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Math Help - Parameterisation

  1. #1
    Super Member Showcase_22's Avatar
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    Parameterisation

    Find a parameterisation of the circle of radius one formed by intersecting the plane x+y+z=0 and the sphere x^2+y^2+z^2=1.

    (Hint: Start by finding two orthogonal unit vectors in the plane).

    Starting with the hint, my first vector is \frac{1}{\sqrt{2}}\begin{pmatrix}<br />
1\\<br />
{-1}\\<br />
0\\<br />
\end{pmatrix} and my second vector is \frac{1}{\sqrt{2}} \begin{pmatrix}<br />
{-1}\\<br />
1\\<br />
0\\ \end{pmatrix}

    Where do I go from here?
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  2. #2
    Senior Member
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    Consider the unit circle: Center O (0,0), Right P (1,0), Up Q (0,1). The standard parameterization is R(t)=O+\cos(t)P+\sin(t)Q=(\cos(t),\sin(t)). The same formula works in your application: Find O,P,Q and use R(t)=O+\cos(t)P+\sin(t)Q
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