Osculating Plane
r[t_] = {t Cos[t], t Sin[t], t};
r1[t_] = D[r[t], t];
r2[t_] = D[r[t], {t, 2}];
p[t_] = r[t] + u r1[t] + v r2[t];
ut[t_] = Normalize[r1[t]];
un[t_] = Normalize[r2[t]];
Manipulate[Show[
Show[
ParametricPlot3D[r[t], {t, -3 Pi, 3 Pi},
PlotRange -> {{-10, 10}, {-10, 10}, {-10, 10}}],
ParametricPlot3D[{x, y, -10}, {x, -8, 8}, {y, -8, 8},
PlotStyle -> White, PerformanceGoal -> "Quality"],
ParametricPlot3D[p[t], {u, -1, 1}, {v, -1, 1},
PerformanceGoal -> "Quality"],
Graphics3D[{Red, PointSize[0.03], Point[r[t]]}],
Graphics3D[{Red, Arrow[{r[t], r[t] + 4 ut[t]}]}],
Graphics3D[{Blue, Arrow[{r[t], r[t] + 4 un[t]}]}]
]
, ViewPoint -> {1.3, -2.4, 2}
], {t, -3 Pi, 3 Pi}]
Helix in Mathematica
r[t_] = {3 Cos[t], 3 Sin[t], t};
r1[t_] = D[r[t], t];
r2[t_] = D[r[t], {t, 2}];
p[t_] = r[t] + u r1[t] + v r2[t];
ut[t_] = Normalize[r1[t]];
un[t_] = Normalize[r2[t]];
Manipulate[Show[
Show[
ParametricPlot3D[r[t], {t, 0, 4 Pi},
PlotRange -> {{-5, 5}, {-5, 5}, {0, 13}}, PlotStyle -> Red,
Boxed -> False, Axes -> False],
ParametricPlot3D[{x, y, 0}, {x, -5, 5}, {y, -5, 5},
PerformanceGoal -> "Quality"],
ParametricPlot3D[{3 Cos[x] , 3 Sin[x], y}, {x, 0, 2 Pi}, {y, 0,
10}, PlotStyle -> Directive[Green, Opacity[0.1]],
PerformanceGoal -> "Quality"],
Graphics3D[{Red, PointSize[0.03], Point[r[t]],
Text[Style["\[Alpha]", 14, Bold], r[t], {-2, -2}]}],
Graphics3D[{Red, Thick,
Line[{{3 Cos[t], 3 Sin[t], 0}, {3 Cos[t], 3 Sin[t], 13}}]}],
Graphics3D[{Red, Thick, Arrow[{r[t], r[t] + 5 ut[t]}],
Text[Style["\!\(\*OverscriptBox[\(a\), \(\[LongRightArrow]\)]\)",
14, Bold], r[t] + 5 ut[t], {1, 0}]}]
]
, ViewPoint -> {1.3, -2.4, 2}
], {t, 0, 11}]
Parametric Curves in Mathematica
r[u_, v_] = {u, v, Sin[u + v]};
r1[u_, v_] = D[r[u, v], u];
r1[u_, v_] = D[r[u, v], v];
ut1[u_, v_] = Normalize[r1[u, v]];
ut2[u_, v_] = Normalize[r2[u, v]];
Manipulate[Show[
Show[
ParametricPlot3D[r[u, v], {u, -2, 2}, {v, -2, 2},
PlotRange -> {{-2, 2}, {-2, 2}, {-3, 5}}, PlotStyle -> Red,
Boxed -> False, Axes -> False, PerformanceGoal -> "Quality"],
ParametricPlot3D[{x, y, -3}, {x, -2, 2}, {y, -2, 2},
PlotStyle -> Directive[Gray, Opacity[0.2]],
PerformanceGoal -> "Quality"],
ParametricPlot3D[r[u, v], {u, -2, 2}, PlotStyle -> {Blue, Thick}]
]
, ViewPoint -> {1.3, -2.4, 2}
], {v, -2, 2}]
Tangent Plane and Normal Line in Mathematica
r[u_, v_] = {u, v, Sin[u + v]};
r1[u_, v_] = Normalize[D[r[u, v], u]];
r2[u_, v_] = Normalize[D[r[u, v], v]];
n[u_, v_] = Cross[r1[u, v], r2[u, v]];
Manipulate[
Show[
ParametricPlot3D[r[u, v], {u, -2, 2}, {v, -2, 2},
PlotRange -> {{-2, 2}, {-2, 2}, {-5, 2}}, PlotStyle -> Red,
Boxed -> False, Axes -> False, PerformanceGoal -> "Quality"],
ParametricPlot3D[{x, y, -5}, {x, -2, 2}, {y, -2, 2},
PlotStyle -> Directive[Gray, Opacity[0.2]],
PerformanceGoal -> "Quality"],
ParametricPlot3D[
r[p, q] + s*r1[p, q] + t*r2[p, q], {s, -1, 1}, {t, -1, 1}],
Graphics3D[{Blue, PointSize[0.04], Point[r[p, q]]}],
Graphics3D[{Black, Thick, Arrow[{r[p, q], r[p, q] + r1[p, q]}]}],
Graphics3D[{Blue, Thick, Arrow[{r[p, q], r[p, q] + r2[p, q]}]}],
Graphics3D[{Red, Thick, Arrow[{r[p, q], r[p, q] + n[p, q]}]}]
], {p, -2, 2}, {q, -2, 2}]
Intersection of two surfaces in Mathematica
f = x^2 + y^2 + z^2 - 4;
fa = (x - 1)^2 + y^2 - 1; ContourPlot3D[{f == 0, fa == 0}, {x, -2,
2}, {y, -2, 2}, {z, -2, 2},
MeshFunctions -> {Function[{x, y, z, ff}, f - fa]},
MeshStyle -> {{Thickness[0.03], Red, Dashed}}, Mesh -> {{0}}]
Intersection of two surfaces in Mathematica
f = x^2 + y^2 + z^2 - 2;
fa = x^3 + y^2 - z^2; ContourPlot3D[{f, fa}, {x, -2, 2}, {y, -2,
2}, {z, -2, 2}, Mesh -> None,
BoundaryStyle -> {1 ->
None, {1, 2} -> {Red, Thickness[0.03], Dashed}}]
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