Descriptions and Samples for the POV-Ray Raytracer by Friedrich A. Lohmüller
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"scale" overview "transform" - transformations

Geometric Transformations

matrix {< , , >}
Matrices and Affine Mapping

(here specially: shearing)

normal vertical shearing
The "matrix"-keyword can be used to define explicit the transformation matrix for objecte or textures.
The syntax of this keyword: 
 matrix <m00, m01, m02,
         m10, m11, m12,
         m20, m21, m22,
         m30, m31, m32>

where the numbers mij may be all kind of float expressions
They form the elements of the first 3 columns of a 4x4-martrix, with the 4th column is defined fixed by <0,0,0,1>.
A point P1(x1/y1/z1) will be mapped to an other point P2(x2/y2/z2) , by following definition:

x2 = m00*x1 + m10*y1 + m20*z1 + m30
y2 = m01*x1 + m11*y1 + m21*z1 + m31
z2 = m02*x1 + m12*y1 + m22*z1 + m32
Somewhat abstract - isn't it? Here some samples to clear this up:

1) The matrix of the identical mapping:
matrix < 1, 0, 0,
         0, 1, 0,
         0, 0, 1,
         0, 0, 0 >
2) This mtrix causes a parallel translation by the vector <2,3,4>:
matrix < 1, 0, 0,
         0, 1, 0,
         0, 0, 1,
         2, 3, 4 >
   The same, only much more effective and better to understand, is descriped by the statement:    translate< 2, 3, 4>.
3) Rotation around the y axis with an angle of Alpha (messured in degrees) will be caused by the following matrix:
(Speciality: sin and cos in POVRAY need their arguments for the angle in radians!)
 
#declare A = radians(Alpha);
matrix<cos(A), 0, sin(A),
          0  , 1,    0  ,
      -sin(A), 0, cos(A),
          0  , 0,    0   >

Rotation around the y axis by 30°
(sin30° = 0.5; cos30° = 0.886): 

matrix <0.886, 0,   0.5,
          0  , 1,     0,
       -0.5  , 0, 0.886,
          0  , 0,     0 >

The same, only much more effective and better to understand, is descriped by the statement: rotate< 0, Alpha, 0> .

 4) The following matrix causes an additional streching by an streching factor of k = 3 with the center in O(0/0/0): 
matrix < 3, 0, 0,
         0, 3, 0,
         0, 0, 3,
         0, 0, 0 >

Also this mapping can be descriped by "scale<3,3,3>" or more simpler by "scale 3" more easily and easier to understand.

These samples have demonstrated that all simple transformations can be descriped by "matrix".

Normally we don't use transformation matrices for all the simple transformations (translation, scaling = centric streching, rotation), because it's hard to understand. But there is an interesting aspect of the matrix statement: It allows also affine mapping of a more general kind which is not to get in another way, e.g. the shearing.
The following matrix causes a shearing of an object or a texture along the y axis:


matrix < 1, 1, 0,
         0, 1, 0,
         0, 0, 1,
         0, 0, 0 >

A shearing along the x direction by 30° is caused by:


  matrix< 1 , 0, 0,
         0.5, 1, 0,
          0 , 0, 1,
          0 , 0, 0 >


normal vertical

shearing

The combination of different mappings by matrices are rarely easy to understand. The following matrix may demonstrate it. This matrix rotates an object around the y axis by 30 degrees and then it is shearing along the y axis and translates parallel in y direction:

 matrix< 0.886, 0.5,  0.5  ,
           0  , 1  ,  0    ,
           0  , 0  ,  0.886,
           0  , 1.5,  0     >


This demonstrates why it is recommanded - if you have no intensiv experience with this - to divide such obscure kind of mappings and to do it step by step:

rotate<0,30,0> 
//Shearing in y direction:
 matrix<1, 0.5, 0,
        0, 1  , 0,
        0, 0  , 1,
        0, 0  , 0 >
 translate<0,1.5,0>

This statements will do the same!

"scale" overview "transform" - transformations
© Friedrich A. Lohmüller, 2004
email email: (legacy email redacted)
homepage:www.f-lohmueller.de