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May 02, 2026

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But \(\sin^2\phi = 1 - \cos^2\phi = 1 - \left(rac{z}{
ight)^2 = 1 - rac{z^2}{x^2 + y^2 + z^2}\). So:
9c^2 \left(1 - rac{z^2}{x^2 + y^2 + z^2}
ight) = x^2 + y^2 + z^2.
Multiply through:
9c^2 - rac{9c^2 z^2}{x^2 + y^2 + z^2} = x^2 + y^2 + z^2.
Multiply both sides by \(x^2 + y^2 + z^2\):
9c^2(x^2 + y^2 + z^2) - 9c^2 z^2 = (x^2 + y^2 + z^2)^2.
9c^2(x^2 + y^2) = (x^2 + y^2 + z^2)^2.
This is the equation of a **torus** centered along the \(z\)-axis with...

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