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The oloid and the sphericon are members of a special family of solids that develop their entire surface when rolling down a flat plane. Planes (trivially); which may be viewed as a cylinder whose cross-section is a line; Tangent developable surfaces; which are constructed by extending the tangent lines of a spatial curve.
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments , half-lines , or lines connecting a common point, the apex, to all of the points on a base that is in a plane that does not contain ...
Development can be generalized further using flat connections. From this point of view, rolling the tangent plane over a surface defines an affine connection on the surface (it provides an example of parallel transport along a curve ), and a developable surface is one for which this connection is flat.
Patches of conical surfaces that avoid the apex are special cases of developable surfaces, surfaces that can be unfolded to a flat plane without stretching. When the directrix has the property that the angle it subtends from the apex is exactly 2 π {\displaystyle 2\pi } , then each nappe of the conical surface, including the apex, is a ...
Conical spiral with an archimedean spiral as floor projection Floor projection: Fermat's spiral Floor projection: logarithmic spiral Floor projection: hyperbolic spiral. In mathematics, a conical spiral, also known as a conical helix, [1] is a space curve on a right circular cone, whose floor projection is a plane spiral.
The cone over a closed interval I of the real line is a filled-in triangle (with one of the edges being I), otherwise known as a 2-simplex (see the final example). The cone over a polygon P is a pyramid with base P. The cone over a disk is the solid cone of classical geometry (hence the concept's name). The cone over a circle given by
The hollow cone pattern is also achievable by the spiral design of nozzle. This nozzle impinges the fluid upon a protruding spiral. This spiral shape breaks the fluid apart into several hollow cone patterns. By altering the topology of the spiral the hollow cone patterns can be made to converge to form a single hollow cone.
However, the vertices of the polyhedron have a different distance structure: the local geometry of a polyhedron vertex is the same as the local geometry at the apex of a cone. Any cone can be formed from a flat sheet of paper with a wedge removed from it by gluing together the cut edges where the wedge was removed.