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5

$begingroup$

Start with a quarter-circle of radius 1 centered at the origin and lying in the xz-plane:

 arc = ParametricPlot3D[{Cos[t], 0, Sin[t]]}, {t, 0, π/2}]

I want to dilate this by a factor of 2 and shift the center to {3, 0, 0}, then show the result graphically.

The following does not work:

shiftAndDilate3D = AffineTransform[{2 IdentityMatrix[3], {3, 0, 0}}]



Graphics3D[GeometricTransformation[arc3D, shiftAndDilate3D]]

The error I get is that "Graphics3DBox is not a Graphics3D primitive or directive."

What am I doing wrong?

graphics3d geometric-transform

share|improve this question

edited 5 hours ago

MarcoB

36.9k556113

asked 5 hours ago

murraymurray

6,1981835

$endgroup$

add a comment | 

5

$begingroup$

Start with a quarter-circle of radius 1 centered at the origin and lying in the xz-plane:

 arc = ParametricPlot3D[{Cos[t], 0, Sin[t]]}, {t, 0, π/2}]

I want to dilate this by a factor of 2 and shift the center to {3, 0, 0}, then show the result graphically.

The following does not work:

shiftAndDilate3D = AffineTransform[{2 IdentityMatrix[3], {3, 0, 0}}]



Graphics3D[GeometricTransformation[arc3D, shiftAndDilate3D]]

The error I get is that "Graphics3DBox is not a Graphics3D primitive or directive."

What am I doing wrong?

graphics3d geometric-transform

share|improve this question

edited 5 hours ago

MarcoB

36.9k556113

asked 5 hours ago

murraymurray

6,1981835

$endgroup$

add a comment | 

$begingroup$

Start with a quarter-circle of radius 1 centered at the origin and lying in the xz-plane:

 arc = ParametricPlot3D[{Cos[t], 0, Sin[t]]}, {t, 0, π/2}]

I want to dilate this by a factor of 2 and shift the center to {3, 0, 0}, then show the result graphically.

The following does not work:

shiftAndDilate3D = AffineTransform[{2 IdentityMatrix[3], {3, 0, 0}}]



Graphics3D[GeometricTransformation[arc3D, shiftAndDilate3D]]

The error I get is that "Graphics3DBox is not a Graphics3D primitive or directive."

What am I doing wrong?

graphics3d geometric-transform

share|improve this question

edited 5 hours ago

MarcoB

36.9k556113

asked 5 hours ago

murraymurray

6,1981835

$endgroup$

 arc = ParametricPlot3D[{Cos[t], 0, Sin[t]]}, {t, 0, π/2}]

shiftAndDilate3D = AffineTransform[{2 IdentityMatrix[3], {3, 0, 0}}]



Graphics3D[GeometricTransformation[arc3D, shiftAndDilate3D]]

share|improve this question

edited 5 hours ago

MarcoB

36.9k556113

asked 5 hours ago

murraymurray

6,1981835

share|improve this question

edited 5 hours ago

MarcoB

36.9k556113

asked 5 hours ago

murraymurray

6,1981835

edited 5 hours ago

MarcoB

36.9k556113

edited 5 hours ago

MarcoB

36.9k556113

asked 5 hours ago

murraymurray

6,1981835

asked 5 hours ago

murraymurray

6,1981835

3 Answers
3

active

oldest

votes

5

$begingroup$

You could work with regions instead. Your arc:

arc = ParametricRegion[{Cos[t], 0, Sin[t]}, {{t, 0, [Pi]/2}}];

The transformed arc:

shiftAndDilate3D = AffineTransform[{2 IdentityMatrix[3], {3, 0, 0}}];

new = TransformedRegion[arc, shiftAndDilate3D];

Visualization:

Show[

    Region[arc, BaseStyle->Red],

    Region[new, BaseStyle->Blue],

    Axes->True

]

share|improve this answer

answered 5 hours ago

Carl WollCarl Woll

69.4k393179

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  How should one know from the Mathematica documentation that ParametricRegion is a "region" suitable for the 1st argument to TransformedRegion? (The documentation involving GeometricTransformatino, Region, etc., is sadly deficient. As with Image, these things seem to have been thrown into Mathematica without sufficient exposition, or perhaps even without sufficient coherence for the various kinds of things, on the one hand, and sufficient distinctions among them, on the other hand.)
  $endgroup$
  – murray
  2 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  On the one hand, I like this answer better than the others because it is more direct. On the other hand, it uses yet a different type of object, namely, a "region".
  $endgroup$
  – murray
  2 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  But I'm annoyed by the different method to style the resulting graphics display, namely, through use of the BaseStyle option of Region instead of the usual PlotStyle option of ParametricPlot3D and so many other graphics and graphics 3D functions. Moreover, why should it be necessary to wrap the result of a ParametricRegion expression with a Region? Either there's something fundamental here I don't understand, or else a whole slew of graphics-like or geometric-like constructs have been thrown into recent versions of Mathematica without sufficient rationalization of the whole domain.
  $endgroup$
  – murray
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Further rant: Why does ParametricRegion take as 2nd argument a list of lists (of parameters and their extent), whereas ParametricPlot3D uses the parameter information as a list, then another list, etc.?
  $endgroup$
  – murray
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @murray Lots of questions. For a brief overview of regions, see reference.wolfram.com/language/guide/GeometricComputation. Basically, a region is a computable object. Region is a function that displays the object (similar to RegionPlot). Show converts a region object to a Graphics/Graphics3D object. ParametricRegion has a different syntax because it supports arguments without bounds, e.g., ParametricRegion[{Cos[t], 0, Sin[t]}, t].
  $endgroup$
  – Carl Woll
  1 hour ago
  

  
  
  
  

 | 
show 2 more comments

5

$begingroup$

You cannot apply those geometric transformations to the results of the plotting; instead, you should apply them to a Graphics primitive, e.g. the Line object generated by ParametricPlot, which you can extract using e.g. Cases:

arcLine = 

 First@Cases[ParametricPlot3D[{Cos[t], 0, Sin[t]}, {t, 0, π/2}], _Line, All]



Graphics3D[{

  Red, arcLine,

  Blue, GeometricTransformation[arcLine, shiftAndDilate3D]

}]

In red in the plot above is your original curve, in blue the transformed one.

share|improve this answer

edited 4 hours ago

answered 5 hours ago

MarcoBMarcoB

36.9k556113

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Where in the documentation does it explain what is, and what is not, a "geometric object"?
  $endgroup$
  – murray
  5 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @murray I was being loose with words there. I really meant a "Graphics primitive", such as those discussed in this guide: Graphics objects. I fixed it in the answer.
  $endgroup$
  – MarcoB
  4 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Alas, the doc page guide/GraphicsObjects does not fully identify or clarify what is, and what is not, a "graphics object". For example, given the answer by Carl Woll to my question, a ParametricRegion seems to qualify as such an object, yet is not listed on that guide page. So the documentation list of graphics objects doesn't specify what they are, and certainly -- and this is perhaps a real design defect -- the Head of these objects doesn't tell you that they are "graphics objects"!
  $endgroup$
  – murray
  2 hours ago
  

  
  
  
  

add a comment | 

4

$begingroup$

Use arc3D[[1]] (which contains all the graphics primitives and their styles) as the first argument of GeometricTransformation:

arc3D = ParametricPlot3D[{Cos[t], 0, Sin[t]}, {t, 0, π/2}]

Show[arc3D, 

 Graphics3D[GeometricTransformation[arc3D[[1]], shiftAndDilate3D] /. l_Line :> {Orange, l}],

 PlotRange -> All]

share|improve this answer

edited 4 hours ago

answered 5 hours ago

kglrkglr

187k10203421

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I cannot reproduce that. Rather, I get an error message: "An improperly formatted option was encountered while reading a Graphics3DBox. The left-hand side of the option was not a symbol or string."
  $endgroup$
  – murray
  4 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @murray, forgot a comma before PlotRange (fixed now).
  $endgroup$
  – kglr
  4 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Sorry, I should have caught that!
  $endgroup$
  – murray
  2 hours ago
  

  
  
  
  

add a comment | 

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5

$begingroup$

You could work with regions instead. Your arc:

arc = ParametricRegion[{Cos[t], 0, Sin[t]}, {{t, 0, [Pi]/2}}];

The transformed arc:

shiftAndDilate3D = AffineTransform[{2 IdentityMatrix[3], {3, 0, 0}}];

new = TransformedRegion[arc, shiftAndDilate3D];

Visualization:

Show[

    Region[arc, BaseStyle->Red],

    Region[new, BaseStyle->Blue],

    Axes->True

]

share|improve this answer

answered 5 hours ago

Carl WollCarl Woll

69.4k393179

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  How should one know from the Mathematica documentation that ParametricRegion is a "region" suitable for the 1st argument to TransformedRegion? (The documentation involving GeometricTransformatino, Region, etc., is sadly deficient. As with Image, these things seem to have been thrown into Mathematica without sufficient exposition, or perhaps even without sufficient coherence for the various kinds of things, on the one hand, and sufficient distinctions among them, on the other hand.)
  $endgroup$
  – murray
  2 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  On the one hand, I like this answer better than the others because it is more direct. On the other hand, it uses yet a different type of object, namely, a "region".
  $endgroup$
  – murray
  2 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  But I'm annoyed by the different method to style the resulting graphics display, namely, through use of the BaseStyle option of Region instead of the usual PlotStyle option of ParametricPlot3D and so many other graphics and graphics 3D functions. Moreover, why should it be necessary to wrap the result of a ParametricRegion expression with a Region? Either there's something fundamental here I don't understand, or else a whole slew of graphics-like or geometric-like constructs have been thrown into recent versions of Mathematica without sufficient rationalization of the whole domain.
  $endgroup$
  – murray
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Further rant: Why does ParametricRegion take as 2nd argument a list of lists (of parameters and their extent), whereas ParametricPlot3D uses the parameter information as a list, then another list, etc.?
  $endgroup$
  – murray
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @murray Lots of questions. For a brief overview of regions, see reference.wolfram.com/language/guide/GeometricComputation. Basically, a region is a computable object. Region is a function that displays the object (similar to RegionPlot). Show converts a region object to a Graphics/Graphics3D object. ParametricRegion has a different syntax because it supports arguments without bounds, e.g., ParametricRegion[{Cos[t], 0, Sin[t]}, t].
  $endgroup$
  – Carl Woll
  1 hour ago
  

  
  
  
  

 | 
show 2 more comments

5

$begingroup$

You could work with regions instead. Your arc:

arc = ParametricRegion[{Cos[t], 0, Sin[t]}, {{t, 0, [Pi]/2}}];

The transformed arc:

shiftAndDilate3D = AffineTransform[{2 IdentityMatrix[3], {3, 0, 0}}];

new = TransformedRegion[arc, shiftAndDilate3D];

Visualization:

Show[

    Region[arc, BaseStyle->Red],

    Region[new, BaseStyle->Blue],

    Axes->True

]

share|improve this answer

answered 5 hours ago

Carl WollCarl Woll

69.4k393179

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  How should one know from the Mathematica documentation that ParametricRegion is a "region" suitable for the 1st argument to TransformedRegion? (The documentation involving GeometricTransformatino, Region, etc., is sadly deficient. As with Image, these things seem to have been thrown into Mathematica without sufficient exposition, or perhaps even without sufficient coherence for the various kinds of things, on the one hand, and sufficient distinctions among them, on the other hand.)
  $endgroup$
  – murray
  2 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  On the one hand, I like this answer better than the others because it is more direct. On the other hand, it uses yet a different type of object, namely, a "region".
  $endgroup$
  – murray
  2 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  But I'm annoyed by the different method to style the resulting graphics display, namely, through use of the BaseStyle option of Region instead of the usual PlotStyle option of ParametricPlot3D and so many other graphics and graphics 3D functions. Moreover, why should it be necessary to wrap the result of a ParametricRegion expression with a Region? Either there's something fundamental here I don't understand, or else a whole slew of graphics-like or geometric-like constructs have been thrown into recent versions of Mathematica without sufficient rationalization of the whole domain.
  $endgroup$
  – murray
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Further rant: Why does ParametricRegion take as 2nd argument a list of lists (of parameters and their extent), whereas ParametricPlot3D uses the parameter information as a list, then another list, etc.?
  $endgroup$
  – murray
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @murray Lots of questions. For a brief overview of regions, see reference.wolfram.com/language/guide/GeometricComputation. Basically, a region is a computable object. Region is a function that displays the object (similar to RegionPlot). Show converts a region object to a Graphics/Graphics3D object. ParametricRegion has a different syntax because it supports arguments without bounds, e.g., ParametricRegion[{Cos[t], 0, Sin[t]}, t].
  $endgroup$
  – Carl Woll
  1 hour ago
  

  
  
  
  

 | 
show 2 more comments

$begingroup$

You could work with regions instead. Your arc:

arc = ParametricRegion[{Cos[t], 0, Sin[t]}, {{t, 0, [Pi]/2}}];

The transformed arc:

shiftAndDilate3D = AffineTransform[{2 IdentityMatrix[3], {3, 0, 0}}];

new = TransformedRegion[arc, shiftAndDilate3D];

Visualization:

Show[

    Region[arc, BaseStyle->Red],

    Region[new, BaseStyle->Blue],

    Axes->True

]

share|improve this answer

answered 5 hours ago

Carl WollCarl Woll

69.4k393179

$endgroup$

arc = ParametricRegion[{Cos[t], 0, Sin[t]}, {{t, 0, [Pi]/2}}];

shiftAndDilate3D = AffineTransform[{2 IdentityMatrix[3], {3, 0, 0}}];

new = TransformedRegion[arc, shiftAndDilate3D];

Show[

    Region[arc, BaseStyle->Red],

    Region[new, BaseStyle->Blue],

    Axes->True

]

share|improve this answer

answered 5 hours ago

Carl WollCarl Woll

69.4k393179

answered 5 hours ago

Carl WollCarl Woll

69.4k393179

answered 5 hours ago

Carl WollCarl Woll

69.4k393179

5

$begingroup$

You cannot apply those geometric transformations to the results of the plotting; instead, you should apply them to a Graphics primitive, e.g. the Line object generated by ParametricPlot, which you can extract using e.g. Cases:

arcLine = 

 First@Cases[ParametricPlot3D[{Cos[t], 0, Sin[t]}, {t, 0, π/2}], _Line, All]



Graphics3D[{

  Red, arcLine,

  Blue, GeometricTransformation[arcLine, shiftAndDilate3D]

}]

In red in the plot above is your original curve, in blue the transformed one.

share|improve this answer

edited 4 hours ago

answered 5 hours ago

MarcoBMarcoB

36.9k556113

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Where in the documentation does it explain what is, and what is not, a "geometric object"?
  $endgroup$
  – murray
  5 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @murray I was being loose with words there. I really meant a "Graphics primitive", such as those discussed in this guide: Graphics objects. I fixed it in the answer.
  $endgroup$
  – MarcoB
  4 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Alas, the doc page guide/GraphicsObjects does not fully identify or clarify what is, and what is not, a "graphics object". For example, given the answer by Carl Woll to my question, a ParametricRegion seems to qualify as such an object, yet is not listed on that guide page. So the documentation list of graphics objects doesn't specify what they are, and certainly -- and this is perhaps a real design defect -- the Head of these objects doesn't tell you that they are "graphics objects"!
  $endgroup$
  – murray
  2 hours ago
  

  
  
  
  

add a comment | 

5

$begingroup$

You cannot apply those geometric transformations to the results of the plotting; instead, you should apply them to a Graphics primitive, e.g. the Line object generated by ParametricPlot, which you can extract using e.g. Cases:

arcLine = 

 First@Cases[ParametricPlot3D[{Cos[t], 0, Sin[t]}, {t, 0, π/2}], _Line, All]



Graphics3D[{

  Red, arcLine,

  Blue, GeometricTransformation[arcLine, shiftAndDilate3D]

}]

In red in the plot above is your original curve, in blue the transformed one.

share|improve this answer

edited 4 hours ago

answered 5 hours ago

MarcoBMarcoB

36.9k556113

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Where in the documentation does it explain what is, and what is not, a "geometric object"?
  $endgroup$
  – murray
  5 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @murray I was being loose with words there. I really meant a "Graphics primitive", such as those discussed in this guide: Graphics objects. I fixed it in the answer.
  $endgroup$
  – MarcoB
  4 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Alas, the doc page guide/GraphicsObjects does not fully identify or clarify what is, and what is not, a "graphics object". For example, given the answer by Carl Woll to my question, a ParametricRegion seems to qualify as such an object, yet is not listed on that guide page. So the documentation list of graphics objects doesn't specify what they are, and certainly -- and this is perhaps a real design defect -- the Head of these objects doesn't tell you that they are "graphics objects"!
  $endgroup$
  – murray
  2 hours ago
  

  
  
  
  

add a comment | 

$begingroup$

You cannot apply those geometric transformations to the results of the plotting; instead, you should apply them to a Graphics primitive, e.g. the Line object generated by ParametricPlot, which you can extract using e.g. Cases:

arcLine = 

 First@Cases[ParametricPlot3D[{Cos[t], 0, Sin[t]}, {t, 0, π/2}], _Line, All]



Graphics3D[{

  Red, arcLine,

  Blue, GeometricTransformation[arcLine, shiftAndDilate3D]

}]

In red in the plot above is your original curve, in blue the transformed one.

share|improve this answer

edited 4 hours ago

answered 5 hours ago

MarcoBMarcoB

36.9k556113

$endgroup$

arcLine = 

 First@Cases[ParametricPlot3D[{Cos[t], 0, Sin[t]}, {t, 0, π/2}], _Line, All]



Graphics3D[{

  Red, arcLine,

  Blue, GeometricTransformation[arcLine, shiftAndDilate3D]

}]

share|improve this answer

edited 4 hours ago

answered 5 hours ago

MarcoBMarcoB

36.9k556113

answered 5 hours ago

MarcoBMarcoB

36.9k556113

answered 5 hours ago

MarcoBMarcoB

36.9k556113

4

$begingroup$

Use arc3D[[1]] (which contains all the graphics primitives and their styles) as the first argument of GeometricTransformation:

arc3D = ParametricPlot3D[{Cos[t], 0, Sin[t]}, {t, 0, π/2}]

Show[arc3D, 

 Graphics3D[GeometricTransformation[arc3D[[1]], shiftAndDilate3D] /. l_Line :> {Orange, l}],

 PlotRange -> All]

share|improve this answer

edited 4 hours ago

answered 5 hours ago

kglrkglr

187k10203421

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I cannot reproduce that. Rather, I get an error message: "An improperly formatted option was encountered while reading a Graphics3DBox. The left-hand side of the option was not a symbol or string."
  $endgroup$
  – murray
  4 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @murray, forgot a comma before PlotRange (fixed now).
  $endgroup$
  – kglr
  4 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Sorry, I should have caught that!
  $endgroup$
  – murray
  2 hours ago
  

  
  
  
  

add a comment | 

4

$begingroup$

Use arc3D[[1]] (which contains all the graphics primitives and their styles) as the first argument of GeometricTransformation:

arc3D = ParametricPlot3D[{Cos[t], 0, Sin[t]}, {t, 0, π/2}]

Show[arc3D, 

 Graphics3D[GeometricTransformation[arc3D[[1]], shiftAndDilate3D] /. l_Line :> {Orange, l}],

 PlotRange -> All]

share|improve this answer

edited 4 hours ago

answered 5 hours ago

kglrkglr

187k10203421

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I cannot reproduce that. Rather, I get an error message: "An improperly formatted option was encountered while reading a Graphics3DBox. The left-hand side of the option was not a symbol or string."
  $endgroup$
  – murray
  4 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @murray, forgot a comma before PlotRange (fixed now).
  $endgroup$
  – kglr
  4 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Sorry, I should have caught that!
  $endgroup$
  – murray
  2 hours ago
  

  
  
  
  

add a comment | 

$begingroup$

Use arc3D[[1]] (which contains all the graphics primitives and their styles) as the first argument of GeometricTransformation:

arc3D = ParametricPlot3D[{Cos[t], 0, Sin[t]}, {t, 0, π/2}]

Show[arc3D, 

 Graphics3D[GeometricTransformation[arc3D[[1]], shiftAndDilate3D] /. l_Line :> {Orange, l}],

 PlotRange -> All]

share|improve this answer

edited 4 hours ago

answered 5 hours ago

kglrkglr

187k10203421

$endgroup$

arc3D = ParametricPlot3D[{Cos[t], 0, Sin[t]}, {t, 0, π/2}]

Show[arc3D, 

 Graphics3D[GeometricTransformation[arc3D[[1]], shiftAndDilate3D] /. l_Line :> {Orange, l}],

 PlotRange -> All]

share|improve this answer

edited 4 hours ago

answered 5 hours ago

kglrkglr

187k10203421

answered 5 hours ago

kglrkglr

187k10203421

answered 5 hours ago

kglrkglr

187k10203421