Lemniscate

lem[t_]:={Cos[t],Sin[t]Cos[t]}/(1+Sin[t]^2);
ParametricPlot[lem[t],{t,-Pi,Pi}]

lemP[u_,v_]:=Join[lem[u],{v}]//Evaluate;
ParametricPlot3D[lemP[u,v],{u,-Pi,Pi},{v,-1,1}]

lemI[x_,y_]:=(x^2+y^2)^2-(x^2-y^2);
ContourPlot[lemI[x,y]==0,{x,-1.1,1.1},{y,-0.4,0.4},
	AspectRatio->Automatic]

ContourPlot3D[lemI[x,y]==0,{x,-1.1,1.1},{y,-0.4,0.4},{z,-1,1},BoxRatios->Automatic]

Problem 1-1 (Dini’s pseudosphere)

p[t_][u_,v_]:={
Cos[t] Sech[v] Cos[u],
Cos[t]Sech[v] Sin[u],
Cos[t ](v-Tanh[v]) + Sin[t] u}

ParametricPlot3D[p[0][u,v],{u,-Pi,Pi},{v,-3,3}]

ParametricPlot3D[p[0.2][u,v],{u,-Pi,Pi},{v,-3,3}]

ListAnimate[Table[
ParametricPlot3D[p[t][u,v],{u,-Pi,Pi},{v,-3,3},
	PlotRange->{{-1,1},{-1,1},{-3,3}}],{t,-Pi/2,Pi/2,Pi/8}]]

Problem 1 - 2

ContourPlot3D[x^4+y^4+z^4-1==0,{x,-1.1,1.1},{y,-1.1,1.1},{z,-1.1,1.1}]