Converting a parametric surface to an implicit (ordinary) equation

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Problem 2331 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.9 "Surfaces. Their Equations"

The parametric representation of surfaces, introduced by Gauss, consists in giving the coordinates of the points of the surface as functions of two parameters: x=φ(u,v)x=\varphi (u,v), y=ψ(u,v)y=\psi (u,v), z=ω(u,v)z=\omega (u,v).

The equation of a sphere can be represented in the form

x=asin⁡θcos⁡φ,y=asin⁡θsin⁡φ,z=acos⁡θ. x = a\sin \theta \cos \varphi , \qquad y = a\sin \theta \sin \varphi , \qquad z = a\cos \theta .

Find a similar form of the equation for an ellipsoid.

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Problem 2332 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.9 "Surfaces. Their Equations"

A surface is given by the equations

x=acos⁡4ucos⁡4v,y=acos⁡4usin⁡4v,z=asin⁡4u. x = a\cos ^{4}u\cos ^{4}v, \qquad y = a\cos ^{4}u\sin ^{4}v, \qquad z = a\sin ^{4}u .

Find its equation in ordinary form.

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Problem 2333 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.9 "Surfaces. Their Equations"

The same question (find the equation in ordinary form) for the surface given by the equations

x=acos⁡3ucos⁡3v,y=acos⁡3usin⁡3v,z=asin⁡3v. x = a\cos ^{3}u\cos ^{3}v, \qquad y = a\cos ^{3}u\sin ^{3}v, \qquad z = a\sin ^{3}v .
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Problem 2334 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.9 "Surfaces. Their Equations"

The same question (find the equation in ordinary form) for the surface given by the equations

x=au2+v2−1u2+v2+1,y=b2uu2+v2+1,z=c2vu2+v2+1. x = a\frac{u^{2}+v^{2}-1}{u^{2}+v^{2}+1}, \qquad y = b\frac{2u}{u^{2}+v^{2}+1}, \qquad z = c\frac{2v}{u^{2}+v^{2}+1} .
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References

[ГюКу03] Гюнтер Н.М., Кузьмин Р.О. Сборник задач по высшей математике. 13-е изд., Издательство «Лань», 2003.