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Cone of a circle. The original space X is in blue, and the collapsed end point v is in green.. In topology, especially algebraic topology, the cone of a topological space is intuitively obtained by stretching X into a cylinder and then collapsing one of its end faces to a point.
One can use the mapping cylinder to construct homotopy colimits: [citation needed] this follows from the general statement that any category with all pushouts and coequalizers has all colimits. That is, given a diagram, replace the maps by cofibrations (using the mapping cylinder) and then take the ordinary pointwise limit (one must take a bit ...
Cone: Tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex: A right circular cone and an oblique circular cone Cylinder: Straight parallel sides and a circular or oval cross section A solid elliptic cylinder A right and an oblique circular cylinder Ellipsoid
On the Sphere and Cylinder (Greek: Περὶ σφαίρας καὶ κυλίνδρου) is a treatise that was published by Archimedes in two volumes c. 225 BCE. [1] It most notably details how to find the surface area of a sphere and the volume of the contained ball and the analogous values for a cylinder , and was the first to do so.
The term cylinder can also mean the lateral surface of a solid cylinder (see cylinder (geometry)). If a cylinder is used in this sense, the above paragraph would read as follows: A plane section of a right circular cylinder of finite length [ 6 ] is a circle if the cutting plane is perpendicular to the cylinder's axis of symmetry, or an ellipse ...
The cylinder is an example of a developable surface. In mathematics, a developable surface (or torse: archaic) is a smooth surface with zero Gaussian curvature. That is, it is a surface that can be flattened onto a plane without distortion (i.e. it can be bent without stretching or compression).
A cone can be generated by moving a line (the generatrix) fixed at the future apex of the cone along a closed curve (the directrix); if that directrix is a circle perpendicular to the line connecting its center to the apex, the motion is rotation around a fixed axis and the resulting shape is a circular cone. [3] The generatrix of a cylinder, a ...
In the case of lines, the cone extends infinitely far in both directions from the apex, in which case it is sometimes called a double cone. Either half of a double cone on one side of the apex is called a nappe. The axis of a cone is the straight line passing through the apex about which the base (and the whole cone) has a circular symmetry.