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  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:

Table: Common types of mapping with their composite classification properties.

Basic Property	A	B	C	D	E	F	G	H	I
LINEAR	$\bullet$	$\bullet$	$\bullet$	$\bullet$	$\times$	$\bullet$	$\bullet$	$-$	$-$
INDEPENDENT	$\circ$	$-$	$\circ$	$\circ$	$\circ$	$\bullet$	$\circ$	$-$	$-$
DIAGONAL	$\bullet$	$-$	$\bullet$	$\bullet$	$\bullet$	$\times$	$\bullet$	$-$	$-$
ISOTROPIC	$\bullet$	$\bullet$	$\bullet$	$-$	$-$	$\bullet$	$\bullet$	$\bullet$	$-$
POSITIVE_DET	$\bullet$	$\bullet$	$\bullet$	?	?	?	?	$\bullet$	$\bullet$
NEGATIVE_DET	$\times$	$\times$	$\times$	?	?	?	?	$\times$	$\times$
CONSTANT_DET	$\circ$	$\circ$	$\circ$	$\circ$	$-$	$\circ$	$\circ$	$-$	$\bullet$
UNIT_DET	$\bullet$	$\bullet$	$-$	$-$	$-$	$\bullet$	$\bullet$	$-$	$-$

Key:

Mapping types:	A	-	Shift of origin
	B	-	Rotation about an axis
	C	-	Magnification about a point
	D	-	Graphical scaling (linear)
	E	-	Graphical scaling (non-linear)
	F	-	Interchange of axes
	G	-	Axis reversal
	H	-	Conformal map projection
	I	-	Equal area map projection

Symbols:	$\bullet$	-	Required
	$\circ$	-	Implied
	$\times$	-	Prohibited
	$-$	-	Irrelevant
	?	-	See note in text

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.