Representación de los vectores posición P(t), velocidad v(t) y aceleración a(t) en una función vectorial en el plano.
f1[t_] := {t^2 - 3, t}
f2[t_] := 2 {Cos[t], Sin[t]}
f3[t_] := {Cos[t] (1 - 2 Cos[t]), (1 - 2 Cos[t]) Sin[t]}
f4[t_] := 2 {Cos[t] Cos[2 t], Cos[2 t] Sin[t]}
fun[f_] :=
ParametricPlot[f[t], {t, -4, 4}, Ticks -> None, PlotRange -> 3,
ImageSize -> {50, 50}]
Manipulate[
Show[ParametricPlot[f[t], {t, -4, 4}, PlotRange -> 4],
Graphics[{Text["P(t)", f[r]/2 + {0, 0.2}],
Text["v(t)", f[r] + 1.1 f'[r]], Text["a(t)", f[r] + f''[r] 1.03],
Red, Arrow[{f[r], f[r] + f'[r]}], Green, Arrow[{{0, 0}, f[r]}],
Pink, Arrow[{f[r], f''[r] + f[r]}]}]], {r, -3,
3}, {{f, f1, "r(t)"}, {f1 -> fun[f1], f2 -> fun[f2], f3 -> fun[f3],
f4 -> fun[f4]}}]
Clear[a]
f1[t_] := {t^2 - 3, t^2, t}
f2[t_] := {2 Cos[t], 2 Sin[t], t/2}
f3[t_] := {3 Cos[t], Sin[t], 2}
f4[t_] := {t Cos[t], t Sin[t], t}/5
fun[f_, p_] :=
ParametricPlot3D[f[t], {t, -p, p}, Ticks -> None, PlotRange -> 3,
ImageSize -> {50, 50}]
func[f_, p_] :=
ParametricPlot3D[f[t], {t, -p, p}, Axes -> True,
AxesOrigin -> {0, 0, 0}, PlotRange -> 4]
g1 = func[f1, 3];
g2 = func[f2, 30];
g3 = func[f3, 4];
g4 = func[f4, 30];
Manipulate[{gra, a} =
Switch[f, f1, {g1, 1.2}, f2, {g2, 3}, f3, {g3, 3}, f4, {g4, 15}];
Show[gra,
Graphics3D[{Text["P(t)", f[r]/2 + {0, 0, 0.2}],
Text["v(t)", f[r] + 1.1 f'[r]], Text["a(t)", f[r] + f''[r] 1.03],
Red, Arrow[{f[r], f[r] + f'[r]}], Green, Arrow[{{0, 0, 0}, f[r]}],
Pink, Arrow[{f[r], f''[r] + f[r]}]}]], {r, -a,
a}, {{f, f1, "r(t)"}, {f1 -> fun[f1, 5], f2 -> fun[f2, 50],
f3 -> fun[f3, 4], f4 -> fun[f4, 50]}}]
Cambio del vector Tangente Unitario - Curvatura
Aunque la función no se encuentra parametrizada con respecto a la longitud de curva, se evidencia el cambio del vector tangente unitario con respecto a la curvatura de la parábola.
graf = ContourPlot[x^2 + y^2 == 1, {x, -2, 2}, {y, -2, 2},
Axes -> True, ContourStyle -> {Green, Dashed}];
Manipulate[cur = 2./(4 r^2 + 1)^(3/2);
Row[{Show[ParametricPlot[{t^2 + 1, t}, {t, -3, 3}],
Graphics[{Red,
Arrow[{{r^2 + 1,
r}, {2 r, 1}/Sqrt[4 r^2 + 1] + {r^2 + 1, r}}]}],
ImageSize -> Small],
Show[graf,
Graphics[{Red, Arrow[{{0, 0}, {2 r, 1}/Sqrt[4 r^2 + 1]}]}],
ImageSize -> Small,
PlotLabel -> Row[{"Curvatura = ", cur}]]}], {{r, -2, "t"}, -3, 3},
ContentSize -> {400, 240}]
Sistema de vectores TNB y Planos osculador y normal
Clear[a, bb]
f1[t_] := {t^2 - 3, t^2, t}
f2[t_] := {2 Cos[t], 2 Sin[t], t/2}
f3[t_] := {3 Cos[t], Sin[t], 2}
f4[t_] := {t Cos[t], t Sin[t], t}/5
fun[f_, p_] :=
ParametricPlot3D[f[t], {t, -p, p}, Ticks -> None, PlotRange -> 3,
ImageSize -> {50, 50}]
func[f_, p_] :=
ParametricPlot3D[f[t], {t, -p, p}, Axes -> True,
AxesOrigin -> {0, 0, 0}, PlotRange -> 4]
g1 = func[f1, 3];
g2 = func[f2, 30];
g3 = func[f3, 4];
g4 = func[f4, 30];
gg = Graphics3D[{Point[{0, 0, -10}]}];
Manipulate[
tt[t_] := Normalize[f'[t]];
nn[t_] := Normalize[f''[t]];
bb[t_] := Normalize@Cross[f'[t], f''[t]];
{gra, a} =
Switch[f, f1, {g1, 1.2}, f2, {g2, 3}, f3, {g3, 3}, f4, {g4, 15}];
Show[gra,
Graphics3D[{Text["T", f[r] + 1.1 tt[r]],
Text["N", f[r] + nn[r] 1.03], Text["B", f[r] + bb[r] 1.03], Red,
Arrow[{f[r], f[r] + tt[r]}], Green, Arrow[{{0, 0, 0}, f[r]}],
Pink, Arrow[{f[r], nn[r] + f[r]}], Orange,
Arrow[{f[r], bb[r] + f[r]}]}],
If[pn, ContourPlot3D[
Dot[f'[r], {x, y, z} - f[r]] == 0, {x, -4, 4}, {y, -4, 4}, {z, -4,
4}, Mesh -> None, ContourStyle -> Opacity[0.6]], gg],
If[po, ContourPlot3D[
Dot[bb[r], {x, y, z} - f[r]] == 0, {x, -4, 4}, {y, -4, 4}, {z, -4,
4}, Mesh -> None, ContourStyle -> Opacity[0.6]], gg]], {r, -a,
a}, {{f, f1, "r(t)"}, {f1 -> fun[f1, 5], f2 -> fun[f2, 50],
f3 -> fun[f3, 4], f4 -> fun[f4, 50]}}, {{pn, False,
"Plano Normal"}, {False, True}}, {{po, False,
"Plano Osculador"}, {False, True}}, ControlPlacement -> Left]
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