Basic Properties

The basic classification properties are defined as follows:

LINEAR:

A mapping has this property if all its output variables are related to its input variables by linear arithmetic expressions. Such a mapping will preserve straight lines. In two dimensions, examples of LINEAR mappings include shifts of origin, rotations, reflections, magnifications and shearing deformations.

• In general, a mapping is LINEAR if all its first derivatives are constant (i.e. do not change from point to point).

INDEPENDENT:

A mapping has this property if a change in each input variable causes a corresponding change in only a single distinct output variable. Such a mapping will preserve the independence of the coordinate axes. A simple example in two dimensions would be the interchange of the two axes.

• In general, a mapping is INDEPENDENT if there is at most one element in each row and column of its Jacobian matrix which is not identically zero.

DIAGONAL:

A mapping has this property if each output variable depends only on the corresponding input variable, so that the coordinate axes are preserved. There are many examples of such mappings in two dimensions, including those normally used for scaling linear (and logarithmic) graphs. Note that a DIAGONAL mapping is more strongly constrained than an INDEPENDENT mapping (above) in which the coordinate axes may be interchanged. A DIAGONAL mapping is necessarily always INDEPENDENT.

• In general, a mapping is DIAGONAL if its Jacobian matrix is square and diagonal (i.e. all its off-diagonal terms are identically zero). If the Jacobian matrix is not square, then the mapping is DIAGONAL if $\frac{\partial y_{i}}{\partial x_{j}}$ is identically zero for all $i \ne j$.

ISOTROPIC:

A mapping has this property if it locally preserves shapes and the angles between lines. Such a mapping may apply a local scale factor to the distances between neighbouring points, but this factor will not depend on the orientation of the line between the two points, although it may vary from point to point. In two dimensions, an ISOTROPIC mapping will convert a circle at any point into another circle (but possibly of a different size and in a different place), whereas a non-ISOTROPIC mapping would produce an ellipse. If the mapping is also LINEAR (see above) then circles of any size will behave in this way, whereas with a non-LINEAR mapping this may only be true for circles of infinitely small size. Isotropy is an important property of conformal map projections.

• In general, a mapping is ISOTROPIC if it introduces a distance scale factor between neighbouring points which does not depend on the relative orientation of the points. This will be true if its Jacobian matrix $\textbf{J}$ satisfies:

  
  

  $$\mathbf{\tilde{J}} \mathbf{J} = \lambda \mathbf{I}$$ (10)

  
  

  where $\mathbf{\tilde{J}}$ denotes the transpose of $\mathbf{J}$, $\lambda$ is a constant and $\mathbf{I}$ is an identity matrix. A mapping cannot have this property if the number of output variables is less than the number of input variables. A mapping with only one input and one output variable is necessarily always ISOTROPIC.

POSITIVE_DET:

A mapping has this property if the determinant of its Jacobian matrix is greater than zero at all points. In two dimensions, such a mapping can locally represent rotations, magnifications and shearing deformations and can globally represent ``rubber-sheet'' distortions, but it will lack any component of reflection. A string of text subjected to such a mapping would remain legible (although possibly highly distorted) and would not be converted into a mirror image of itself.

• This property can only apply to mappings with an equal number of input and output variables.

NEGATIVE_DET:

A mapping has this property if the determinant of its Jacobian matrix is less than zero at all points. In two dimensions, such a mapping will locally include a component of reflection (possibly also combined with rotation, magnification and shearing deformation) and can globally represent ``rubber-sheet'' distortion combined with a reflection. A string of text subjected to such a mapping would be converted into a mirror image of itself (in addition to any other distortion present).

• This property can only apply to mappings with an equal number of input and output variables.

N.B. A mapping may not have both the POSITIVE_DET and NEGATIVE_DET properties simultaneously. It is also possible that neither of these properties may apply if the determinant is positive at some points and negative at others.

CONSTANT_DET:

A mapping has this property if its area (or volume) scale factor has the same value at all points. If the mapping has an equal number of input and output variables, then this will be true if the determinant of its Jacobian matrix has the same value at all points. Mappings which are LINEAR (see above) necessarily have the CONSTANT_DET property, but it can also apply to non-LINEAR mappings and is an important property of equal area map projections.

• A mapping cannot have this property if the number of output variables is less than the number of input variables.

UNIT_DET:

A mapping has this property if the absolute value of its area (or volume) scale factor is unity (and it has the same sign) at all points. If the mapping has an equal number of input and output variables, then this will be true if the determinant of its Jacobian matrix has an absolute value of unity (and the same sign) at all points. This is a stronger constraint than the CONSTANT_DET property (above) and a mapping with the UNIT_DET property necessarily has the CONSTANT_DET property also. In addition, one of the two properties POSITIVE_DET or NEGATIVE_DET will apply.

• A mapping cannot have this property if the number of output variables is less than the number of input variables.