... polar¹

Note that Equations 8 and 11 require knowlege of which quadrant the angle lies in, because $\theta$ ranges over $[0,2\pi)$.

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... cone²

double cone means that it is two ``standard'' cones joined at their apices.

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... equation³

The equation itself is rather fierce and you would not be expected to do the full expansion in an exam. For those who are interested, it looks like this:
t⁴ ( x_D⁴ + y_D⁴ + z_D⁴ + 2 x_D² y_D² + 2 x_D² z_D² + 2 y_D² z_D² ) + t³ ( 4 x_D³ x_E + 4 y_D³ y_E + 4 z_D³ z_E + 4 x_D² y_D y_E + 4 x_D² z_D z_E + 4 x_D x_E y_D² + 4 y_D² z_D z_E + 4 x_D x_E z_D² + 4 y_D y_E z_D² ) + t² ( - 2 R² x_D² - 2 R² y_D² + 2 R² z_D² - 2 r² x_D² - 2 r² y_D² - 2 r² z_D² + 6 x_D² x_E² + 2 x_E² y_D² + 8 x_D x_E y_D y_E + 2 x_D² y_E² + 6 y_D² y_E² + 2 x_E² z_D² + 2 y_E² z_D² + 8 x_D x_E z_D z_E + 8 y_D y_E z_D z_E + 2 x_D² z_E² + 2 y_D² z_E² + 6 z_D² z_E² ) + t ( - 4 R² x_D x_E - 4 R² y_D y_E + 4 R² z_D z_E - 4 r² x_D x_E - 4 r² y_D y_E - 4 r² z_D z_E + 4 x_D x_E³ + 4 x_E² y_D y_E + 4 x_D x_E y_E² + 4 y_D y_E³ + 4 x_E² z_D z_E + 4 y_E² z_D z_E + 4 x_D x_E z_E² + 4 y_D y_E z_E² + 4 z_D z_E³ ) + ( R⁴ - 2 R² x_E² - 2 R² y_E² + 2 R² z_E² + r⁴ - 2 r² R² - 2 r² x_E² - 2 r² y_E² - 2 r² z_E² + x_E⁴ + y_E⁴ + z_E⁴ + 2 x_E² y_E² + 2 x_E² z_E² + 2 y_E² z_E² ) = 0

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... defined⁴

The disc is also arbitrarily defined if you want a circular disc, but it could be transformed to provide an ellipse.

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... definition⁵

The sphere has unit radius; the cylinder has unit radius and is aligned with the z-axis; the cone is similarly aligned and has unit slope; the torus is aligned with the z-axis.

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... transformation⁶

Where ${\bf T}$ is a translation, ${\bf R}$ a rotation, and ${\bf S}$ a scaling.

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... where⁷

If you are wondering where r has disappeared to, remember that, in a unit sphere, r=1.

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... being⁸

If you try this at home you will find that you will need to divide through by the factor $-1/(lmn\cos\phi)$.

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... doubles⁹

It doesn't quite double the number of points when the sequence is open and of finite length, but we will gloss over that at the moment.

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