B Classification Properties

This Appendix describes the classification properties which may be declared and associated with a transformation (Section 5.1 shows how this is done and how the information may subsequently be retrieved). In order to be precise, the definitions given here are necessarily mathematical. Readers who require simpler and more specific information about how to classify a particular transformation may find Table 4 helpful.

B.1 General

Many of the properties described here depend on the nature of a Jacobian matrix associated with a transformation; there are potentially two of these matrices, corresponding with the forward and inverse mappings. Using the notation of Equation 1, the Jacobian matrix ${\mathbf{J}}_F$ associated with the forward mapping is the $n\times m$ matrix of partial derivatives:

while that associated with the inverse mapping ${\mathbf{J}}_I$ is the equivalent $m\times n$ matrix obtained by inter-changing input and output variables ($x$ and $y$) throughout.

The significance of these matrices can be seen by considering a simple linear mapping in two dimensions. Such a mapping is capable of representing a combination of a shift of origin, magnification, rotation, reflection and shearing deformation:

It may be re-written as the matrix equation:

The Jacobian matrix $\mathbf{J}$ therefore contains the coefficients which define this mapping and determine its character, apart from a shift of origin. The determinant of $\mathbf{J}$ ($det\mathbf{J}=\; ae-bd$) is the signed “area scale factor” which the mapping introduces (i.e. the area of the parallelogram produced when the mapping acts on a unit square). In more than two dimensions, $det\mathbf{J}$ would be the equivalent “volume scale factor”.

If the mapping is not linear, then the Jacobian matrix will vary from point to point. Nevertheless, it may still be regarded as a local linear approximation to the true mapping (apart from re-location of the origin) and $det\mathbf{J}$ can still be interpreted as the local area (or volume) scale factor, which may now change from point to point.

Changing dimensionality. For transformations with an equal number of input and output variables ($m\; =\; n$), the Jacobian matrices ${\mathbf{J}}_F$ and ${\mathbf{J}}_I$ associated with the forward and inverse mappings (if specified) will both be square. If the transformation functions are correctly formulated, then these two matrices will be mutually inverse and will satisfy:

where $$I is an identity matrix. Their determinants will also be related by:

As a consequence of this (and the definitions of the basic classification properties given below) any property which applies to one of a transformation’s two mappings will necessarily apply to the complementary mapping also.

If the transformation affects a change of dimensionality, however, so that $m\ne n$, then it is possible that certain properties may only apply to one of its two mappings. It is still acceptable to associate such properties with the transformation, however, because the TRANSFORM software will take account of the number of input/output variables, and will omit properties which it knows cannot apply when information about a particular mapping is requested. In general, therefore, a classification property may be declared for a transformation if either of its mappings has that property.

B.2 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 $\frac{\partial yi}{\partial xj}$ 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 $$J satisfies:

  $$\mathbf{<mover\; accent="true"><mrow>J</mrow>̃</mover>}\mathbf{J}=\; \lambda \mathbf{I}$$

  (10)

  where $\mathbf{<mover\; accent="true"><mrow>J</mrow>̃</mover>}$ 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.

B.3 Composite Properties

Many important mapping properties are composite; i.e. they depend on the presence of several of the basic properties above in combination. Table 4 lists the more important of these and the following notes augment the information in this Table. The presence of a possible shift of origin is disregarded throughout:

A – Shift of origin.

The mapping implements a simple shift of coordinate origin, the nature of which must be determined by transforming a test point.

B – Rotation about an axis.

The mapping represents a simple rotation about an axis (a point in two dimensions) without associated magnification or distortion. If the DIAGONAL property also applies, then the amount of rotation will be zero, so the mapping reduces to a shift of origin (see A above).

C – Magnification about a point.

The mapping applies a simple positive magnification (a zoom) factor about a point without any associated rotation or other form of distortion. If the magnification factor is negative, then a component of reflection will be introduced if the number of input/output variables is odd. In this case the POSITIVE_DET property should be replaced by NEGATIVE_DET.

D – Graphical scaling (linear).

This type of mapping is commonly used to scale the axes of a graph, with different scale factors being applied to each axis. Either POSITIVE_DET or NEGATIVE_DET will also apply depending on the sign of the scale factors in use and whether they result in a mirror image. POSITIVE_DET will apply if the number of negative scale factors is even and NEGATIVE_DET will apply if this number is odd.

E – Graphical scaling (non-linear).

This type of mapping is commonly used to non-linearly scale the axes of graphs (to produce a log-log plot for instance). Since the non-linear functions used are normally monotonic, either the POSITIVE_DET or NEGATIVE_DET property will usually apply, depending on the sign of the scaling functions’ derivatives along each axis. POSITIVE_DET will apply if the number of negative derivatives is even and NEGATIVE_DET will apply if this number is odd.

F – Interchange of axes.

The mapping simply interchanges coordinate values. The property POSITIVE_DET will apply if the resulting axis permutation is cyclic and NEGATIVE_DET will apply if the permutation is non-cyclic.

G – Axis reversal.

The mapping reverses one or more of the axes (i.e. changes the sign of the coordinates with or without the addition of a constant). The POSITIVE_DET property will apply if the number of axes reversed is even, while NEGATIVE_DET will apply if this number is odd.

H – Conformal map projection.

This implements a conformal map projection which locally preserves shapes and angles but may introduce a scale factor which varies from point to point.

I – Equal area map projection.

The mapping implements an equal area map projection in which the area scale factor does not vary from point to point, although shapes and the angles between lines may be distorted.