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Ruled surface generated by two Bézier curves as directrices (red, green) A surface in 3-dimensional Euclidean space is called a ruled surface if it is the union of a differentiable one-parameter family of lines. Formally, a ruled surface is a surface in is described by a parametric representation of the form
Castelnuovo's theorem implies that to construct a minimal model for a smooth surface, we simply contract all the −1-curves on the surface, and the resulting variety Y is either a (unique) minimal model with K nef, or a ruled surface (which is the same as a 2-dimensional Fano fiber space, and is either a projective plane or a ruled surface ...
A sphere is the surface of a solid ball, here having radius r. In mathematics, a surface is a mathematical model of the common concept of a surface.It is a generalization of a plane, but, unlike a plane, it may be curved; this is analogous to a curve generalizing a straight line.
A ruled surface is one which can be generated by the motion of a straight line in E 3. [46] Choosing a directrix on the surface, i.e. a smooth unit speed curve c(t) orthogonal to the straight lines, and then choosing u(t) to be unit vectors along the curve in the direction of the lines, the velocity vector v = c t and u satisfy
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.
This is a list of surfaces in mathematics. They are divided into minimal surfaces, ruled surfaces, non-orientable surfaces, quadrics, pseudospherical surfaces, algebraic surfaces, and other types of surfaces.
In geometry, a right conoid is a ruled surface generated by a family of straight lines that all intersect perpendicularly to a fixed straight line, called the axis of the right conoid. Using a Cartesian coordinate system in three-dimensional space , if we take the z -axis to be the axis of a right conoid, then the right conoid can be ...
Whitney umbrella as a ruled surface, generated by a moving straight line Whitney umbrella made with a single string inside a plastic cube. Whitney's umbrella is a ruled surface and a right conoid. It is important in the field of singularity theory, as a simple local model of a pinch point singularity.
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