Q: Is curve a generalization of a line? ¶
A: Yes, and in that curvature is not necessarily zero.
Q: Is curve defined by a polynomial f of total degree d, then wdf simplifies to a homogeneous polynomial g of degree d? ¶
A: Yes, The values of u, v, w such that g = 0 are the homogeneous coordinates of the points of the completion of the curve in the projective plane and the points of the initial curve are those such w is not zero.
Q: Are curves the curves considered in algebraic geometry? ¶
Q: Is curve the locus of the points of coordinates x, y such that f = 0, where f is a polynomial in two variables defined over some field F? ¶
A: Yes, Algebraic geometry normally looks not only on points with coordinates in F but on all the points with coordinates in an algebraically closed field K. If C is a curve defined by a polynomial f with coefficients in F, the curve is said defined over F. The points of the curve C with coordinates in a field G are said rational over G and can be denoted C ). When G is the field of the rational numbers, one simply talks of rational points).
Q: Is curve said to be a complete intersection? ¶
Q: Is curve the parabola? ¶
A: Yes, and shown to the right.
Q: Is curve a curve with finite length? ¶
Q: Is curve a topological space which is locally homeomorphic to a line? ¶
Q: Is curve said to be regular if its derivative never vanishes? ¶
Q: Are curves called Riemann surfaces? ¶
Q: Is curve also called a Jordan curve? ¶
Q: Is curve a curve that forms a path whose starting point is also its ending point—that is? ¶
A: Yes, and a path from any of its points to the same point.
Q: Are curves the conics? ¶
A: Yes, and which are nonsingular curves of degree two and genus zero, and elliptic curves, which are nonsingular curves of genus one studied in number theory and which have important applications to cryptography.
Q: Is curve a set of points which? ¶
A: Yes, and near each of its points, looks like a line, up to a deformation.
Q: Are curves the advent of analytic geometry in the seventeenth century? ¶
Q: Is curve another unusual example? ¶