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Affine Transformations
A transformation changes the positions of points in the plane. A
set of points, when transformed, may as a result acquire a different
shape.
The transformations that move lines into lines, while preserving their
intersection properties, are
special and interesting, because they will move all polylines into polylines and
all polygons into polygons. These are the affine transformations.
Every affine transformation can be expressed as a transformation that fixes
some special point (the "origin") followed by a simple translation of
the entire plane. These pointfixing transformations are the linear
ones.
The theory of vector spaces demonstrates that linear transformations
can be represented by matrices: twobytwo arrays of numbers (threebythree in
three dimensions).
Other representations are possible. The commonest
alternatives use threebythree arrays in two dimensions and fourbyfour arrays
in three dimensions. For a clear, accurate explanation of this matrix
approach, please see http://www.css.tayloru.edu/~btoll/s99/424/res/mtu/Notes/geometry/geotran.htm
. Some more information about these alternatives also appears the Applications
section below.
The most general linear transformation of the plane can further be broken
down into a series of transformations with definite geometric
interpretations. For example, half of all linear transformations can be
expressed as a skew transformation followed by a "squeeze"
(change of aspect ratio) followed by a rotation. At any point in
this sequence a uniform scaling may be applied. These operations
are defined and illustrated below.
The other half of
the transformations are obtained by first reflecting the points about a
line. A simple way to accomplish this is to introduce a coordinate system
and interchange the x and y coordinates of every point: that is a reflection
about the line y=x.
Affine transformations and GIS
Transformations play many important roles in GIS development and use:

The change from one projection to another is a
transformation. So is a change of datums or a change of coordinate
systems. 

The "warping" or
"rubbersheeting" of images, to georegister them, is a
transformation. 
 Often, applying a suitable transformation to a configuration of shapes can
simplify the analysis of their geometric relationships. 
Every change of map projection is locally (over sufficiently small areas) well
approximated by an affine transformation. Thus, for example, changes among
equalarea projections must have linear components involving skewings,
rotations, and squeezes but
not
scalings: only the scalings change areas.
Similarly, conformal (shapepreserving) projections must locally not have any squeeze or skew
components, because these operations change shapes: only the rotations and uniform scalings
preserve shapes. Thus, a good geometric understanding of affine
transformations helps to conceptualize changes of projection.
Practically, affine transformations provide a useful family of operations one
can apply to georeference map features. Changes of coordinate system,
changes of units of measurement, and referencing scanned paper maps are common
applications.
The following sections describe linear transformationsthe interesting parts
of affine transformationsin more detail. They are followed by a brief
section on GIS applications and another section
specific to the ArcView 3.x GIS software.
Taxonomy of linear transformations

This is the reference
figure. The following figures will show what happens to this
figures as it undergoes various transformations.
The origin is a light blue point. The lines
form a regular square grid. The interior colors distinguish
different vertical bands of square cells. 
To
describe transformations in detail, we use a coordinate
system. Let the coordinates of the origin in the reference
figure be (0,0). Establish units of measurement that make the
squares exactly one unit on a side. 

The blue vector, e_{1},
extends one unit in the horizontal direction. The red vector, e_{2},
extends one unit in the vertical direction. Every point in the
plane has a unique expression as the sum of multiples of e_{1}
and e_{2}. If we write p = x * e_{1}
+ y * e_{2}, then x and y are the (usual) coordinates
of the point p. The red and blue vectors are the basis
vectors for the coordinate system. 
To
understand or describe a transformation, it suffices to specify what the
transformation does to the basis vectors. This follows from some
simple logic: If p is any point and T denotes transformation,
linearity means that T(p) = T( x * e_{1}
+ y * e_{2}) = x * T(e_{1}) + y * T(e_{2}).
Thus, once we know T(e_{1}) and T(e_{2}),
we can easily determine T(p) for any point p.
Let
numbers A, B, C, and D stand for the coefficients for the destinations
of the two basis vectors. Specifically, suppose the transformation
moves e_{1} to A * e_{1} + C * e_{2}
and e_{2} to B * e_{1} + D * e_{2}
. Conventionally these coefficients are written as the matrix
The effect of the transformation on an arbitrary point p = x * e_{1}
+ y * e_{2 }is then the product of this matrix and the column
vector (x, y): T(p)
= T(x * e_{1}
+ y * e_{2})
= x * T(e_{1}) + y * T(e_{2})
= x * (A * e_{1} + C * e_{2}) + y * (B * e_{1} + D * e_{2})
= (x*A + y*B) e_{1} + (x*C + y*D) e_{2}. 
Matrices
are also useful for describing affine transformations.
This is done with a geometric trick. The usual
(Euclidean) plane can be embedded in three dimensional Euclidean space
as the set of all points (x, y, 1) at unit height above the xy
plane. The most general three dimensional linear
transformation sends such a point to the point at location (A*x + B*y +
C, D*x + E*y + F, G*x + H*y + I) where A, B, ..., I are the coefficients
of the threebythree matrix for the transformation. When G = H = 0 and I = 1, the third coordinate of the new point will also be
1, so that the new point also lies in the original plane. Such
matrices with G = H = 0 and I = 1 therefore describe linear transformations of
the embedded plane.
Evidently coefficients A, B, D, and E determine a linear
transformation and coefficients C and F determine a parallel
translation: that is, such threebythree matrices describe affine twodimensional
transformations. Here is what they look like in full matrix form:
Incidentally, even when G or H are nonzero
or I is not 1, we can still create a transformation of the embedded
plane by rescaling the new point until its height above the xyplane
is 1. That is, the transformation determined by the most general
threebythree matrix is of the form
x > (A*x + B*y + C)/(G*x + H*y + I)
y > (D*x + E*y + F)/(G*x + H*y + I)
These are the projective transformations of the plane. 
Skew transformations


This is the result of
a skew transformation of the reference figure. The original reference lines are overlaid
in gray. The transformed basis vectors are shown in their original
colors. Evidently,
e_{1} was not moved while e_{2} was moved
to e_{1} + e_{2}. More formally, the
coordinates for the transformation of e_{1} are (1, 0)
and for the transformation of e_{2} are (1, 1).
This is written as the matrix

The first column of numbers specifies where e_{1}
goes and the second column specifies where e_{2} goes.
All skew
transformations "look" like this one in the sense that
 They preserve some line: every point on this line is left unmoved.
 Lines parallel to the fixed line are sent to themselves.
Points along these parallel lines are shifted by an amount
proportional to the (signed) distance from the fixed line.
 The areas of shapes are unchanged.
If you choose a basis with e_{1} on the fixed line,
then the matrix for any skew transformation will look like
where n is some number. Evidently, skew transformations change
angles and distances in general. (Distances along the preserved
line are not changed, but most distances are changed.) 
"Squeezes" (aspect ratio changes)


A "squeeze"
changes the aspect ratio of all horizontal rectangles without changing their areas.
The illustrated squeeze has shrunk
horizontal distances by onethird (they were all multiplied by 2/3)
while expanding all vertical distances by onehalf (they were all
multiplied by 3/2) to compensate.
The matrix for this squeeze is

The most general squeeze looks like
this, always expanding one direction to compensate for shrinking the
other so that areas do not change.
If you choose a basis with e_{1} along one direction
of a squeeze and e_{2} in the perpendicular direction,
then the matrix will look like
where t is a positive number.
Squeezes change angles and distances, but preserve areas. 


A rotation preserves
distance, angle, orientation, and area. All shapes are transformed
into congruent shapes.
(A modern view of geometry is that it is the study of
groups of transformations. From this point of view, rotations and
translations define congruence. That is, congruent shapes
are those that can be transformed into each other by some sequence of
rotations and translations. Furthermore, what we mean by
"shape" is whatever properties happen to be preserved by
exactly these operations.) 
This figure shows a 30 degree counterclockwise rotation. It is
considered a "positive" rotation because it moves the first
basis
vector e_{1}
toward the second, e_{2}. Its matrix is
approximately
0.866025 
0.5 
0.5 
0.866025 
The value 0.866025 represents onehalf the square root of three.
When the
basis vectors are chosen perpendicular and of equal length, all rotation
matrices are of the form
Cos(w) 
Sin(w) 
Sin(w) 
Cos(w) 
where w is the angle of rotation. This is practically the definition
of the sine and cosine functions. In particular, given any two
numbers c and s for which c^{2} + s^{2} = 1, there
exists an angle w for which c = cos(w) and s = sin(w). The angle
is unique up to a multiple of a full circle (360 degrees). 


A (uniform) scaling
simply expands or shrinks all distances by a constant amount. This
figure shows the result of scaling by 1.5. Its matrix is
No matter how the basis is selected, the matrix for a uniform scaling
is always of the form
for some number r. 
Any
scaling sends all shapes to similar shapes. It preserves
all angles and even all directions, too. It changes areas, but in
a uniform way: the area of any figure will be r^{2} times its
original area after it is scaled by r. Scalings change all nonzero
distances. 
The
combination of a squeeze of size t and a scaling of size r will
differentially scale the two basis directions. The matrix for the
combination is
The scaling factor in the first (x) direction is r*t and in
the second (y) direction is r/t. Conversely, if these two nonzero
scaling factors A and D, respectively, are independently specified, then
you can determine that r = sqrt(A*D) and t = sqrt(A/D). Thus
squeezes and uniform scalings are just another way to describe possibly
independent rescalings in the two basis directions. 
There are other linear transformations, too, such as linear projections (all
points in the plane are collapsed onto a single line), but these are not invertible: they
cannot be undone, because two or more points end up at the same location.
The invertible linear transformations GL(2,R) form a group: any
succession of invertible linear transformations is an invertible linear
transformation. The identity in this group is the transformation
that fixes all points. Its matrix is
It is simultaneously a skew transformation (n=0), a squeeze (t=1), a rotation
(w=0), and a scaling (r=1).
The order in which linear operations are applied matters. (Uniform
scalings are the exception: you can apply a uniform scaling at any point in a
sequence of transformations without changing the result.) For any other
kind of transformation, however, the results will usually be different.
For example, a squeeze of size t=2 followed by a 90 degree rotation will send
the point (1,0) to the point (0,2), whereas a 90 degree rotation followed by the
same squeeze sends (1,0) to (0, 1/2).
You
can compute the effect of a series of linear transformations by
multiplying their matrices right to left. Our assertion
above that every linear transformation is (after possibly switching x
and y) the result of a skewing, squeezing, rotation, and scaling is
easily proven this way. Simply do the multiplication:
r 
0 
* 
c 
s 
* 
t 
0 
* 
1 
n 
= 
crt 
cnrt  sr/t 
0 
r 

s 
c 

0 
1/t 

0 
1 

srt 
snrt + cr/t 
(where, for brevity, c = Cos(w) and s = Sin(w)). If we now write the four
coefficients crt = A, cnrt  sr/t = B, srt = C, and snrt + cr/t = D we
can solve for w, n, r, and t. Specifically, let's assume A*D  B*C
is nonzero;
the case A*D  B*C = 0 (for noninvertible transformations) can be treated
specially. The solution, which you can check with simple algebra,
is:

If A*D  B*C < 0, first switch the
coordinates. This makes A*D  B*C > 0. (This
operation effectively switches the columns of the matrix.) A*D
 B*C is the determinant of the linear transformation.

Let F = 1/(A*A + C*C). (This
exists because if A*A + C*C = 0, then A = C = 0, implying A*D  B*C
= 0, which is the special case we excluded above.) Using the
definitions of A and C in terms of c, s, r, and t, compute
that F = 1/(r^{2}t^{2}).

n = (A*B + C*D) * F.

t = 1/Sqrt(F * (A*D  B*C))

r = Sqrt(A*D  B*C)

c = A/(r*t), s = C/(r*t).
From (2) it is evident that c^{2} + s^{2} = (A^{2}
+ C^{2}) / (A^{2} + C^{2}) = 1, so the
rotation matrix (and therefore its angle w) is
determined.
For example, you can check that the
matrix A = 0, B = 1, C = 2, D = 1 corresponds to:

A reflection about y=x (x > y, y > x), 

followed by a horizontal skew (y > y + n*x) of size
n=1, 

followed by a rescaling (x > r*x, y > r*y) of r=1.4142135624, 

followed by a rotation in the positive (counterclockwise) direction of
45 degrees. 
No squeeze is listed because the squeeze in
this case is the identity operation, which does nothing. We might
say that there is no squeeze component to this particular
transformation. Likewise, other transformations may omit one or
more of their possible components. 
Georeferencing images
Raster data sets are fundamentally arrays of numbers. Their natural
coordinates are the array indices. These place the numbers at points of a
unit square grid. Georeferencing a raster data set therefore involves
applying an affine transformation to move the grid points onto the desired data
locations. This is usually done by specifying the matrix coefficients in a
"header" or "world" file.
We will continue to write the threebythree transformation matrix as
representing the transformation
x > A*x + B*y + C
y > D*x + E*y + F
With this notation a standard "world file" for an image is an
ASCII (text) file with six lines. On each line is a number representing
one of the coefficients. They appear in the order A, D, B, E, C, F
(that is, arranged by columns, not rows, with the bottom row suppressed).
(Other conventions also exist for "world files," depending on the file
format and the software, but they generally contain the same six matrix
coefficients in some order.)
Transforming features
Sometimes map features need transformation to adjust units of measurement,
rotate a map, change the origin of a coordinate system, or remove a uniform
distortion. Frequently an affine transformation will provide sufficient
accuracy.
Figuring out the transformation is amazingly easy. Recall that,
for a linear transformation, the first column of the matrix contains the
coefficients of the point where the first basis vector (1, 0) is sent and the
second column contains the coefficients of the point where the second basis
vector (0, 1) is sent. Evidently the third column contains the translation
part of the affine transformation.
Therefore, simply write down where (1, 0), (0, 1), and (0, 0) must go.
Suppose these are the points (a,d), (b,e), and (C, F), respectively. This
implies that before everything is translated by (C,F), point (1,0) must go to
(A, D) = (aC, dF) and point (0, 1) must go to (B, E) = (bD, eF).
Values A, B, C, D, E, F provide the threebythree matrix coefficients.
For example, suppose you need to transform a set of
features so that
 Units of measurement are converted from meters to feet (that is, multiplied
by 39.37/12),
 then rotated 90 degrees clockwise,
 then shifted by (500000, 150000) to a new origin.
Before the shift is applied, step (1) first converts basis vector (1, 0) to
(39.37/12, 0) and then step (2) rotates that to (0, 39.37/12); similarly, basis vector (0, 1) is
converted to (0, 39.37/12) and then rotated to (39.37/12, 0). Therefore
A=0, B = 39.37/12, C = 500000, D = 39.37/12, E = 0, and F = 1500000.
This example illustrates the power of the geometric interpretation of
transformations and matrices: the entire transformation matrix was computed
without doing a single matrix multiplication.
The ArcView Transform2D Class
The rest of this article is specific to the ArcView 3.x GIS software.
ArcView GIS software in versions 3.2 and later supports affine transformation
of shapes through the Transform2D Class. We have created a user interface
to this class and posted it on the ArcScripts
pages for free download. The interface represents the Avenue requests rather than
the natural geometric interpretation outlined above.

This is the Transform 2D dialog. The
buttons correspond to Avenue "requests." Requests are
operations that potentially modify a transformation.
The fillin text lines in the top half of the dialog specify
parameters for the buttons: scale factors,
translation components, and rotation angle (in degrees
counterclockwise). Filling in the parameters does nothing; you
have to press a button to apply the desired transformation to the
matrix.
The "Invert" and "Reset" buttons need no
parameters. "Invert" computes the affine transformation
that undoes the current one. "Reset" replaces the matrix
by the identity (the transformation that keeps all points in place). 
The checkboxes offer two independent reflections. The
"horizontal" reflection is the transformation x > x, y > y
and the "vertical" reflection is x > x, y > y. Again,
no reflection is applied until the "Reflect" button is pressed.
The fillin text lines in the bottom half of the dialog show the
threebythree affine transformation matrix.
Pressing the buttons applies the specified transformation to the current
matrix, rather than replacing the matrix. In this way you can build a
transformation piecebypiece by accumulating simpler transformations.
You can also specify a transformation by typing matrix coefficients
directly. This is necessary for skew transformations because ArcView does
not provide a skew request. Because the Transform2D class uses all nine
coefficients of the matrix, nine coefficients are displayed. You will
discover, however, that ArcView almost ignores the bottom row of values: it
does not do projective transformations. (The
exception concerns the lower right entry, the "I" coefficient.
ArcView's Transform2D object uses it to scale the translation!
For further information, you will have to experimentthere is no
documentation.)
Actions causing a change in the matrix are followed by an audible
confirmation (a beep). Undesired changes are readily undone
("Undo").
The bottom third of the dialog provides some information about the current
transformation matrix. The "Scaling is uniform," "X
Scale," and "Y Scale" values show the results of ArcView
requests. However, these results do not correspond to the scale
factors r*t and r/t described above. It is unclear how ArcView
calculates them. They do not appear to have a consistent geometric
meaning.
The "Info" button therefore computes the geometric
components of the current matrix for you.

Consider the previous example.
This figure shows the dialog after the scale, rotation, and move
transformations are applied (in that order).
Pressing the "Info" button produces this report:
The transformation is:
A rescaling (x > r*x, y > r*y) of r=3.28083333
Followed by a rotation in the positive (counterclockwise) direction of 90 degrees
Followed by a translation (x > x + e, y > y + f) of (e,f) = (500000, 1500000).
NB: This is just one of many possible geometric interpretations.
This report is computed directly from the matrix coefficients. 
Pressing "OK" to this dialog will apply the transformation shown by
the matrix to all selected shapes in the active ArcView feature theme.
Modification history:
Please contact our webmaster with comments and suggestions about this
page. We warmly welcome your help.
20 September 2002: Minor amplifications, including
pointing out which geometric propertiesdistance, angle, area, directionare
preserved by which types of transformations..
10 April 2002: Systematic, minor clarifications
introduced throughout. Minor changes in formatting. Link to Taylor
U. added.
30 November 2001: Minor corrections.
21 September 2000: Minor corrections.
8 June 2000: Some minor typographical errors were corrected. Inadvertent transposition
of "squeezes" and "scalings" in one sentence was rectified.
22 May 2000: First posted. 