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The net has to be such that the straight line is fully within it, and one may have to consider several nets to see which gives the shortest path. For example, in the case of a cube , if the points are on adjacent faces one candidate for the shortest path is the path crossing the common edge; the shortest path of this kind is found using a net ...
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.
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 cone of curves is defined to be the convex cone of linear combinations of curves with nonnegative real coefficients in the real vector space () of 1-cycles modulo numerical equivalence. The vector spaces N 1 ( X ) {\displaystyle N^{1}(X)} and N 1 ( X ) {\displaystyle N_{1}(X)} are dual to each other by the intersection pairing, and the nef ...
More generally, given a vector bundle (finite-rank locally free sheaf) E on X, if R=Sym(E *) is the symmetric algebra generated by the dual of E, then the cone is the total space of E, often written just as E, and the projective cone is the projective bundle of E, which is written as ().
The (usually infinite) collection of all these triangles can be (partially) depicted in the shape of a cone with the apex N. The cone ψ is sometimes said to have vertex N and base F. One can also define the dual notion of a cone from F to N (also called a co-cone) by reversing all the arrows above. Explicitly, a co-cone from F to N is a family ...
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A projective cone (or just cone) in projective geometry is the union of all lines that intersect a projective subspace R (the apex of the cone) and an arbitrary subset A (the basis) of some other subspace S, disjoint from R.