How to make a tight crop of a 3d plot?

Question

I like Mathematica, but it's syntax baffles me.

I am trying to figure out how to minimize the whitespace around a graphic.

For example,

ParametricPlot3D[{r Cos[t], r Sin[t], r^2}, {r, 0, 1}, {t, 0, 2 \[Pi]}, 
 Boxed -> True, Axes -> False]

Puts the 3d bounding box at the limits of the view. But if I don't show the 3d bounding box,

ParametricPlot3D[{r Cos[t], r Sin[t], r^2}, {r, 0, 1}, {t, 0, 2 \[Pi]}, 
 Boxed -> False, Axes -> False]

there is all this white space around the actual object.

Is there some way (syntax) that can put the view just around the visible objects?

Edit in response to answers

Ok, from the below answers, I have two solutions; 1) use ImageCrop, or 2) use Method->{"ShrinkWrap" -> True}. However both of these options do a little something strange to the plot I want (maybe it is just a problem with the plot itself).

So the actual plot I am after is,

Module[{r = 1, \[Theta] = \[Pi]/2, \[CurlyPhi] = \[Pi]/6, \[Psi] = \[Pi]/12},
Framed@Show[
 Graphics3D[
  {
   {Arrowheads[.025],
    Arrow[{{0, 0, 0}, {1.1, 0, 0}}], Text["x", {1.2, 0, 0}],
    Arrow[{{0, 0, 0}, {0, 1.1, 0}}], Text["y", {0, 1.2, 0}],
    Arrow[{{0, 0, 0}, {0, 0, 1.1}}], Text["z", {0, 0, 1.2}],
    Arrow[{{0, 0, 0}, r {Cos[\[Theta]] Sin[\[CurlyPhi]], 
      Sin[\[Theta]] Sin[\[CurlyPhi]], Cos[\[CurlyPhi]]}}]},
   {Specularity[White, 50], Opacity[.1], Sphere[{0, 0, 0}, r]}
   },
  Boxed -> False,
  ImageSize -> 600,
  PlotRange -> 1.1 {{-r, r}, {-r, r}, {0, r}}
  ]]
 ]

Which has too much whitespace. If I replace Framed@Show[ with Framed@ImageCrop@Show[ I get,

which actually crops some of the (hemi)sphere. If just use Method -> {"ShrinkWrap" -> True}, in the Show options, I get,

which looks almost correct, but the x and z textboxes have now not included. Seems like I can't win!

Answer 1

Actually, there isn't white space at all:

Show[RegionPlot3D[True, {x, -1, 1}, {y, -1, 1}, {z, 0, 1}, 
                  PlotStyle -> Directive[Yellow, Opacity[0.5]], Mesh -> None, 
                  Boxed -> False, Axes -> False, PlotRangePadding -> 0], 
     ParametricPlot3D[{r Cos[t], r Sin[t], r^2}, {r, 0, 1}, {t, 0, 2 Pi} ]]

Edit

If you want to crop the image in 2D:

p = ParametricPlot3D[{r Cos[t], r Sin[t], r^2}, {r, 0, 1}, {t, 0, 2 Pi}, 
                     Boxed -> False, Axes -> False, PlotRangePadding -> 0];
Framed@ImageCrop@p

Edit

For your plot. Use .2 as Opacity. It has been reported elsewhere in this site that lowering the opacity too much makes other functions unable to detect the object.

Module[{r = 
   1, \[Theta] = \[Pi]/2, \[CurlyPhi] = \[Pi]/6, \[Psi] = \[Pi]/12},
 Framed@ImageCrop@Show[
    Graphics3D[
     {{Specularity[White, 50], Opacity[.2], 
       Sphere[{0, 0, 0}, r]}, {Arrowheads[.025], 
       Arrow[{{0, 0, 0}, {1.1, 0, 0}}], Text["x", {1.2, 0, 0}], 
       Arrow[{{0, 0, 0}, {0, 1.1, 0}}], Text["y", {0, 1.2, 0}], 
       Arrow[{{0, 0, 0}, {0, 0, 1.1}}], Text["z", {0, 0, 1.2}], 
       Arrow[{{0, 0, 0}, 
         r {Cos[\[Theta]] Sin[\[CurlyPhi]], 
           Sin[\[Theta]] Sin[\[CurlyPhi]], Cos[\[CurlyPhi]]}}]}}, 
     Boxed -> False, ImageSize -> 600, 
     PlotRange -> 1.1 {{-r, r}, {-r, r}, {0, r}}]]]

Answer 2

I think what you are looking for is ViewAngle option. The graph below compares default Automatic versus custom setting for ViewAngle. The image are framed intentionally to see clearly the removal of surrounding white space.

Framed[#, FrameMargins -> 0] & /@ 
     (ParametricPlot3D[{r Cos[t], r Sin[t], r^2}, {r, 0, 1}, {t, 0, 2 \[Pi]}, 
        Boxed -> False, Axes -> False, PlotRangePadding -> 0, 
        ImageSize -> 350 {1, 1}, ViewAngle -> #] & /@ {Automatic, .31}) 

Mathematica has rich set of options for so called "simulated camera" that "views" all 3D objects. To learn in detail about this i suggest downlading notebbok and watching free video of the @Yu-SungChang course Lights, Camera, Graphics! . Here is a illustrious diagram from that course explaining how ViewAngle option (measured in your example in 0.31 radians) sets up the view of a 3D objects.

You actually can change the ViewAngle setting interactively by holding CTRL or ALT key and dragging graphics with mouse - this effectively zooms in or out. See this tutorial for details.

Answer 3

ImageCrop seems to be a bit buggy (at least right here in Version 8.04, Win 64). It tends to crop lightly coloured areas rather agressively. You could try the following work-around, which works more reliably:

imcrop[img_] := ImagePad[img, -BorderDimensions[img, 0]]

g = Graphics3D[{Specularity[White, 50], Opacity[.1], 
    Sphere[{0, 0, 0}, 1]}, Boxed -> False, 
   PlotRange -> 1.1 {{-1, 1}, {-1, 1}, {0, 1}}
   ];
Column[Framed /@ {g, ImageCrop[g], imcrop[g]}]

For your graphics it seems to work without additional changes to Opacity or similar:

g = Module[{r = 
     1, \[Theta] = Pi/2, \[CurlyPhi] = Pi/6, \[Psi] = Pi/12},
    Show[Graphics3D[{{Arrowheads[.025], 
       Arrow[{{0, 0, 0}, {1.1, 0, 0}}], Text["x", {1.2, 0, 0}], 
       Arrow[{{0, 0, 0}, {0, 1.1, 0}}], Text["y", {0, 1.2, 0}], 
       Arrow[{{0, 0, 0}, {0, 0, 1.1}}], Text["z", {0, 0, 1.2}], 
       Arrow[{{0, 0, 0}, 
         r {Cos[\[Theta]] Sin[\[CurlyPhi]], 
           Sin[\[Theta]] Sin[\[CurlyPhi]], 
           Cos[\[CurlyPhi]]}}]}, {Specularity[White, 50], Opacity[.1],
        Sphere[{0, 0, 0}, r]}}, Boxed -> False, ImageSize -> 600, 
     PlotRange -> 1.1 {{-r, r}, {-r, r}, {0, r}}]]];    

Framed@imcrop[g]