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# Topic: Ruled surface

###### In the News (Tue 23 Jul 19)

 Surface - LoveToKnow 1911 These leading ideas apply to surfaces, but the ideas peculiar to surfaces are scarcely of the like fundamental nature, being rather developments of the former set in their application to a more advanced portion of geometry; there is consequently less occasion for the historical mode of treatment. A plane may meet the surface in a curve having (r) a cusp (spinode) or (2) a pair of double points; in each case there is a singly infinite system of such singular tangent planes, and the locus of the points of contact is the curve, the envelope of the tangent planes the torse. Again, a ruled surface may evidently be deformed by first rotating round a generator, the portion of the surface lying to one side of this generator, then round the consecutive generator, the portion of the surface lying beyond this again, and so on. www.1911encyclopedia.org /Surface   (8339 words)

 PlanetMath: ruled surface   (Site not responding. Last check: 2007-10-13) Such a surface may be formed by using two auxiliary curves given e.g. Other ruled surfaces are for example all cylindrical surfaces (plane included), conical surfaces, one-sheeted hyperboloid. This is version 8 of ruled surface, born on 2005-08-28, modified 2005-08-29. planetmath.org /encyclopedia/RuledSurface.html   (234 words)

 Ruled surface - Wikipedia, the free encyclopedia The last two are special cases of ruled quadrics (which also include the hyperbolic paraboloid, the hyperboloid of one sheet, and the conical surface with elliptical directrix). The properties of being ruled or doubly-ruled are preserved by projective maps, and therefore are concepts of projective geometry. Analogues for algebraic surfaces are studied in algebraic geometry. en.wikipedia.org /wiki/Ruled_surface   (378 words)

 Index: Ruled Surfaces A ruled surface (or scroll) is a surface swept out by a straight line as it moves through space. Some quadratic surfaces are ruled: hyperboloids of one sheet, hyperbolic paraboloids, and quadratic cones and cylinders. A general way to form a ruled surface is to take three curves in space, and move a straight line so that it intersects all three curves at all times. math.arizona.edu /~models/Ruled_Surfaces   (194 words)

 PlanetMath: developable surface   (Site not responding. Last check: 2007-10-13) A generatrix of a ruled surface is torsal, if in each of its points there is one and the same tangent plane of the surface. Gauss has proved that a surface is developable if and only if it is a torsal ruled surface. This is version 2 of developable surface, born on 2005-08-28, modified 2005-08-28. planetmath.org /encyclopedia/DevelopableSurface.html   (158 words)

 Six types of Ruled Surfaces Scrolls, or ruled surfaces, are surfaces generated by straight lines or rulings and have been studied for centuries by geometers such as the Jesuits Roger Boscovich and Andre Tacquet as well as by their famous students such as Gaspar Monge and Phillippe de Lahire. Hence the surface, which is called a helicoid, is transformed into itself by a screw motion of the pitch d with the line of striction as axis. This surface represents an important point in applied mathematics called the "saddle point" found at its center which is a maximum point in one plane but a minimum point in another plane, but actually is neither for the surface. www.faculty.fairfield.edu /jmac/rs/sixmodels.htm   (1452 words)

 Bin's On-Line Dictionary   (Site not responding. Last check: 2007-10-13) To establish or settle by, or as by, a rule; to fix by universal or general consent, or by common practice. To mark with lines made with a pen, pencil, etc., guided by a rule or ruler; to print or mark with lines by means of a rule or other contrivance effecting a similar result; as, to rule a sheet of paper of a blank book. Ruled surface (Geom.), any surface that may be described by a straight line moving according to a given law; -- called also a scroll. language.bin.org /ref/dict/?t=Ruled   (175 words)

 Polygon Mesh Models   (Site not responding. Last check: 2007-10-13) A surface is ruled if through every one of its points, passes at least one line which lies entirely on its surface. Ruled surfaces are created by dragging a line segment through space and the line remains parallel with itself at all times. A cylinder is a ruled surface, formed by dragging a line segment around a circle. glasnost.itcarlow.ie /~powerk/Graphics/Notes/node5.html   (1677 words)

 Week8 - Basic Three-Dimensional Modeling Limited conversion between model types is available from solids to surfaces and from surfaces to wireframes; however, you cannot convert from wireframes to surfaces or from surfaces to solids. If the curve is a circle, the ruled surface begins at the 0-degree quadrant point, as determined by the current X axis plus the current value of the SNAPANG system variable. The ruled surface is constructed as a 2 × N polygon mesh. darkwing.uoregon.edu /~arch/landcad/week8.htm   (1500 words)

 Ruled surface   (Site not responding. Last check: 2007-10-13) The first two are special cases of ruled quadrics (which also include the hyperbolic paraboloid, the hyperboloid of one sheet, and the conical surface with elliptical diretrix). move with constant speed along two skew straight lines, the surface is a hyperbolic paraboloid, or a piece of an hyperboloid of one sheet. A developable surface — one that can be (locally) unrolled onto a flat plane without tearing or stretching — is necessarily ruled, but the converse is not always true. ruled-surface.peernet.sk   (267 words)

 Curves & Surfaces > Surfaces > Special Surfaces > Geometry Tangent surfaces arise from the tangent vectors to curves. An example of a surface of revolution is the pseudosphere, which is the surface of revolution of a tractrix. Kuen's surface and Dini's surface, as well as the sphere and pseudosphere all have constant Gaussian curvature. www.math.hmc.edu /~gu/curves_and_surfaces/surfaces/_geometry.html   (277 words)

 Fillet Surfaces The Fillet Surface command creates either the fillet surface, or three curves which show the boundaries of the fillet surface, or both. The curves created with the Fillet Surface feature, can be used to create a Ruled surface which might be used for a check surface. All three curves may be selected for the ruled surface, however, this may increase the time it takes to draw the surface on the screen, and thus, slow down the operation somewhat. www.ezcam.com /web/products/help/ezmill/surfaces/fillet_surfaces.htm   (156 words)

 Ruled Surface -- from Wolfram MathWorld A ruled surface is a surface that can be swept out by moving a line in space. The rulings of a ruled surface are asymptotic curves. Furthermore, the Gaussian curvature on a ruled regular surface is everywhere nonpositive. mathworld.wolfram.com /RuledSurface.html   (195 words)

 ruled surface A cylinder, for example, is a ruled surface of parallel straight lines. A cone is a ruled surface of straight lines that meet at the apex of the cone. Also known as scrolls, ruled surfaces have been studied for centuries by geometers such as the Jesuits Roger Boscovich and Andre Tacquet as well as by their famous students, including Gaspar Monge and Phillippe de Lahire. www.daviddarling.info /encyclopedia/R/ruled_surface.html   (189 words)

 [No title]   (Site not responding. Last check: 2007-10-13) Constructs a piecewise ruled surface approximation to the given trimmed surface, TSrf, in the given direction, Dir, that is close to the surface to within Epsilon. The given trimmed ruled surface might be non-developable, in which case approximation will be of a surface with no twist. Returns a new trimmed surface that is the reversed surface of Srf by flipping the U and the V directions of the surface, as well as flipping them in the trimming curves. www.cs.technion.ac.il /~irit/prog_man/trim_lib.html   (3188 words)

 Images and Pictures of Recent Research Projects:   (Site not responding. Last check: 2007-10-13) Ruled tracing of adaptive isoparametric curves is a novel rendering technique to directly scan convert freeform parametric surfaces. An artistic rendering method of free-form surfaces with the aid of half-toned text that is laid-out on the given surface is presented. Once the layout is constructed on the surface, a shading process is applied to the text, affecting the width of the symbols as well as their color, according to some shader function. www.cs.technion.ac.il /~irit/irit_images.html   (4342 words)

 Glossary: Ruled Surface   (Site not responding. Last check: 2007-10-13) Given a parameterized curve c(t) and a sequence of unit vectors z(t) that varies continuously with t, the parametric surface defined by Less formally, it is the surface swept out by moving a straight line along the curve c so that it points in the direction z(t) at each time t. For example, the saddle surfaces z = x y and z = x^2 - y^2 are both ruled surfaces, as is the hyperboloid of one sheet, the helicoid, a cylinder and a cone. www.geom.uiuc.edu /docs/research/RP2-handle/Glossary/RuledSurface.html   (84 words)

 Chapter 2: Landform   (Site not responding. Last check: 2007-10-13) Such as surface is sometimes referred to as a 2.5D surface, because although there is a third dimension, it is severely constrained by the first two (this term is also used sometimes to refer to extruded forms, which have a similar characteristic). Surfaces, simple solids, and solids formed from surfaces, all share a similar characteristic: they represent continuous surfaces, without gaps, holes, or overhangs, and are sufficient for the majority of topographic models. Writing rules, or procedures, that transform each cell value a step at a time, as a function of similarly located cells in other layers, or neighboring cells in the same layer, can be used as a way to create dynamic models. www.landscapemodeling.org /html/ch2/ch2text.htm   (8475 words)

 VTK: vtkRuledSurfaceFilter Class Reference A surface is generated by connecting the points defining each pair of lines with straight lines. Note that there are several different approaches for creating the ruled surface, the method for creating the surface can either use the input points or resample from the polylines (using a user-specified resolution). An important use of this filter is to combine it with vtkStreamLine to generate stream surfaces. noodle.med.yale.edu /vtk/classvtkRuledSurfaceFilter.html   (1207 words)

 ruled This command creates a surface using two or more curves, each of which define a cross section of the constructed surface at a specific location. Select the first curve for the surface from the screen or double-click on the curve name in the dialog box. Ruled surface is the same as Cross Sectional, except that the Ruled link type is line. www.ezcam.com /web/products/help/ezmill/menu_surfaces/ruled.htm   (177 words)

 SurfaceWorks surfaces A Mirrored Surface is a surface entity created by reflecting a given parent surface across a specified mirror entity. A Ruled Surface is created from two curves by connecting points having equal t parameter values with straight lines (the "rulings"). A Tangent Boundary Surface is defined by four boundary curves (and/or snakes and/or points), a set of control values that influence the interior shape of the surface, and a set of edge conditions. www.aerohydro.com /products/industrial/surfaces.htm   (636 words)

 Half Twist Ruled Surfaces The surface is usually formed by giving a half twist to a narrow strip of paper but all its properties are not obvious. The equation for such a surface is given in cylindrical coordinates (r, t, z). A Moebius surface is generated by a line PQP', which pivots about point Q, making an angle t/2 with the vertical z axis as Q moves around the circumference of circle C of radius a in the xy plane. www.faculty.fairfield.edu /jmac/rs/halftw.htm   (1681 words)

 Hyperbolic paraboloid Of course, this is directly related to the "saddle shaped" appearance of the surface. Another interesting property of this surface is that it is a ruled surface. The other smooth quadric ruled surface, the hyperboloid of one sheet, also contains two 1-parameter families of lines. www.math.umn.edu /~roberts/java.dir/JGV/paraboloid1.html   (214 words)

 Mathematics 535 (Fall 2002) Information   (Site not responding. Last check: 2007-10-13) We saw this Cayley ruled surface in Week 4 as the second from the right among the projections of the Veronese. Similarly the line joining 2 singular points on a cubic surface is entirely contained in the surface, since a line hitting a cubic surface 4 times is wholly in the surface. But the intersection of a plane and a cubic surface is the zero set of a degree 3 polynomial, so it can not contain 4 lines unless the plane is in the surface. www.math.rutgers.edu /courses/535/535-f02/Movie4.html   (405 words)

 Simple Swept Surfaces The swept surface defined by these two curves will be shown on the drawing canvas of the surface system. In this case, the swept surface is simply a ruled surface. In the following figure, the profile curve is the one used to design a vase in the discussion of surface revolution, and the trajectory curve is a line segment on the x-axis. www.cs.mtu.edu /~shene/COURSES/cs3621/LAB/surface/swept.html   (558 words)

 VS3D / VScad3 Documentation A ruled surface is basically a surface that spans two lines or curves. The color of the resulting ruled surface is determined by the current main attribute color setting, which can be set via the main "Resources / Attributes" menu. If fewer than two entities were selected prior to the dialog creation, then XYZ coordinates can be typed into this field manually as three columns of numbers separated by spaces and/or commas. www.designscomputed.com /vs3d/docs/guide/guide_2_4_5G.html   (259 words)

 Developable surface - Wikipedia, the free encyclopedia A developable surface is a surface with a metric that can be flattened onto a plane without distortion (i.e. Formally, in mathematics, a developable surface is a surface with zero Gaussian curvature. One consequence of this is that all developable surfaces embedded in 3D space are ruled surfaces, but hyperboloids are examples of ruled surfaces that are not developable. en.wikipedia.org /wiki/Developable_surface   (294 words)

 A hyperboloid of one sheet The hyperboloid of one sheet is a quadric ruled surface, i.e., a surface of degree 2 that contains infinitely many lines. A drawing of the hyperboloid in which some of these lines are explicitly shown. The other smooth quadric ruled surface, the hyperbolic paraboloid, also contains two 1-parameter families of lines. www.math.umn.edu /~roberts/java.dir/JGV/hyperboloid1a.html   (190 words)

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