point of the given curve is the starting point of a circular helix. If the profile curve is contained in a plane through the axis, it is called the meridian of the generalized helicoid. Simple examples of generalized helicoids are the helicoids. The meridian of a helicoid is a line which intersects the axis orthogonally. Essential types of generalized helicoids are * ruled generalized helicoids. Their profile curves are lines and the surfaces are ruled surfaces. *circular generalized helicoids. Their profile curves are circles. In mathematics helicoids play an essential role as minimal surfaces. In the technical area generalized helicoids are used for staircases, slides, screws, and pipes. Analytical representation Screw motion of a point Moving a point on a screwtype curve means, the point ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Catalan Surface
In geometry, a Catalan surface, named after the Belgian mathematician Eugène Charles Catalan, is a ruled surface all of whose rulings are parallel to a fixed plane. Equations The vector equation of a Catalan surface is given by :''r'' = ''s''(''u'') + ''v'' ''L''(''u''), where ''r'' = ''s''(''u'') is the space curve and ''L''(''u'') is the unit vector of the ruling at ''u'' = ''u''. All the vectors ''L''(''u'') are parallel to the same plane, called the '' directrix plane'' of the surface. This can be characterized by the condition: the mixed product 'L''(''u''), ''L' ''(''u''), ''L" ''(''u'')= The parametric equations of the Catalan surface ar x=f(u)+vi(u),\quad y=g(u)+vj(u),\quad z=h(u)+vk(u) \, Special cases If all the rulings of a Catalan surface intersect a fixed Line (geometry), line, then the surface is called a conoid. Catalan proved that the helicoid and the plane were the only ruled minimal surfaces. See also * Generalized helicoid In geometry, a generali ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conoid
In geometry a conoid () is a ruled surface, whose rulings (lines) fulfill the additional conditions: :(1) All rulings are parallel to a plane, the '' directrix plane''. :(2) All rulings intersect a fixed line, the ''axis''. The conoid is a right conoid if its axis is perpendicular to its directrix plane. Hence all rulings are perpendicular to the axis. Because of (1) any conoid is a Catalan surface and can be represented parametrically by :\mathbf x(u,v)= \mathbf c(u) + v\mathbf r(u)\ Any curve with fixed parameter is a ruling, describes the ''directrix'' and the vectors are all parallel to the directrix plane. The planarity of the vectors can be represented by :\det(\mathbf r,\mathbf \dot r,\mathbf \ddot r)=0 . If the directrix is a circle, the conoid is called a circular conoid. The term ''conoid'' was already used by Archimedes in his treatise '' On Conoids and Spheroides''. Examples Right circular conoid The parametric representation : \mathbf x(u,v)=(\cos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helix
A helix () is a shape like a corkscrew or spiral staircase. It is a type of smooth space curve with tangent lines at a constant angle to a fixed axis. Helices are important in biology, as the DNA molecule is formed as two intertwined helices, and many proteins have helical substructures, known as alpha helices. The word ''helix'' comes from the Greek word ''ἕλιξ'', "twisted, curved". A "filled-in" helix – for example, a "spiral" (helical) ramp – is a surface called '' helicoid''. Properties and types The ''pitch'' of a helix is the height of one complete helix turn, measured parallel to the axis of the helix. A double helix consists of two (typically congruent) helices with the same axis, differing by a translation along the axis. A circular helix (i.e. one with constant radius) has constant band curvature and constant torsion. A '' conic helix'', also known as a ''conic spiral'', may be defined as a spiral on a conic surface, with the distance to the ape ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Channel Surface
In geometry and topology, a channel or canal surface is a surface formed as the envelope of a family of spheres whose centers lie on a space curve, its '' directrix''. If the radii of the generating spheres are constant, the canal surface is called a pipe surface. Simple examples are: * right circular cylinder (pipe surface, directrix is a line, the axis of the cylinder) * torus (pipe surface, directrix is a circle), * right circular cone (canal surface, directrix is a line (the axis), radii of the spheres not constant), * surface of revolution (canal surface, directrix is a line), Canal surfaces play an essential role in descriptive geometry, because in case of an orthographic projection its contour curve can be drawn as the envelope of circles. *In technical area canal surfaces can be used for ''blending surfaces'' smoothly. Envelope of a pencil of implicit surfaces Given the pencil of implicit surfaces :\Phi_c: f(,c)=0 , c\in _1,c_2/math>, two neighboring surfaces \Phi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conoid
In geometry a conoid () is a ruled surface, whose rulings (lines) fulfill the additional conditions: :(1) All rulings are parallel to a plane, the '' directrix plane''. :(2) All rulings intersect a fixed line, the ''axis''. The conoid is a right conoid if its axis is perpendicular to its directrix plane. Hence all rulings are perpendicular to the axis. Because of (1) any conoid is a Catalan surface and can be represented parametrically by :\mathbf x(u,v)= \mathbf c(u) + v\mathbf r(u)\ Any curve with fixed parameter is a ruling, describes the ''directrix'' and the vectors are all parallel to the directrix plane. The planarity of the vectors can be represented by :\det(\mathbf r,\mathbf \dot r,\mathbf \ddot r)=0 . If the directrix is a circle, the conoid is called a circular conoid. The term ''conoid'' was already used by Archimedes in his treatise '' On Conoids and Spheroides''. Examples Right circular conoid The parametric representation : \mathbf x(u,v)=(\cos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangent Developable
In the mathematical study of the differential geometry of surfaces, a tangent developable is a particular kind of developable surface obtained from a curve in Euclidean space as the surface swept out by the tangent lines to the curve. Such a surface is also the envelope of the tangent planes to the curve. Parameterization Let \gamma(t) be a parameterization of a smooth space curve. That is, \gamma is a twice-differentiable function with nowhere-vanishing derivative that maps its argument t (a real number) to a point in space; the curve is the image of \gamma. Then a two-dimensional surface, the tangent developable of \gamma, may be parameterized by the map :(s,t)\mapsto \gamma(t) + s\gamma(t). The original curve forms a boundary of the tangent developable, and is called its directrix or edge of regression. This curve is obtained by first developing the surface into the plane, and then considering the image in the plane of the generators of the ruling on the surface. The en ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helix
A helix () is a shape like a corkscrew or spiral staircase. It is a type of smooth space curve with tangent lines at a constant angle to a fixed axis. Helices are important in biology, as the DNA molecule is formed as two intertwined helices, and many proteins have helical substructures, known as alpha helices. The word ''helix'' comes from the Greek word ''ἕλιξ'', "twisted, curved". A "filled-in" helix – for example, a "spiral" (helical) ramp – is a surface called '' helicoid''. Properties and types The ''pitch'' of a helix is the height of one complete helix turn, measured parallel to the axis of the helix. A double helix consists of two (typically congruent) helices with the same axis, differing by a translation along the axis. A circular helix (i.e. one with constant radius) has constant band curvature and constant torsion. A '' conic helix'', also known as a ''conic spiral'', may be defined as a spiral on a conic surface, with the distance to the ape ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
