Homotopy Examples

Example 1

Homotopy between f:[0,2]->ℝ defined by f(x)=1+(((x^2)(x-2))^2) and g:[0,2]->ℝ defined by g(x)=1.

Manipulate[Plot[{1+(1-t)(x^2)(((x-2))^2),1+(x^2)(((x-2))^2),1},{x,0,2},PlotRange->{0,3},AxesLabel->{"ℝ","ℝ"},PlotStyle->{Thick,Thin,Thin}],{t,0,1}]

Plot3D[1+(1-t)(x^2)(((x-2))^2),{x,0,2},{t,0,1},AxesLabel->Automatic,BoxRatios->{2,1,1}]

Example 2

Homotopy between the inclusion map f:(S^1)->(ℝ^2) defined f(x,y)=(x,y) and the constant map g:(S^1)->(ℝ^2) defined by g(x,y)=(0,0).

Manipulate[Graphics[{Orange,Circle[],Blue, Thick,Circle[{0,0},1-t]},Axes->True],{t,0,1}]

Example 3

Homotopy between any two continuous functions  f:ℝ->ℝ and g:ℝ->ℝ.

f[x_]:=Sin[x];
g[x_]:=Exp[x];

Manipulate[Plot[{(1-t)f[x]+t g[x],f[x],g[x]},{x,-5,5},PlotRange->{-5,5},AxesLabel->{"ℝ","ℝ"},PlotStyle->{Thick,Thin,Thin}],{t,0,1}]

Plot3D[1+(1-t)(x^2)(((x-2))^2),{x,0,2},{t,0,1},AxesLabel->Automatic,BoxRatios->{2,1,1}]