Manipulate[ParametricPlot[{(1-t)u/3+ Sin[u t],Cos[u t]-1},{u,-π,π}],{t,0.01,1}]

Manipulate[ParametricPlot3D[{(1-t)u/3+ Sin[u t],-Cos[u t]-1,v},{u,-π,π},{v,-π,π}],{t,0.01,1}]

Manipulate[ParametricPlot3D[{(R+r( (1-t)θ +Cos[t θ]))Cos[ϕ],(R+r( (1-t)θ +Cos[t θ]))Sin[ϕ], r Sin[θ]}/.R->3/.r->1,{θ,-π,π},{ϕ,-π,π}],{t,0,1}]

Manipulate[ParametricPlot3D[{(R-r Cos[θ t])Cos[ϕ],(R-r Cos[t θ])Sin[ϕ],(1-t)θ+ r Sin[t θ]}/.R->2/.r->1,{θ,-π,π},{ϕ,-π,π},PlotRange->{{-π,π},{-π,π},{-π,π}}],{t,0,1}]

[{(1-t)u/3+ Sin[u t],-Cos[u t]-1,v}

Manipulate[ParametricPlot3D[{(1-t)u+ Sin[u t],-Cos[u t]-1,v},{u,-π,π},{v,-π,π}],{t,0.01,1}]

{(1-t)u+ Sin[u t],-Cos[u t]-1,v}

Manipulate[ParametricPlot3D[
{

(R-r  ((1-a)+a*Cos[θ t]))((1-a)ϕ+Sin[a ϕ]),
(R-r Cos[t θ])(-Cos[a ϕ]),
((1-t)θ+ r Sin[t θ])

}/.R->2/.r->1,{θ,-π,π},{ϕ,-π,π},PlotRange->{{-3π,3π},{-π,π},{-π,π}}],{a,0,1},{t,0,1}]

π/4

Manipulate[ParametricPlot3D[
{
(R-(r-a z) ((1-a)+a*Cos[θ t]))((1-a)ϕ+Sin[a ϕ]),
(R-(r-z) Cos[t θ])(-Cos[a ϕ]),
((1-t)θ+ (r-z) Sin[t θ])

}

/.R->2/.r->1,{θ,-π,π},{ϕ,-π,π},PlotRange->{{-π,π},{-π,π},{-π,π}}],{a,0,1},{t,0,1},{z,-1,1}]

PolarWarp[x_,y_,z_,

Manipulate[ParametricPlot3D[
{

(R-r Cos[θ t])((1-a)ϕ+Sin[a ϕ]),
(R-r Cos[t θ])(-Cos[a ϕ]),
(1-t)θ+ r Sin[t θ]

}/.R->2/.r->1,{θ,-π,π},{ϕ,-π,π},PlotRange->{{-1.2π,1.2π},{-π,π},{-π,π}}],{a,0,1},{t,0,1}]

Manipulate[ParametricPlot3D[{(R-r Cos[θ t])Cos[ϕ],(R-r Cos[t θ])Cos[ϕ],(1-t)θ+ r Sin[t θ]}/.R->2/.r->1,{θ,-π,π},{ϕ,-π,π},PlotRange->{{-π,π},{-π,π},{-π,π}}],{t,0,1}]