Re: how does Rotate in 2D work?

*To*: mathgroup at smc.vnet.net*Subject*: [mg127662] Re: how does Rotate in 2D work?*From*: "Alexander Elkins" <alexander_elkins at hotmail.com>*Date*: Tue, 14 Aug 2012 04:19:41 -0400 (EDT)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*Delivered-to*: l-mathgroup@wolfram.com*Delivered-to*: mathgroup-newout@smc.vnet.net*Delivered-to*: mathgroup-newsend@smc.vnet.net*References*: <jvt3om$luo$1@smc.vnet.net>

The answer to the question "How can I use the Rotate command so that the _origin_ of the arrow or the Line I want to rotate is where the rotation occur around?" is given by the following replacement function: altRotate[g_, \[Theta]_, {x_, y_}] := Rotate[g, \[Theta], ({x, y} - RotationTransform[-\[Theta]][{x, y}])/( 2 - 2 Cos[\[Theta]])] /; Mod[\[Theta], \[Pi]] != 0 Here it is using the example given: Graphics[{altRotate[Arrow[{{0, 0}, {1, 0}}], 90 Degree, {1, 0}]}, Axes -> True] To understand why this is the correct answer, note that Rotate[g_, \[Theta]_, {x_, y_}] gives the same transformation as GeometricTransformation[g, RotationTransform[\[Theta], {x, y}]] as shown by the following two examples: Manipulate[ Graphics[{{Dashed, Circle[pt, Norm[pt]]}, Table[Rotate[{Hue[\[Theta]/2/\[Pi]], Arrow[{{0, 0}, {1, 0}}], Line[{{-2, -2}, {-2, 2}, {2, 2}, {2, -2}, {-2, -2}}]}, \[Theta] , pt], {\[Theta], 0, 2 \[Pi], \[Pi]/3}]}, Axes -> True, PlotRange -> {{-8, 8}, {-8, 8}}], {{pt, {-2, 2}}, Locator}] Manipulate[ Graphics[{{Dashed, Circle[pt, Norm[pt]]}, Table[GeometricTransformation[{Hue[\[Theta]/2/\[Pi]], Arrow[{{0, 0}, {1, 0}}], Line[{{-2, -2}, {-2, 2}, {2, 2}, {2, -2}, {-2, -2}}]}, RotationTransform[\[Theta], pt]], {\[Theta], 0, 2 \[Pi], \[Pi]/3}]}, Axes -> True, PlotRange -> {{-8, 8}, {-8, 8}}], {{pt, {-2, 2}}, Locator}] Then further note that: RotationTransform[\[Theta], {x, y}] == Composition[TranslationTransform[{x, y}], RotationTransform[\[Theta] ], TranslationTransform[-{x, y}]] == Composition[TranslationTransform[{x, y}], TranslationTransform[RotationTransform[\[Theta] ][-{x, y}]], RotationTransform[\[Theta] ]] == Composition[TranslationTransform[{x, y} - RotationTransform[\[Theta] ][{x, y}]], RotationTransform[\[Theta] ]] Since the desired transformation is equivalent to Composition[TranslationTransform[{u, v}], RotationTransform[\[Theta] ] ] then it is clear that we must solve for {x, y} in terms of {u, v}: In[174]:= Simplify[{x, y} /. Solve[{x, y} - RotationTransform[\[Theta] ][{x, y}] == {u, v}, {x, y}]][[1]] Out[174]= {1/2 (u - v Cot[\[Theta]/2]), 1/2 (v + u Cot[\[Theta]/2])} Which is the same result as given by: In[176]:= FullSimplify[({u, v} - RotationTransform[-\[Theta]][{u, v}])/ (2 - 2 Cos[\[Theta]])] Out[176]= {1/2 (u - v Cot[\[Theta]/2]), 1/2 (v + u Cot[\[Theta]/2])} Note also that: Composition[TranslationTransform[{x, y}], RotationTransform[\[Theta]]] == AffineTransform[{RotationMatrix[\[Theta] ], {x, y}}] So we could also create another very simple function to replace Rotate: tRotate[g_, \[Theta]_, {x_, y_}] := GeometricTransformation[g, AffineTransform[{RotationMatrix[\[Theta] ], {x, y}}]] And use it like so: Graphics[{tRotate[Arrow[{{0, 0}, {1, 0}}], 90 Degree, {1, 0}]}, Axes -> True] Hope this helps... "Nasser M. Abbasi" <nma at 12000.org> wrote in message news:jvt3om$luo$1 at smc.vnet.net... > > Suppose I have an arrow Arrow[{{0,0},{0,1}}] and I want > to rotate it 90 degrees, but taking the origin of the arrow > to be the point {0.5,0} when doing the rotation, instead of > the point {0,0} as it is above. > > i.e. given > > | > | > +-----> > (0,0) (1,0) > > Now if I do Rotate on the above, by 90 degrees, with {0,0} as > origin, it gives, using the command > > Graphics[{ > Rotate[Arrow[{{0, 0}, {1, 0}}], 90 Degree, {0, 0}] > }, Axes -> True] > > > (0,1) ^ > | > | > +--- > (0,0) > > > So far so good. Now, I want to obtain this > > ^ > | > | | > +------+ > (0,0) (1,0) > > i.e. I want the rotation to be around (1,0), and not (0,0). > I thought I can do it using the same command above, by just > changing {0,0} to {1,0} like this > > Graphics[{ > Rotate[Arrow[{{0, 0}, {1, 0}}], 90 Degree, {1, 0}] > }, Axes -> True] > > But the above gave > > | (1,0) > +------+-- > (0,0) ^ > | > | > > It is more strange when asking for rotation around say (.5,0), > > Graphics[{ > Rotate[Arrow[{{0, 0}, {1, 0}}], 90 Degree, {.5, 0}] > }, Axes -> True] > > Now it gives > ^ > | | > +-----|------ > (0,0) |(0.5,0) > | > | > > So, I think this has to do what that 'bounding box' that help talks about, > but ofcourse help does not say how to change this or anything, and no > examples. > > question is: > How can I use the Rotate command so that the _origin_ of the arrow > or the Line I want to rotate is where the rotation occur around? i.e > I want the arrow to be based from that rotation point before the > rotation start. > > (I know there other ways to do this, using RotationMatrix and such, > but I wanted to find how to do it using Rotate). I think the > problem is with the Bounding Box thing, which I do not > understand now how to change in this case. > > ps. Please WRI, improve your help pages more. Add more 'words' and > do not be so brief and cryptic in the description and add more > examples and add links to things you mention. > > thanks. > --Nasser > > > > > >