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单词 RuledSurface
释义

ruled surface


A straight line g moving continuously in space sweeps a ruled surfaceMathworldPlanetmath.  Formally:  A surface S in 3 is a ruled surface if it is connected and if for any point p of S, there is a line g such that  pgS.

Such a surface may be formed by using two auxiliary curves given e.g. in the parametric forms

r=a(t),r=b(t).

Using two parameters s and t we express theposition vector (http://planetmath.org/PositionVector) of an arbitrary point of the ruled surface as

r=a(t)+sb(t).

Here  r=a(t)  is a curve on the ruled surface and is called or the of the surface, while  r=b(t)  is the director curve of the surface.  Every position of g is a generatrix or ruling of the ruled surface.

Examples

1.  Choosing the z-axis (r=ctk,  c0) as the and the unit circle (r=icost+jsint)as the director curve we get the helicoid (“screw surface”; cf. the circular helix)

r=ctk+s(icost+jsint)=(scostssintct).

2.  The equation

z=xy

presents a hyperbolic paraboloidMathworldPlanetmath (if we rotate the coordinate systemMathworldPlanetmath (http://planetmath.org/RotationMatrix) 45 about the z-axis using the formulae  x=(x-y)/2, y=(x+y)/2,  the equation gets the form  x2-y2=2z).  Since the position vector of any point of the surface may be written using the parameters s and t as

r=(0t0)+s(10t),

we see that it’s a question of a ruled surface with rectilinear directrixPlanetmathPlanetmath and director curve.

3.  Other ruled surfaces are for example all cylindricalsurfacesMathworldPlanetmath (plane included), conical surfaces,one-sheeted hyperboloid (http://planetmath.org/QuadraticSurfaces).

Titleruled surface
Canonical nameRuledSurface
Date of creation2016-03-03 17:28:55
Last modified on2016-03-03 17:28:55
Ownerpahio (2872)
Last modified bypahio (2872)
Numerical id19
Authorpahio (2872)
Entry typeTopic
Classificationmsc 51M20
Classificationmsc 51M04
Related topicEquationOfPlane
Related topicGraphOfEquationXyConstant
Definesdirectrix
Definesbase curve
Definesdirector curve
Definesgeneratrix
Definesgeneratrices
Definesruling
Defineshelicoid
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更新时间:2025/5/4 9:11:22