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A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) one full revolution around an axis of rotation (normally not intersecting the generatrix, except at its endpoints). [1] The volume bounded by the surface created by this revolution is the solid of revolution.
The following table gives definitions of such objects; Monge patches is perhaps the most visually intuitive, as it essentially says that a regular surface is a subset of ℝ 3 which is locally the graph of a smooth function (whether over a region in the yz plane, the xz plane, or the xy plane).
The surface formed is a surface of revolution; it encloses a solid of revolution. Solids of revolution ( Matemateca Ime-Usp ) In geometry , a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the axis of revolution ), which may not intersect the generatrix (except at its boundary).
The graph of a continuous function of two variables, defined over a connected open subset of R 2 is a topological surface. If the function is differentiable, the graph is a differentiable surface. A plane is both an algebraic surface and a differentiable surface. It is also a ruled surface and a surface of revolution.
In particular, for some placements of the two points the optimal solution is generated by a discontinuous function that is nonzero at the two points and zero everywhere else. This function leads to a surface of revolution consisting of two circular disks, one for each point, connected by a degenerate line segment along the axis of revolution.
A parametric surface need not be a topological surface. A surface of revolution can be viewed as a special kind of parametric surface. If f is a smooth function from R 3 to R whose gradient is nowhere zero, then the locus of zeros of f does define a surface, known as an implicit surface. If the condition of non-vanishing gradient is dropped ...
If the axis of revolution is tangent to the circle, the surface is a horn torus. If the axis of revolution passes twice through the circle, the surface is a spindle torus (or self-crossing torus or self-intersecting torus). If the axis of revolution passes through the center of the circle, the surface is a degenerate torus, a double-covered sphere.
In geometry, a catenoid is a type of surface, arising by rotating a catenary curve about an axis (a surface of revolution). [1] It is a minimal surface, meaning that it occupies the least area when bounded by a closed space. [2] It was formally described in 1744 by the mathematician Leonhard Euler.