Bezier Curves

Math 3335

Helmut  Knaust - Department of Mathematical Sciences, University of Texas at El Paso, hknaust@utep.edu
9/15/16

2D - 1 control point

b0={1,1}
b1={0,2}
b2={2,3}

bc1[t_]=t b0+(1-t)b1
bc2[t_]=t b1+(1-t)b2

bc[t_]=t bc1[t]+(1-t) bc2[t]

ParametricPlot[{bc1[t],bc2[t],bc[t]},{t,0,1}]

3D -1 control point

b0={1,1,2}
b1={0,2,1}
b2={2,3,-1}

bc1[t_]=t b0+(1-t)b1
bc2[t_]=t b1+(1-t)b2

bc[t_]=t bc1[t]+(1-t) bc2[t]

ParametricPlot3D[{bc1[t],bc2[t],bc[t]},{t,0,1}]

2D - 2 control points

Manipulate[b0={1,2};b3={-2,2};bc1[t_]=t b0+(1-t)b1;bc2[t_]=t b1+(1-t)b2;
bc3[t_]=t b2+(1-t)b3;
c1[t_]=t bc1[t]+(1-t) bc2[t];
c2[t_]=t bc2[t]+(1-t) bc3[t];
bc[t_]=t c1[t]+(1-t)c2[t];ParametricPlot[{bc1[t],bc2[t],bc3[t],bc[t]},{t,0,1},PlotStyle->{Red,Red,Red,{AbsoluteThickness[3],Blue}},PlotRange->{{-3,3},{-3,3}},Axes->None,ImageSize->500,Epilog->{Blue,AbsolutePointSize[10],Point[b0],Point[b3]}],
{{b1,{-1,-1}}},{{b2,{-2,0}},{-2,-2},{2,2}},ControlType->Locator]

3D - 2 control points

b0={1,1,2}
b1={0,2,1}
b2={2,3,4}
b3={0,1,3}

bc1[t_]=t b0+(1-t)b1
bc2[t_]=t b1+(1-t)b2
bc3[t_]=t b2+(1-t)b3

c1[t_]=t bc1[t]+(1-t) bc2[t]
c2[t_]=t bc2[t]+(1-t) bc3[t]

bc[t_]=t c1[t]+(1-t)c2[t]

Expand[%]

ParametricPlot3D[{bc1[t],bc2[t],bc3[t],bc[t]},{t,0,1},PlotStyle->{Red,Red,Red,{AbsoluteThickness[3],Blue}}]