Results 1 to 2 of 2

Math Help - finding equation of closure of a cone

  1. #1
    Member
    Joined
    Aug 2008
    Posts
    249

    finding equation of closure of a cone

    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?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Aug 2008
    Posts
    249
    never mind, i know how to find the equation now. what i still don't understand is what my book means by closure of a cone. is this the same closure as in the closure of a set X which is a subset of R^n? if X is described by y = x^3 then apparently the y axis is in the closure of the cone. i cannot see this geometrically as it is a slightly complicated surface. is there another way to explain the closure of a cone?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Relation between topological closure and algebraic closure
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: November 4th 2010, 02:45 PM
  2. Finding height of cone
    Posted in the Geometry Forum
    Replies: 1
    Last Post: September 8th 2009, 11:52 AM
  3. Replies: 6
    Last Post: February 11th 2009, 12:56 PM
  4. Finding the closure of sets in a given topology
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: February 7th 2009, 10:01 AM
  5. [SOLVED] [SOLVED] Finding diameter of a cone
    Posted in the Geometry Forum
    Replies: 2
    Last Post: May 21st 2006, 08:45 PM

Search Tags


/mathhelpforum @mathhelpforum