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 ParametricSurfaceTangentPlane

Source Notebook

Compute the tangent plane of a parametric surface

Contributed by: Wolfram Staff (original content by Alfred Gray)

ResourceFunction["ParametricSurfaceTangentPlane"][s,{u,v},{u0,v0}]

computes the tangent plane at the point (u0,v0) of surface s parametrized by variables u and v.

Details and Options

ResourceFunction["ParametricSurfaceTangentPlane"] gives an InfinitePlane object.
The equation for the tangent plane of a two-variable function at a particular point can be written as T(x,y)=f(x0,y0)+fx(x0,y0)(x-x0)+fy(x0,y0)(y-y0).
The plane is spanned by two independent vectors normal to the surface normal.
Tangent planes to a surface are planes that touch the surface locally at one point. Globally, more complex intersections may exist.

Examples

Basic Examples (2) 

Definition of a sphere:

In[1]:=
Entity["Surface", "Sphere"]["ParametricEquations"][1][u, v]
Out[1]=

Tangent plane of the sphere:

In[2]:=
ResourceFunction["ParametricSurfaceTangentPlane"][ Entity["Surface", "Sphere"]["ParametricEquations"][1][u, v], {u, v}, {u0, v0}]
Out[2]=

Plot the tangent plane at any point on a sphere:

In[3]:=
Manipulate[ With[{p = {Cos[u0] Sin[v0], Sin[u0] Sin[v0], Cos[v0]}, p1 = {-(u - u0) Sin[u0] Sin[v0], (u - u0) Cos[u0] Sin[v0], 0}, p2 = {(v - v0) Cos[u0] Cos[v0], (v - v0) Cos[v0] Sin[ u0], -(v - v0) Sin[v0]}}, Graphics3D[{Opacity[.5], Sphere[], PointSize[.025], Point[Normalize[p1]/2 + p], Arrow[{p, Normalize[p1]/2 + p}], Point[Normalize[p2]/2 + p], Arrow[{p, Normalize[p2]/2 + p}], Red, Point[p], Blue, Opacity[.25], ResourceFunction["ParametricSurfaceTangentPlane"][ Entity["Surface", "Sphere"]["ParametricEquations"][1][u, v], {u, v}, {u0, v0}]} /. {u -> Cos[u0], v -> Sin[v0]}, Axes -> True]], {{u0, -1}, -\[Pi], \[Pi]}, {{v0, 1.}, -\[Pi], \[Pi]}]
Out[3]=

Choose a surface:

In[4]:=
surf = {u, v, Sin[u] Cos[v]}
Out[4]=

Plot it:

In[5]:=
plot = ParametricPlot3D[ Evaluate[surf], {u, 0, 3 \[Pi]}, {v, 0, 3 \[Pi]}, Mesh -> False, PlotStyle -> Opacity[.75], PlotRange -> {{0, 3 \[Pi]}, {0, 3 \[Pi]}, {-1.1, 1.4}}, SphericalRegion -> True]
Out[5]=

Compute the tangent plane of the surface:

In[6]:=
tp = ResourceFunction["ParametricSurfaceTangentPlane"][ surf, {u, v}, {u0, v0}]
Out[6]=

Plot the tangent plane at any point (in blue) on the surface; the red curves are the intersection of the plane and the surface:

In[7]:=
Manipulate[ Show[plot, Graphics3D[{Blue, Opacity[.5], ResourceFunction["ParametricSurfaceTangentPlane"][ surf, {u, v}, {u0, v0}] /. {u -> Cos[u0], v -> Sin[v0]}, Red, Thickness[.0075], Cases[Normal[ ContourPlot[ Cos[v] Sin[u] == Cos[v0] Sin[u0] + (u - u0) Cos[u0] Cos[v0] - (v - v0) Sin[ u0] Sin[v0], {u, 0, 3 \[Pi]}, {v, 0, 3 \[Pi]}]], _Line, \[Infinity]] /. {u_Real, v_Real} -> {u, v, Sin[u] Cos[v]}, PointSize[.035], Green, Point[{u0, v0, Sin[u0] Cos[v0]}]}]], {{u0, 2.82}, 0, 3 \[Pi]}, {{v0, 4.43}, 0, 3 \[Pi]}]
Out[7]=

Use ClipPlanes to view the surface only at one side of the plane:

In[8]:=
Manipulate[ Show[ParametricPlot3D[ Evaluate[surf], {u, 0, 3 \[Pi]}, {v, 0, 3 \[Pi]}, PlotStyle -> Opacity[.75], ClipPlanes -> (ResourceFunction["ParametricSurfaceTangentPlane"][ surf, {u, v}, {u0, v0}] /. {u -> Cos[u0], v -> Sin[v0]}), ClipPlanesStyle -> Directive[Opacity[.5], Blue], Mesh -> False, PlotRange -> {{0, 3 \[Pi]}, {0, 3 \[Pi]}, {-1.1, 1.4}}], Graphics3D[{PointSize[.035], Green, Point[{u0, v0, Sin[u0] Cos[v0]}]}]], {{u0, 2.82}, 0, 3 \[Pi]}, {{v0, 4.43}, 0, 3 \[Pi]}]
Out[8]=

Publisher

Enrique Zeleny

Version History

  • 1.0.0 – 16 July 2020

Source Metadata

  • Citation:
    • Gray, A., Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, 1997.

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