In[]:=
SpaceCurve::usage="The parametric equations are indicated below. References: Hollis, Selwyn. CalcLabs with Mathematica, Multivariable Calculus. Cengage Learning: 4a Ed: p.46, 2010; Personal. ";​​f[x_,y_]:=2
2
x
+2
2
y
-3;g[x_,y_]:=
2
x
-
2
y
;​​Show[surf1=Plot3D[f[x,y],{x,-1.7,1.7},{y,-1.7,1.7},ColorFunction"BlueGreenYellow",MeshFalse],surf2=Plot3D[g[x,y],{x,-1.7,1.7},{y,-1.7,1.7},ColorFunction"LightTemperatureMap",MeshFalse],AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic]​​f[rCos[t],rSin[t]]==g[rCos[t],rSin[t]];​​roft=r/.First@Solve[%,r]//Simplify;​​z[t_]:=g[roftCos[t],roftSin[t]];​​γ[t_]:={roftCos[t],roftSin[t],z[t]};​​​​
Out[]=
In[]:=
Animateγ[t_]:=-
3
Cos[t]
2-Cos[2t]
,-
3
Sin[t]
2-Cos[2t]
,
3
2
Cos[t]
2-Cos[2t]
-
3
2
Sin[t]
2-Cos[2t]
;​​ParametricPlot3D[γ[t],{t,0,θ1},AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},MeshFalse,AxesTrue,ColorFunctionFunction[{x,y,z},Hue[z]],BoxedFalse,PlotRange{{-2,2},{-1,3}},PerformanceGoal"Quality"],​​{θ1,0.1,2π,0.1},AnimationRunningFalse
Out[]=
θ1