How to convert parametric form to implicit form?

Question

I'm looking to convert generic shapes' parametric form into their implicit form.

For instance, a sphere's:

Parametric form: $P(u,v) = (( ρ cos θ sin ϕ , ρ sin θ sin ϕ , ρ cos ϕ ))$

Implicit form: $F(x, y, z) = x^2 + y^2 + z^2 - 1$.

I need to do this for cones, cylinders, etc. but I'm at a loss as to where to start and nothing in the equations is popping out as obvious to me. I can figure out the other shapes if I can get a little direction for the sphere.

Thanks

The parametric form for the sphere is incorrect. You may want to check Wiki — GNUSupporter 8964民主女神地下教會, Dec 19 '17 at 04:53
This might be useful : https://math.stackexchange.com/a/150941/310635 — jonsno, Dec 19 '17 at 04:59

Narasimham · Accepted Answer · 2017-12-19T06:17:59.837

0

Plug in parametrization into implicit form to check by some trial/error if it tallies or not, seeing how sin,cos get eliminated. Basic conicoids and axisymmetric/closed/open 3D shapes have to be learnt up.

Parametrization you gave reminds one of a four cornered "samosa" snack.

edited Dec 19 '17 at 06:17

answered Dec 19 '17 at 06:10

Narasimham

40,495

ahh, samsoa, i want to eat this figure now! – Soumil Gupta Dec 17 '21 at 13:18

How to convert parametric form to implicit form?

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