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Thread: Parametric equation of the square? (or the cube in 3D)

  1. #1
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    Parametric equation of the square? (or the cube in 3D)

    Hi,

    I need to sample data in space on the faces of a 3D cube.
    Is there a parametric equation that approximates the square (in 2D) or the cube (3D) ?

    Thanks a lot!
    Benoit
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  2. #2
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    A parametric equation that describes a surface must have two parameters, call them $\displaystyle u$ and $\displaystyle v$. Suppose you want a plane that runs through a point $\displaystyle P_{0}$ which contains two nonparallel lines $\displaystyle l$ and $\displaystyle m$. (In the case of a square in the first quadrant of a Cartesian plane with a corner at the origin, your point will be anywhere in the square and the lines can be $\displaystyle y = x$ and $\displaystyle y = 2x$.) Suppose the equations of the lines are given as $\displaystyle a_{1}x + b_{1}y + c_{1}z = d_{1}$ and $\displaystyle a_{2}x + b_{2}y + c_{2}z = d_{2}$ for constants $\displaystyle a_{1}, a_{2}, b_{1}..., d_{2}$ and suppose the point's coordinates are $\displaystyle (x_{1}, y_{1}, z_{1})$. Then the parametric equations of the plane will be $\displaystyle x = x_{1} + ua_{1} + va_{2}, y = y_{1} + ub_{1} + vb_{2}, z = z_{1} + uc_{1} + vc_{2}$.

    Probably the lesson here is that, whatever you're doing, you don't want to be doing it with parametric equations. I could be wrong.
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  3. #3
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    Hi Ragnar,

    Thank you very much for your answer! In fact i do need to sample data on the surface of the cube (or more precisely on a $\displaystyle C^1$ surface very close to the cube).
    That's to find an optimal configuration of points on the cube for a given energy ( minimization with least square criteria).
    The best would be maybe to look at spherical coordinates, it might be more easy...

    Thanks again
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