let f(x,y) = 0 be the equation of a curve X in R^2, and suppose that Df(x,y) =/= 0 for all (x,y) in X.
a) find an equation for the closure of the cone CX in R^3 over X, when CX is the union of all the lines through the origin and points (x,y,1) with (x,y) in X.
b) if X has equation y = x^3 what is the equation of the closure of CX?
since f(x,y) = 0 and Df(x,y) =/= 0, the implicit function theorem implies that this curve is a smooth manifold and thus one variable can be expressed as a function of the other: x = g(y).
the problem i am having is generating the equation of the closure of the cone. i know the individual parts of it. it is the union of lines that go both through the origin and ones of the points (x,y,1) such that (s,y) is in X. i am having trouble formalizing this information into 1 equation. i changed (x,y,1) to (a,b,1) to avoid confusion between variables. a parametrization of a line that goes through that point and the origin would then be (sa, sb, s) so x = az and y = bz. then i defined F(x/z. y/z) = (a,b,1). would this be a valid equation for the closure?
then when X is described by y = x^3, the equation would be come F(x/z, y/z) = (a,a^3,1). is this correct?