Slime <noo### [at] hotmailcom> wrote:
> I believe all orientations are covered by just

> rotate 360*<rand(R1),rand(R2),rand(R3)>

  So give me a "rotate <A,B,C>" command which, when replacing the two rotates
with it in the scene below, will give the exact same image:

camera { location <1,2,-3> look_at y*.5 angle 35 }
light_source { <100,250,-50>, 1 }
plane { y,0 pigment { checker rgb 1, rgb .5 } }

#declare Object =
  intersection
  { cone { 0,.5, y,.2 }
    box { <-.5,-.01,0><.5,1.01,.5> }
    pigment { rgb x+y }
  }

object
{ Object
  rotate y*45 rotate x*45
}


-- 
#macro N(D)#if(D>99)cylinder{M()#local D=div(D,104);M().5,2pigment{rgb M()}}
N(D)#end#end#macro M()<mod(D,13)-6mod(div(D,13)8)-3,10>#end blob{
N(11117333955)N(4254934330)N(3900569407)N(7382340)N(3358)N(970)}//  - Warp -

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Warp wrote:
>
>   So give me a "rotate <A,B,C>" command which, when replacing the two rotates
> with it in the scene below, will give the exact same image:
>
>   rotate y*45 rotate x*45

I believe this is close enough. Don't have time to calculate
the exact transform.

rotate<53,30,35>

_____________
Kari Kivisalo

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Kari Kivisalo <ray### [at] engineercom> wrote:
> rotate<53,30,35>

  I didn't think of it that way.

  Is any combination of free rotates achievable with a rotate <A,B,C>?
Somehow it's very hard to grasp.
  Some kind of intuitive, graphical or mathematical proof could help.

-- 
#macro M(A,N,D,L)plane{-z,-9pigment{mandel L*9translate N color_map{[0rgb x]
[1rgb 9]}scale<D,D*3D>*1e3}rotate y*A*8}#end M(-3<1.206434.28623>70,7)M(
-1<.7438.1795>1,20)M(1<.77595.13699>30,20)M(3<.75923.07145>80,99)// - Warp -

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Warp wrote:
>
>   Is any combination of free rotates achievable with a rotate <A,B,C>?

I think it was Euler who proved that any orientation in space could be
represented by a single rotation about an arbitrary axis, and that this
rotation is equivalent to three successive rotations about the x, y and
z axes.


_____________
Kari Kivisalo

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Kari Kivisalo <ray### [at] engineercom> wrote:
> I think it was Euler who proved that any orientation in space could be
> represented by a single rotation about an arbitrary axis

  That's even more radical!

  But thinking hard about it, it starts to be intuitive...

-- 
#macro M(A,N,D,L)plane{-z,-9pigment{mandel L*9translate N color_map{[0rgb x]
[1rgb 9]}scale<D,D*3D>*1e3}rotate y*A*8}#end M(-3<1.206434.28623>70,7)M(
-1<.7438.1795>1,20)M(1<.77595.13699>30,20)M(3<.75923.07145>80,99)// - Warp -

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let me summarize this please, to understand this  ...

We proved Warp, to be wrong ?

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Jan Walzer <jan### [at] lzernet> wrote:
> We proved Warp, to be wrong ?

  So?

-- 
#macro M(A,N,D,L)plane{-z,-9pigment{mandel L*9translate N color_map{[0rgb x]
[1rgb 9]}scale<D,D*3D>*1e3}rotate y*A*8}#end M(-3<1.206434.28623>70,7)M(
-1<.7438.1795>1,20)M(1<.77595.13699>30,20)M(3<.75923.07145>80,99)// - Warp -

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Jan Walzer wrote:
> 
> We proved Warp, to be wrong ?

But it was educational. It was the 1st time I tried in practice
an abstract piece of information learned long time ago. If Warp
hadn't asked the question I wouldn't have learned that arbitrary
orientation can be also defined as x y x rotation. When building
a robot this might be good to know :)


_____________
Kari Kivisalo

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