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.