Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 ProjectGraphics3D

Source Notebook

Project 3D graphics onto a plane

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

ResourceFunction["ProjectGraphics3D"][graphics3D,p]

projects an image of graphics3D onto a plane through point p and perpendicular to the line from the center of the graphics3D to p.

ResourceFunction["ProjectGraphics3D"][graphics3D,{e1,e2},p]

projects an image of graphics3D onto a plane with basis vectors {e1, e2} at p, along the line from the origin to p.

ResourceFunction["ProjectGraphics3D"][graphics3D,{e1,e2},p,center]

projects along the line from center to p, as seen from Infinity.

Details and Options

The center of the graphic is computed using PlotRange.
ResourceFunction["ProjectGraphics3D"] takes the same options as Graphics3D plus PlotStyle.

Examples

Basic Examples (3) 

Project a curve onto a plane:

In[1]:=
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-> {"Highlight" -> {"ratio" -> 2}, "HighlightPoint" -> {"ratio" -> 2}, "Droplines" -> {"freeformCursorMode" -> True, "placement" -> {"x" -> "All", "y" -> "None"}}}}}, PlotRange->{{-0.9999998592131705, 0.9999998782112116}, {-0.9999996658276197, 0.9999993650500513}, {-0.9999998830731719, 0.9999999999999918}}, PlotRangePadding->{{ Scaled[0.05], Scaled[0.05]}, { Scaled[0.05], Scaled[0.05]}, { Scaled[0.05], Scaled[0.05]}}, Ticks->{Automatic, Automatic, Automatic}]\), {0, 1, 1}]
Out[1]=

Project a polyhedron:

In[2]:=
ResourceFunction["ProjectGraphics3D"][ Graphics3D@ ResourceFunction["PrimitiveToPolygons"][Dodecahedron[]], {1, 1, -1}]
Out[2]=

Project a surface:

In[3]:=
ResourceFunction["ProjectGraphics3D"][ Plot3D[Sin[x y], {x, 0, Pi}, {y, 0, Pi}], {1, -1, 1}]
Out[3]=

Scope (3) 

A curve not defined at some ranges:

In[4]:=
ParametricPlot3D[{Cos[u], Sin[u], Sqrt[Sin[2 u]]}, {u, 0, 2 Pi}]
Out[4]=

Show it with a projection:

In[5]:=
Show[%, ResourceFunction["ProjectGraphics3D"][%, {0, 0, 1}, PlotStyle -> {Red, Thickness[.01]}]]
Out[5]=

Define a graphic using a callout wrapper and show it along with its projection:

In[6]:=
ParametricPlot3D[ Callout[{Sin[u], Sin[2 u], Cos[u]}, "original curve"], {u, 0, 2 Pi}]; Show[%, ResourceFunction["ProjectGraphics3D"][%, {0, 0, 1}, PlotStyle -> {Red, Thickness[.01]}]]
Out[7]=

Show the correspondence between points in the helix and its projection:

In[8]:=
helix = Entity["SpaceCurve", "Helix"]["ParametricEquations"][1, 1][ t]; points = Table[ResourceFunction["ProjectGraphics3D"][ Graphics3D[ Point[{Cos[t], Sin[t], t}]], {{1, 0, 0}, {0, 1, 1}}, {0, 0, 0}, {0, 0, 0}][[1, 1]], {t, 0, \[Pi], \[Pi]/12}]
Out[9]=
In[10]:=
Show[gr, ResourceFunction["ProjectGraphics3D"][ gr, {{1, 0, 0}, {0, 1, 1}}, {0, 0, 0}, {0, 0, 0}, PlotStyle -> {Red, Thickness[.01]}], Graphics3D[{Arrow /@ N[Thread[{Table[helix, {t, 0, \[Pi], \[Pi]/12}], points}]], Opacity[.5], InfinitePlane[{{0, 0, 0}, {1, 0, 0}, {0, 1, 1}}]}]]
Out[10]=

Project the curve in orthogonal planes:

In[11]:=
Show[gr, ResourceFunction["ProjectGraphics3D"][ gr, # , {0, 0, 0}, PlotStyle -> {Red, Thickness[.01]}] & /@ {{{1, 0, 0}, {0, 1, 0}}, {{1, 0, 0}, {0, 0, 1}}, {{0, 1, 0}, {0, 0, 1}}}, Graphics3D[{Arrow[{{0, 0, 0}, #}] & /@ {{0, 1, 0}/2, {0, 0, 1}}, Opacity[.5], InfinitePlane[{{0, 0, 0}, {-1, 1, 0}, {1, 1, 0}}], InfinitePlane[{{0, 0, 0}, {-1, 0, 1}, {1, 0, 1}}], InfinitePlane[{{0, 0, 0}, {0, -1, 1}, {0, 1, 1}}]}]]
Out[11]=

The rectifying plane:

In[12]:=
rp = ResourceFunction["RectifyingPlane"][helix, t]
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The normal plane:

In[13]:=
np = ResourceFunction["NormalPlane"][helix, t]
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The osculating plane:

In[14]:=
op = ResourceFunction["OsculatingPlane"][helix, t] // Simplify
Out[14]=

Show the projections of the curve in the normal and rectifying planes (the vectors are the tangent, normal and binormal vectors):

In[15]:=
Manipulate[ Show[gr, ResourceFunction["ProjectGraphics3D"][ gr, #, helix /. t -> tf, PlotStyle -> Red] & /@ {rp[[2]], np[[2]], op[[2]]} /. t -> tf, Graphics3D[{{Arrowheads[.05], Blue, Arrow[{helix, helix + rp[[2, 1]]}], Arrow[{helix, helix + rp[[2, 2]]}], Arrow[{helix, helix + np[[2, 1]]}]}, {Opacity[.5], rp, np, op}} /. t -> tf], PlotRange -> All], {{tf, 1}, 0, \[Pi]}, SaveDefinitions -> True]
Out[15]=

Options (1) 

ProjectGraphics3D supports the same options as Graphics3D:

In[16]:=
ParametricPlot3D[{(2 + Cos[8 u]) Cos[u], (2 + Cos[8 u]) Sin[u], Sin[8 u]}, {u, 0, 2 \[Pi]}, PlotStyle -> {Green, Thickness[.01]}]
Out[16]=
In[17]:=
Show[ResourceFunction["ProjectGraphics3D"][%, {0, 1, 1}, PlotStyle -> {Red, Thickness[.01]}, FaceGrids -> All, FaceGridsStyle -> Directive[Orange, Dashed], PlotStyle -> {Red, Thickness[.01]}], %, PlotRange -> All]
Out[17]=

Properties and Relations (3) 

Get the equation for a helix:

In[18]:=
helix = Entity["SpaceCurve", "Helix"]["ParametricEquations"][1, 1][t]
Out[18]=

Define a graphic output and compute its center:

In[19]:=
gr = ParametricPlot3D[helix, {t, 0, \[Pi]}, PlotStyle -> {Green, Thickness[.01]}, PlotRange -> All]; c = Mean /@ AbsoluteOptions[gr, PlotRange][[1, 2]]
Out[20]=

Show the curve (green), its projection (red), and the plane of projection:

In[21]:=
With[{p = {0, 0, 1}}, Show[gr, ResourceFunction["ProjectGraphics3D"][gr, p, PlotStyle -> {Red, Thickness[.01]}], Graphics3D[{Point[c], Arrow[{c, p}], Opacity[.5], InfinitePlane[{{1, 0, 0} + p, p, {0., -0.75221, 0.6589} + p}]}]]]
Out[21]=

Neat Examples (2) 

Hover the mouse over the curve:

In[22]:=
grw = ParametricPlot3D[{Sin[u], Sin[2 u], Cos[u]}, {u, 0, 2 Pi}];
In[23]:=
ResourceFunction["ProjectGraphics3D"][ ParametricPlot3D[ Mouseover[{Sin[u], Sin[2 u], Cos[u]}, First@grw], {u, 0, 2 Pi}, PlotStyle -> {Red, Thickness[.01]}], {0, 0, 1}, PlotRange -> {{-1, 1}, {-1, 1}, {-1, 1}}]
Out[23]=

Move the spanning vectors and the location point:

In[24]:=
helix = Entity["SpaceCurve", "Helix"]["ParametricEquations"][1, 1][ t]; gr = ParametricPlot3D[helix, {t, 0, \[Pi]}, PlotStyle -> {Green, Thickness[.01]}, PlotRange -> All]; c = Mean /@ AbsoluteOptions[gr, PlotRange][[1, 2]];
In[25]:=
Manipulate[ Show[ResourceFunction["ProjectGraphics3D"][ gr, {{x, y, z}, {x1, y1, z1}}, {x2, y2, z2}, PlotStyle -> Red], gr, Graphics3D[{Point[c], Arrow[{{0, 0, 0}, {x2, y2, z2}}], Arrow[{{x2, y2, z2}, # + {x2, y2, z2}}] & /@ {{x, y, z}, {x1, y1, z1}}, Opacity[.5], InfinitePlane[{x2, y2, z2}, {{x, y, z}, {x1, y1, z1}}]}], PlotRange -> All], {{x, 1}, -1, 1, ImageSize -> Tiny}, {{y, 0}, -1, 1, ImageSize -> Tiny}, {{z, 0}, -1, 1, ImageSize -> Tiny}, Delimiter, {{x1, 0}, -1, 1, ImageSize -> Tiny}, {{y1, 1}, -1, 1, ImageSize -> Tiny}, {{z1, 0}, -1, 1, ImageSize -> Tiny}, Delimiter, {{x2, 0}, -1, 1, ImageSize -> Tiny}, {{y2, 0}, -1, 1, ImageSize -> Tiny}, {{z2, 0}, -1, 1, ImageSize -> Tiny}, SaveDefinitions -> True]
Out[25]=

Publisher

Enrique Zeleny

Version History

  • 1.0.0 – 22 April 2021

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