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Cylinder
We have the cylinder x2 + (y-3)2 = r2 where r=3 and z ϵ [0, 5].
In the parameterization of a surface in 3D, we need two parameters.
In the parameterization of cylinder, we use cylinder coordinates with the two parameters θ and z:
x = rcos(θ)
y = rsin(θ)
z = z
z ϵ [0, zmax] = [0, 5]
θ ϵ [0, θmax] = [0,2π]
x2 + (y-3)2 = 32
x2 + y2 - 6y + 9 = 9
x2 + y2 - 6y = 0
r2 - 6rsin(θ) = 0
r(r - 6sin(θ))= 0
r = 0 v r - 6sin(θ) = 0
r = 6sin(θ)
A vector from origin to the point P = (x,y,z) is then given by:
r = [rcos(θ), rsin(θ), z]
= [6sin(θ)cos(θ), 6sin(θ)sin(θ), z]
= [3sin(2θ),6sin2(θ), z]
Use the sliders z and θ to move the point P on the cylinder surface.
Use the sliders zmax and θmax to see how these two parameters generate the cylinder surface.
Follow this link if you want to see the algebra code of this application (select 'Open in App').
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