Function Repository Resource:

ApproximatedSurface

Approximate a parametric surface with different graphics primitives

Contributed by: Enrique Zeleny

ResourceFunction["ApproximatedSurface"][surf,{u,u₀,u_f,m},{v,v₀,v_f,n},"prim"]

compute m×n points of a parametric surface varying u from u₀ to u_f and v from v₀ to v_f using graphics primitives of the type "prim".

ResourceFunction["ApproximatedSurface"][surf,{u,u₀,u_f,m},{v,v₀,v_f,n},{"prim",c₁,c₂}]

compute the parametric surface varying a parameters c₁ and c₂, when available.

Details

Possible values of "prim" are:

"Coordinates"

coordinates of mesh points

"Point"

points

"Arrow"

arrows

"PointLine"

points and lines

"LineSphere"

lines and spheres

"SphereTube"

spheres and tubes

"Sphere"

spheres

"Cuboid"

cuboids

"Cone"

cones

"Cylinder"

cylinders

"Line"

lines

"XTransversalLine"

lines in the x direction

"YTransversalLine"

lines in the y direction

"BezierCurve"

Bézier curves

"Polygon"

a mesh made with polygons

"TransversalPolygon"

polygons in the x and y direction

"XTransversalPolygon"

polygons in the x direction

"YTransversalPolygon"

polygons in the y direction

"Tube"

squares with circular hole

"XTransversalTube"

tubes in the x direction

"YTransversalTube"

tubes in the y direction

"Mesh"

squared mesh

"RandomColorMesh"

randomly colored mesh

"XTransversalStrip"

strips in the x direction

"YTransversalStrip"

strips in the y direction

"Holed"

delete alternating polygons

"Rhombus"

rhombuses

"RhombusHoled"

delete alternating rhombuses

"RandomPolygon"

randomly deleted polygons in the mesh

"DiskHoled"

squares with a circular hole

"Triangle"

squares subdivided into triangles

"XTransversalTriangle"

triangles in the x direction

"YTransversalTriangle"

triangles in the y direction

"ShrinkPolygon"

shrink polygons

"RotateScalePolygon"

rotate and scale polygons in the mesh

"PerturbedMesh"

randomly translate polygons

"PerturbedVertices"

randomly translate vertices

"Distorted"

translate vertices around an axis

"BSplineSurfaceOpen"

an open B–spline surface

"BSplineSurfaceClosed"

a closed B–spline surface

"ShortestTour"

shortest tour through the mesh points

"OutlinePolygons"

outlined polygons

"HexagonalMesh"

hexagonal mesh

"TriangulatedHexagonalMesh”

triangulated hexagonal mesh

"HexagonalMeshHoled"

triangulated hexagonal mesh with a deleted triangle

"OpacitySphereHexagonalMesh"

hexagonal mesh and spheres with opacity

"MeshRegion"

mesh regions

"DiscretizeRegion"

discretized polygons

"Points"

region of points

"SamplePoints"

sample points

"Lines"

region of lines

"Polygons"

region of polygons

"SphereAndTube"

region of spheres and tubes

"SmoothShading"

smooth shading region

"FaceNormals"

region with face normals

"CuboidRegion"

region with cuboids

"DelaunayMesh"

Delaunay mesh

"ConvexHullMesh"

convex hull mesh

texture

an image to be used as a texture

simple polyhedra

a simple polyhedra

custom polygon

an arbitrary polygon

Examples

Basic Examples (2)

Parametric equations for a Klein bottle:

In[1]:=

Out[1]=

Get the coordinates of the mesh of the surface:

In[2]:=

$ResourceFunction[ "ApproximatedSurface"][kb, {u, 0, 2 \[Pi], 35}, {v, 0, 2 \[Pi], 35}, "Coordinates"]$

Out[2]=

A surface plotted with ParametricPlot3D:

In[3]:=

$ParametricPlot3D[ Evaluate@Entity["Surface", "KleinBottle"][ EntityProperty["Surface", "ParametricEquations"]][1, 2, 3][u, v], {u, 0, 2 \[Pi]}, {v, 0, 2 \[Pi]}, Boxed -> False, Axes -> False]$

Out[3]=

Using points:

In[4]:=

$Graphics3D[ ResourceFunction["ApproximatedSurface"][ kb, {u, 0, 2 \[Pi], 35}, {v, 0, 2 \[Pi], 35}, "Point"], Boxed -> False, Axes -> False]$

Out[4]=

Using arrows:

In[5]:=

$Graphics3D[{Arrowheads[.015], ResourceFunction["ApproximatedSurface"][ kb, {u, 0, 2 \[Pi], 35}, {v, 0, 2 \[Pi], 35}, "Arrow"]}, Boxed -> False, Axes -> False]$

Out[5]=

With lines and points:

In[6]:=

$Graphics3D[ ResourceFunction["ApproximatedSurface"][ kb, {u, 0, 2 \[Pi], 35}, {v, 0, 2 \[Pi], 35}, "PointLine"], Boxed -> False, Axes -> False]$

Out[6]=

With lines and spheres:

In[7]:=

$Graphics3D[ ResourceFunction["ApproximatedSurface"][ kb, {u, 0, 2 \[Pi], 35}, {v, 0, 2 \[Pi], 35}, "LineSphere"], Boxed -> False, Axes -> False]$

Out[7]=

With spheres and tubes:

In[8]:=

$Graphics3D[ ResourceFunction["ApproximatedSurface"][ kb, {u, 0, 2 \[Pi], 35}, {v, 0, 2 \[Pi], 35}, {"SphereTube", .1, .05}], Boxed -> False, Axes -> False]$

Out[8]=

Using only spheres:

In[9]:=

$Graphics3D[ ResourceFunction["ApproximatedSurface"][ kb, {u, 0, 2 \[Pi], 35}, {v, 0, 2 \[Pi], 35}, "Sphere"], Boxed -> False, Axes -> False]$

Out[9]=

Modify the radius of the spheres:

In[10]:=

$Graphics3D[ ResourceFunction["ApproximatedSurface"][ kb, {u, 0, 2 \[Pi], 35}, {v, 0, 2 \[Pi], 35}, {"Sphere", .1}], Boxed -> False, Axes -> False]$

Out[10]=

Using cuboids:

In[11]:=

$Graphics3D[ ResourceFunction["ApproximatedSurface"][ kb, {u, 0, 2 \[Pi], 35}, {v, 0, 2 \[Pi], 35}, {"Cuboid", .2}], Boxed -> False, Axes -> False]$

Out[11]=

Using other geometrical objects:

In[12]:=

$Graphics3D[ ResourceFunction["ApproximatedSurface"][ kb, {u, 0, 2 \[Pi], 35}, {v, 0, 2 \[Pi], 35}, {"Cone", .1, .2}], Boxed -> False, Axes -> False]$

Out[12]=

In[13]:=

$Graphics3D[ ResourceFunction["ApproximatedSurface"][ kb, {u, 0, 2 \[Pi], 35}, {v, 0, 2 \[Pi], 35}, {"Cylinder", .1, .2}], Boxed -> False, Axes -> False]$

Out[13]=

In[14]:=

$Graphics3D[ ResourceFunction["ApproximatedSurface"][ kb, {u, 0, 2 \[Pi], 35}, {v, 0, 2 \[Pi], 35}, Dodecahedron[]], Boxed -> False, Axes -> False]$

Out[14]=

Using lines:

In[15]:=

$GraphicsRow[ Graphics3D[ ResourceFunction["ApproximatedSurface"][ kb, {u, 0, 2 \[Pi], 35}, {v, 0, 2 \[Pi], 35}, #], Boxed -> False, Axes -> False] & /@ {"XTransversalLine", "YTransversalLine", "Line"}]$

Out[15]=

Using BezierCurve:

In[16]:=

$Graphics3D[ ResourceFunction["ApproximatedSurface"][ kb, {u, 0, 2 \[Pi], 35}, {v, 0, 2 \[Pi], 35}, "BezierCurve"], Boxed -> False, Axes -> False]$

Out[16]=

Using transversal polygons:

In[17]:=

$GraphicsRow[ Graphics3D[ ResourceFunction["ApproximatedSurface"][ kb, {u, 0, 2 \[Pi], 35}, {v, 0, 2 \[Pi], 35}, #], Boxed -> False, Axes -> False] & /@ {"XTransversalPolygon", "YTransversalPolygon"}]$

Out[17]=

In[18]:=

$Graphics3D[ ResourceFunction["ApproximatedSurface"][ kb, {u, 0, 2 \[Pi], 35}, {v, 0, 2 \[Pi], 35}, "TransversalPolygon"], Boxed -> False, Axes -> False]$

Out[18]=

Using tubes:

In[19]:=

$GraphicsRow[ Graphics3D[ ResourceFunction["ApproximatedSurface"][ kb, {u, 0, 2 \[Pi], 35}, {v, 0, 2 \[Pi], 35}, #], Boxed -> False, Axes -> False] & /@ {"XTransversalTube", "YTransversalTube", "Tube"}]$

Out[19]=

Using polygons:

In[20]:=

$Graphics3D[ ResourceFunction["ApproximatedSurface"][ kb, {u, 0, 2 \[Pi], 70}, {v, 0, 2 \[Pi], 70}, "Mesh"], Boxed -> False, Axes -> False]$

Out[20]=

With a texture:

In[21]:=

$Graphics3D[{EdgeForm[], ResourceFunction["ApproximatedSurface"][ kb, {u, 0, 2 \[Pi], 35}, {v, 0, 2 \[Pi], 35}, \!\(\* GraphicsBox[ TagBox[RasterBox[CompressedData[" 1:eJzt1bGrl1Ucx/GDZYUEEgghhCFCGEFElyCi+3tKCCOE0AIJJLiFEihlilRG 4OLScpcWG1p0aHJpuUuLDi02NLm03KXFhpbb0NQV+jzP+Z4/IV4HEt8pt+9z vpxehz/67NS5Pa21y0/s/nJq44s3L13a+Oq9xx7G2Y0Lay9feGT3t2f/++fR 9vBcm1o5S++7f+T12jdLb97ZGf78yHrfv947sV7//pXSz5/ZXB9+fuk/T24N P+9e6dvHt4efv1N669DeVf3v7S/9/Y2Dq2H+0l8feHFV53m19JnNY6vhe0uv 7zu9qvNulH72+vnVcD+ruo9rq/o910t/vv7t8H2bpbevfjd8743Sp7Z+GL7/ Zum7Oz8O93G79NraT8P9bJW+dfHn4b7ulD5w+5fh/u6Vvv7gt+E+75f+++jv w/1ulz537o/hvh+Uvn/zr+H+d0q/vf3PsI821e/ZM9X97J3q9z0+1X3tm+r3 PjnV/e2f6vc/NdV9HpjqfTw91f0enOr9PDPVfR+a6n0dnur+j0z1/p4rfXfn 6FTv84XSu+9lqvf7Uum1tbWp3vcrpXff11Tv/7XSty6uT3Ufbwz7ODbs461h H8eHfbwz7OPEsI93h32cHPbx/rCP08M+Phj2cWbYx4fDPjaGfXw87OPcsI9P hn2cH/bx6bCPi8M+Lg/7uDLs48thH1eHfXwz7OPa0Pl3+R6e8IQnbTnD/794 whOetPnwhCc8SfOEJzzhSTcPT3jSvT+e8IQnaZ7whCc8SfOEJzzhyTIvT3jC kzRPeMITnqR5whOe8KTNhyc84UmaJzzhCU+6eXjCk+798YQnPEnzhCc84Uma JzzhCU+WeXnCE56kecITnvAkzROe8IQnbT484QlP0jzhCU940s3DE550748n POFJmic84QlP0jzhCU94sszLE57wJM0TnvCEJ2me8IQnPGnz4QlPeJLmCU94 wpNuHp7wpHt/POEJT9I84QlPeJLmCU94wpNlXp7whCdpnvCEJzxJ84QnPOFJ mw9PeMKTNE94whOedPPwhCfd++MJT3iS5glPeMKTNE94whOeLPPyhCc8SfOE JzzhSZonPOEJT9p8eMITnqR5whOe8KSbhyc86d4fT3jCkzRPeMITnqR5whOe 8GSZlyc84UmaJzzhCU/SPOEJT3jS5sMTnvAkzROe8IQn3Tw84Un3/njCE56k ecITnvAkzROe8IQny7w84QlP0jzhCU94kuYJT3jCkzYfnvCEJ2me8IQnPOnm 4QlPuvfHE57wJM0TnvCEJ2me8IQnPFnm5QlPeJLmCU94wpM0T3jCE560+fCE JzxJ84QnPOFJNw9PeNK9P57whCdpnvCEJzxJ84QnPOHJMi9PeMKTNE94whOe pHnCE57wpM2HJzzhSZonPOEJT7p5eMKT7v3xhCc8SfOEJzzhSZonPOEJT5Z5 ecITnqR5whOe8CTNE57whCdtPjzhCU/SPOEJT3jSzcMTnnTvjyc84UmaJzzh CU/SPOEJT3iyzMsTnvAkzROe8IQnaZ7whCc8afPhCU94kuYJT3jCk24envCk e3884QlP0jzhCU94kuYJT3jCk2VenvCEJ2me8IQnPEnzhCc84UmbD094wpM0 T3jCE5508/CEJ9374wlPeJLmCU94wpM0T3jCE54s8/KEJzxJ84QnPOFJmic8 4QlP2nx4whOepHnCE57wpJuHJzzp3h9PeMKTNE94whOepHnCE57wZJmXJzzh SZonPOEJT9I84QlPeNLmwxOe8CTNE57whCfdPDzhSff+eMITnqR5whOe8CTN E57whCfLvDzhCU/SPOEJT3iS5glPeMKTNh+e8IQnaZ7whCc86ebhCU+698cT nvAkzROe8IQnaZ7whCc8WeblCU94kuYJT3jCkzRPeMITnrT58IQnPEnzhCc8 4Uk3D0940r0/nvCEJ2me8IQnPEnzhCc84ckyL094wpM0T3jCE56kecITnvCk zYcnPOFJmic84QlPunl4wpPu/fGEJzxJ84QnPOFJmic84QlPlnl5whOepHnC E57wJM0TnvCEJ20+POEJT9I84QlPeNLNwxOedO+PJzzhSZonPOEJT9I84QlP eLLMyxOe8CTNE57whCdpnvCEJzxp8+EJT3iS5glPeMKTbh6e8KR7fzzhCU/S POEJT3iS5glPeMKTZV6e8IQnaZ78/zz5F/JBkaE= "], {{0, 101.}, {101., 0}}, { 0., 1.}, ColorFunction->RGBColor], BoxForm`ImageTag["Real32", ColorSpace -> "RGB", Interleaving -> True], Selectable->False], DefaultBaseStyle->"ImageGraphics", ImageSizeRaw->{101., 101.}, PlotRange->{{0, 101.}, {0, 101.}}]\)]}, Boxed -> False, Axes -> False]$

Out[21]=

A randomly colored mesh:

In[22]:=

$Graphics3D[ ResourceFunction["ApproximatedSurface"][ kb, {u, 0, 2 \[Pi], 70}, {v, 0, 2 \[Pi], 70}, "RandomColorMesh"], Boxed -> False, Axes -> False, Lighting -> "Neutral"]$

Out[22]=

With strips:

In[23]:=

$Graphics3D[ ResourceFunction["ApproximatedSurface"][ kb, {u, 0, 2 \[Pi], 34}, {v, 0, 2 \[Pi], 35}, "XTransversalStrip"], Boxed -> False, Axes -> False]$

Out[23]=

With holes:

In[24]:=

$Graphics3D[ ResourceFunction["ApproximatedSurface"][ kb, {u, 0, 2 \[Pi], 35}, {v, 0, 2 \[Pi], 35}, "Holed"], Boxed -> False, Axes -> False]$

Out[24]=

Using rhombuses:

In[25]:=

$Graphics3D[ ResourceFunction["ApproximatedSurface"][ kb, {u, 0, 2 \[Pi], 35}, {v, 0, 2 \[Pi], 34}, "Rhombus"], Boxed -> False, Axes -> False]$

Out[25]=

In[26]:=

$Graphics3D[ ResourceFunction["ApproximatedSurface"][ kb, {u, 0, 2 \[Pi], 35}, {v, 0, 2 \[Pi], 34}, "RhombusHoled"], Boxed -> False, Axes -> False, PlotRange -> All]$

Out[26]=

Randomly remove polygons:

In[27]:=

$Graphics3D[ ResourceFunction["ApproximatedSurface"][ kb, {u, 0, 2 \[Pi], 35}, {v, 0, 2 \[Pi], 35}, "RandomPolygon"], Boxed -> False, Axes -> False]$

Out[27]=

Use transparency in a texture to put in holes:

In[28]:=

$Graphics3D[ ResourceFunction["ApproximatedSurface"][ kb, {u, 0, 2 \[Pi], 70}, {v, 0, 2 \[Pi], 70}, "DiskHoled"], Boxed -> False, Axes -> False]$

Out[28]=

Using triangles:

In[29]:=

$Graphics3D[ ResourceFunction["ApproximatedSurface"][ kb, {u, 0, 2 \[Pi], 40}, {v, 0, 2 \[Pi], 70}, "XTransversalTriangle"], Boxed -> False, Axes -> False]$

Out[29]=

In[30]:=

$Graphics3D[ ResourceFunction["ApproximatedSurface"][ kb, {u, 0, 2 \[Pi], 25}, {v, 0, 2 \[Pi], 25}, "Triangle"], Boxed -> False, Axes -> False]$

Out[30]=

Shrink the polygons:

In[31]:=

$Graphics3D[ ResourceFunction["ApproximatedSurface"][ kb, {u, 0, 2 \[Pi], 35}, {v, 0, 2 \[Pi], 35}, "ShrinkPolygon"], Boxed -> False, Axes -> False]$

Out[31]=

Rotate and scale the polygons:

In[32]:=

$Graphics3D[ ResourceFunction["ApproximatedSurface"][ kb, {u, 0, 2 \[Pi], 25}, {v, 0, 2 \[Pi], 25}, {"RotateScalePolygon", \[Pi]/6., .7}], Boxed -> False, Axes -> False]$

Out[32]=

Randomly translate polygons:

In[33]:=

$Graphics3D[ ResourceFunction["ApproximatedSurface"][ kb, {u, 0, 2 \[Pi], 70}, {v, 0, 2 \[Pi], 70}, "PerturbedMesh"], Boxed -> False, Axes -> False]$

Out[33]=

Randomly translate vertices:

In[34]:=

$Graphics3D[{EdgeForm[], ResourceFunction["ApproximatedSurface"][ kb, {u, 0, 2 \[Pi], 35}, {v, 0, 2 \[Pi], 35}, {"PerturbedVertices", .1}]}, Boxed -> False, Axes -> False]$

Out[34]=

Distort polygons around an axis of rotation with a varying parameter:

In[35]:=

$Graphics3D[ ResourceFunction["ApproximatedSurface"][ kb, {u, 0, 2 \[Pi], 70}, {v, 0, 2 \[Pi], 70}, {"Distorted", 4/11, {0, 0, 1}}], Boxed -> False, Axes -> False]$

Out[35]=

Use a BSplineSurface:

In[36]:=

$Graphics3D[ ResourceFunction["ApproximatedSurface"][ kb, {u, 0, 2 \[Pi], 15}, {v, 0, 2 \[Pi], 15}, "BSplineSurfaceClosed"], Boxed -> False, Axes -> False]$

Out[36]=

Shortest tour through the mesh:

In[37]:=

$Graphics3D[ ResourceFunction["ApproximatedSurface"][ kb, {u, 0, 2 \[Pi], 70}, {v, 0, 2 \[Pi], 70}, "ShortestTour"], Boxed -> False, Axes -> False]$

Out[37]=

For a surface with edges:

In[38]:=

$ParametricPlot3D[{(1 + v^2) Cos[u], (1 + v^2) Sin[u], 2 v}, {u, 0, \[Pi]}, {v, -1, 1}]$

Out[38]=

An open B-spline:

In[39]:=

$Graphics3D[{Opacity[.5], ResourceFunction[ "ApproximatedSurface"][{(1 + v^2) Cos[u], (1 + v^2) Sin[u], 2 v}, {u, 0, \[Pi], 15}, {v, -1, 1, 15}, "BSplineSurfaceOpen"]}, Boxed -> False, Axes -> False]$

Out[39]=

A closed B-spline:

In[40]:=

$Graphics3D[{Opacity[.5], ResourceFunction[ "ApproximatedSurface"][{(1 + v^2) Cos[u], (1 + v^2) Sin[u], 2 v}, {u, 0, \[Pi], 15}, {v, -1, 1, 15}, "BSplineSurfaceClosed"]}, Boxed -> False, Axes -> False]$

Out[40]=

Scope (12)

Parametric equations for a Klein bottle:

In[41]:=

Out[41]=

With a perforated polygon:

In[42]:=

$Graphics3D[{EdgeForm[], ResourceFunction["ApproximatedSurface"][ kb, {u, 0, 2 \[Pi], 35}, {v, 0, 2 \[Pi], 35}, Polygon[{{0., 0., 0.}, {0., 1., 0.}, {0.2, 0.2, 0.}, {0.2, 0.8, 0.}, {0.8, 0.2, 0.}, {0.8, 0.8, 0.}, {1., 0., 0.}, {1., 1., 0.}}, {1, 7, 8, 2} -> {{4, 6, 5, 3}}]]}, Boxed -> False]$

Out[42]=

With outlined polygons:

In[43]:=

$Graphics3D[ ResourceFunction["ApproximatedSurface"][ kb, {u, 0, 2 \[Pi], 35}, {v, 0, 2 \[Pi], 35}, "OutlinePolygons"], Boxed -> False, Axes -> False]$

Out[43]=

With disks:

In[44]:=

$Graphics3D[ ResourceFunction["ApproximatedSurface"][ kb, {u, 0, 2 \[Pi], 35}, {v, 0, 2 \[Pi], 35}, Polygon[{{0.5, 0., 0}, {0.47552825814757677`, 0.15450849718747373`, 0}, {0.4045084971874737, 0.29389262614623657`, 0}, {0.29389262614623657`, 0.4045084971874737, 0}, {0.15450849718747373`, 0.47552825814757677`, 0}, {0., 0.5, 0}, {-0.15450849718747373`, 0.47552825814757677`, 0}, {-0.29389262614623657`, 0.4045084971874737, 0}, {-0.4045084971874737, 0.29389262614623657`, 0}, {-0.47552825814757677`, 0.15450849718747373`, 0}, {-0.5, 0., 0}, {-0.47552825814757677`, -0.15450849718747373`, 0}, {-0.4045084971874737, -0.29389262614623657`, 0}, {-0.29389262614623657`, -0.4045084971874737, 0}, {-0.15450849718747373`, -0.47552825814757677`, 0}, {0., -0.5, 0}, {0.15450849718747373`, -0.47552825814757677`, 0}, {0.29389262614623657`, -0.4045084971874737, 0}, {0.4045084971874737, -0.29389262614623657`, 0}, {0.47552825814757677`, -0.15450849718747373`, 0}, {0.5, 0., 0}}]], Boxed -> False]$

Out[44]=

With a 3D disk using BSplineSurface:

In[45]:=

$Graphics3D[ ResourceFunction["ApproximatedSurface"][ kb, {u, 0, 2 \[Pi], 35}, {v, 0, 2 \[Pi], 35}, BSplineSurface[.35 {{{1, 0, 0}, {1, 1, 0}, {0, 1, 0}}, {{1, -1, 0}, {0, 0, 0}, {-1, 1, 0}}, {{0, -1, 0}, {-1, -1, 0}, {-1, 0, 0}}}, SplineKnots -> {{0, 0, 0, 1, 1, 2}, {0, 0, 0, 1, 1, 2}}, SplineWeights -> {{1, 1/Sqrt[2], 1}, {1/Sqrt[2], 1, 1/Sqrt[2]}, {1, 1/Sqrt[2], 1}}]], PlotRange -> {{-4.1, 2.5}, {-3., 5.1}, {-3., 3.}}]$

Out[45]=

A hexagonal mesh:

In[46]:=

$Graphics3D[ ResourceFunction["ApproximatedSurface"][ kb, {u, 0, 2 \[Pi], 30}, {v, 0, 2 \[Pi], 29}, "HexagonalMesh"], Boxed -> False, Axes -> False]$

Out[46]=

A hexagonal mesh subdivided with triangles:

In[47]:=

$Graphics3D[ ResourceFunction["ApproximatedSurface"][ kb, {u, 0, 2 \[Pi], 16}, {v, 0, 2 \[Pi], 15}, "TriangulatedHexagonalMesh"], Boxed -> False, Axes -> False]$