bin[N_, k_] := Binomial[n, k]

bezier[V_] := Block[{n, k, out},
      (n = Length[V] - 1;
       out = Sum[V[[k + 1]] bin[ n, k] t^k(1 - t)^(n - k),
             {k, 0, n}];
       Return[Expand[out]])]

cur = bezier[{{0, 16}, {2, 0}, {45, 45}, {0, 2}, {16, 0}}]

dispbez[curve_] := ParametricPlot[curve, {t, 0, 1}, AspectRatio -> Automatic]

dispbez[cur]

revolve[curve_] := Block[{y, z},
              (y = curve[[1]];
               z = curve[[2]];
               Return[{y Cos[alpha], y Sin[alpha], z}])]

sur = revolve[cur]

PP3[surfaces_] := 
  param[surfaces, {t, 0, 1}, {alpha, 0, 2 Pi}, AspectRatio -> 1,
                Axes -> True, PlotRange -> All] /. param -> ParametricPlot3D


PP3[sur]

VPO3[surfaces_, a_, b_, c_] := 
  param[surfaces, {t, 0, 1}, {alpha, 0, 2Pi}, AspectRatio -> 1,
                            ViewPoint -> {a, b, c}, Axes -> True] /. 
    param -> ParametricPlot3D


VPO3[sur, -1, .6, .6]

VPO3[sur, -1, -.6, -.6]

chihat = revolve[{3t^2, Log[2 - t]}]

pichat = revolve[{3t^2, 7 Log[2 - t]^2}]

VPO3[chihat, 4, 3, .5]

VPO3[pichat, 4, 3, .5]


VPO3[pichat, 3, 3, -1]


sphere[C_, R_] := C + revolve[{R Cos[Pi (t - 1/2)], R Sin[Pi (t - 1/2)]}]

ball = sphere[{0, 0, 0}, 1]

PP3[ball]

VPO3[{pichat, ball}, 3, 3, -1]

ball2 = sphere[{0, 0, -4}, 3]

VPO3[{pichat, ball, ball2}, 3, 3, -1]

ParametricPlot[{2 + Sin[2 Pi t], 2 Pi t}, {t, 0, 1}]

amphor=revolve[{2 + Sin[2  Pi t], 2 Pi t}]

PP3[amphor]


abso[V_]:=Sqrt[V[[1]]^2+V[[2]]^2+V[[3]]^2]

TT[cur_] := Block[{ut, den, te},
		      (te = D[cur, t];
			     den = abso[te];
			     ut = te/den;
			Return[ut])]

NN[cur_] := Block[{ut, den, te, UTG},
		      (UTG = TT[cur];
		       ut = TT[UTG];
			  Return[ut])]

curve1[a_, r_, b_] := {r Cos[2 Pi b t], r  Sin[2 Pi  b t], a t}

Cross[{a1, a2, a3}, {b1, b2, b3}]

ra := Random[]

raunit := Block[{v, ut},
			      (v = {ra, ra, ra};
				     ut = v/abso[v];
				    Return[ut])]

gopher[cur_, r_] :=
   Block[{T, N, B, ut},
		       ( T = TT[cur]; 
			        N = TT[T]; 
						    B = Cross[T, N]; 
			      ut = cur + r  Cos[alpha] N + r  Sin[alpha] B;
			     Return[ut])]
			     

disp[curve_] := 
  para[curve, {t, 0, 1}, AspectRatio -> Automatic, Axes -> True] /. 
    para -> ParametricPlot

disp3[curve_] :=  para[curve, {t, 0, 1}, 
AspectRatio -> Automatic, Axes -> True] /.para ->ParametricPlot3D

body = bezier[{3, -1, 9, -3, 2}]

xaxis = bezier[{{0, 0}, {15, 0}}]

disp[{{15 t, body}, xaxis}]

disp[{t, t^2, t^3}]

cur[5, 3, 2]

GG1 = gopher[{5t, 8 t^2, 5 t^3}, body]


dispsurf[surfaces_, a_, b_, c_, m_] := 
  param[surfaces, {t, 0, 1}, {alpha, 0, 2 Pi}, AspectRatio -> Automatic,
			Axes -> True, ViewPoint -> {a, b, c}, PlotPoints -> m] /. 
    param -> ParametricPlot3D


dispsurf[GG1, 14, 2, 13, 40]

dispsurf[GG1, 14, 12, 13,30]

dispsurf[GG1, 14, -12, 13,20]