A generalized transform object represented internally
as a 4x4 double-precision floating point matrix. The mathematical representation
is row major, as in traditional matrix mathematics. A Transform3D is used to
perform translations, rotations, and scaling and shear effects.
A transform has an associated type, and all type classification is left to the
Transform3D object. A transform will typically have multiple types, unless it
is a general, unclassifiable matrix, in which case it won't be assigned a type.
The Transform3D type is internally computed when the transform object is constructed
and updated any time it is modified. A matrix will typically have multiple types.
For example, the type associated with an identity matrix is the result of ORing
all of the types, except for ZERO and NEGATIVE_DETERMINANT, together. There
are public methods available to get the ORed type of the transformation, the
sign of the determinant, and the least general matrix type. The matrix type
flags are defined as follows:* ZERO - zero matrix. All of the elements in the
matrix have the value 0.
* IDENTITY - identity matrix. A matrix with ones on its main diagonal and zeros
every where else.
* SCALE - the matrix is a uniform scale matrix - there are no rotational or
translation components.
* ORTHOGONAL - the four row vectors that make up an orthogonal matrix form a
basis, meaning that they are mutually orthogonal. The scale is unity and there
are no translation components.
* RIGID - the upper 3 X 3 of the matrix is orthogonal, and there is a translation
component-the scale is unity.
* CONGRUENT - this is an angle- and length-preserving matrix, meaning that it
can translate, rotate, and reflect about an axis, and scale by an amount that
is uniform in all directions. These operations preserve the distance between
any two points, and the angle between any two intersecting lines.
* AFFINE - an affine matrix can translate, rotate, reflect, scale anisotropically,
and shear. Lines remain straight, and parallel lines remain parallel, but the
angle between intersecting lines can change.A matrix is also classified by the
sign of its determinant:NEGATIVE_DETERMINANT - this matrix has a negative determinant.
An orthogonal matrix with a positive determinant is a rotation matrix. An orthogonal
matrix with a negative determinant is a reflection and rotation matrix.The Java
3D model for 4 X 4 transformations is:
[ m00 m01 m02 m03 ] [ x ] [ x' ] [ m10 m11 m12 m13 ] . [ y ] = [ y' ] [ m20 m21 m22 m23 ] [ z ] [ z' ] [ m30 m31 m32 m33 ] [ w ] [ w' ] x' = m00 . x+m01 . y+m02 . z+m03 . w y' = m10 . x+m11 . y+m12 . z+m13 . w z' = m20 . x+m21 . y+m22 . z+m23 . w w' = m30 . x+m31 . y+m32 . z+m33 . w
Note: When transforming a Point3f or a Point3d, the input w is set to 1. When
transforming a Vector3f or Vector3d, the input w is set to 0.