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Math Help - Equation for an oblique cone?

  1. #1
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    Equation for an oblique cone?

    For my own purposes I have been trying to write a program that draws a radial gradient. In computer graphics, a radial gradient blends a colour from the edge of a circle to a focus point within. When the focus is at the centre of the circle, it is simple enough to treat the gradient as a right circular cone with a height of 1.0. Given a pixel at position x and y, z will represent the ratio with which to interpolate between colours. The equation I used is:

    {x^2 \over a^2} + {y^2 \over b^2} = {z^2 \over c^2}

    A gradient with a non-centred focus is an oblique cone, but I am having a difficult time finding the equation used to represent an oblique cone. The extent of my math education is single variable calculus about a dozen years ago, so I am not up to the task of trying to derive the equation myself. Does such an equation exist? Or is an oblique cone simply a right cone whose base is an inclined section? Any help would be appreciated.

    Thank you.
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  2. #2
    Member pflo's Avatar
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    Re: Equation for an oblique cone?

    Is this what you're looking for?

    x^2+y^2=\frac{a^2(h-z)^2}{h^2}
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  3. #3
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    Re: Equation for an oblique cone?

    Quote Originally Posted by pflo View Post
    x^2+y^2=\frac{a^2(h-z)^2}{h^2}
    This is a right cone. See the picture here.
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  4. #4
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    Re: Equation for an oblique cone?

    Unfortunately, no. As I mentioned, I'm beginning to think that an oblique cone is merely a right cone rotated such that an inclined elliptical section becomes its base.
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