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Math Help - help with ellipse equation

  1. #1
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    help with ellipse equation

    Hi,
    I've got an ellipse defined by a central (steep) radius of curvature (R0) and a p-value (shape factor) that is p=1-e^2
    I've derived most of the formulas I need for calculating the local radius of curvature at given points. One problem I don't seem to be able to solve is the definition of the ellipse via the flat radius of curvature.
    The p-values I've got define the radius of the ellipse at its steep apex and from there it becomes flatter.
    I would want to define the ellipse via its flat radius and become steeper from there.
    I know that
    a = R0 * p
    b = SQRT(R0^2*p)
    Rflat = a^2/b
    so Rflat = (R0 * p)^2 / SQRT R0*p)

    can someone derive the expression for Rsteep as a function of Rflat and p?
    thanks heaps
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  2. #2
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    Quote Originally Posted by doceye View Post
    Hi,
    I've got an ellipse defined by a central (steep) radius of curvature (R0) and a p-value (shape factor) that is p=1-e^2
    I've derived most of the formulas I need for calculating the local radius of curvature at given points. One problem I don't seem to be able to solve is the definition of the ellipse via the flat radius of curvature.
    The p-values I've got define the radius of the ellipse at its steep apex and from there it becomes flatter.
    I would want to define the ellipse via its flat radius and become steeper from there.
    I know that
    a = R0 * p
    b = SQRT(R0^2*p)
    Rflat = a^2/b
    so Rflat = (R0 * p)^2 / SQRT R0*p)

    can someone derive the expression for Rsteep as a function of Rflat and p?
    thanks heaps
    R_{flat} = \frac{(R_0 p)^2}{\sqrt{R_0^2 p}}
    if I have that right.

    Then
    R_{flat} = \frac{R_0^2 p^2}{R_0 \sqrt{p}} = R_0 p^{3/2}

    R_0 = \frac{R_{flat}}{p^{3/2}}

    But I'm having a hard time believing you can do the rest of what you did and not see how to do this. Is this what you wanted?

    -Dan
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