AffineTransformation Tutorial¶
Welcome¶
Welcome to this tutorial about affine transformations which are used to convert coordinates from one domain to another. As an example we will convert world coordinates to pixel (screen or image) coordinates. All of the mathematics required will be introduced along the way and you will soon be making use of the power of these kind of calculations.
About Your Instructor
Christian Mueller is a GeoTools developer and is working for a big customer to build up a GIS infrastructure. In the GeoTools project he is best known for his work on Image Mosaic JDBC
Definitions¶
Before we start, some important definitions.
Abbreviation 
Meaning 


lower left corner 

upper left corner 

upper right corner 

lower right corner 
We have a rectangle in world coordinates, the origin is at the LLC, x = 2000, y = 3000,width = 8000
units an height=9000
units.
The resulting corner points are:
Point 
world x,y coordinates 


2000, 3000 

2000, 12000 

10000, 12000 

10000, 3000 
Next, we want to map this rectangle to a screen, using pixel coordinates. The available screen
size=400x300
pixels. Unfortunately, pixel coordinates have their origin in the ULC
normally,
not in the LLC
. The next table shows the mappings.:
Point 
world x,y coordinates 
pixel x,y coordinates 


2000, 3000 
0,300 

2000,12000 
0, 0 

10000,12000 
400, 0 

10000, 3000 
400,300 
The challenge is to find a method how to transform each point within the world rectangle to a point in the pixel rectangle. (And the other way around)
Mathematical Background¶
Since most people I know dislike mathematics I will reduce this section to an absolute minimum. An affine transformation is based on a matrix.
Here we need a
3x3
matrix like this one:[ m00 m01 m02 ] [ m10 m11 m12 ] [ 0 0 1 ]
The numbers are row and column indices, e.g. m12 is the matrix element in row 1, column 2. The values in the bottom row are always as shown.
A point is represented as a column vector:
[ x ] [ y ] [ 1 ]
Once again, the bottom element is a constant (always 1).
Transforming the point coordinates involves multiplying the point’s column vector by the affine transform matrix:
[ x_new] = [ m00 m01 m02 ] [ x ] = [ m00x + m01y + m02 ] [ y_new] = [ m10 m11 m12 ] [ y ] = [ m10x + m11y + m12 ] [ 1 ] = [ 0 0 1 ] [ 1 ] = [ 0 + 0 + 1 ]
Note
It is very important to understand this, but don’t worry if you are unfamiliar with matrix arithmetic because the steps are explained in detail below.
Identify Matrix
A quick test. Why is the matrix:
[ 1 0 0 ] [ 0 1 0 ] [ 0 0 1 ]
called the identity matrix ?
Answer:
[ 1 0 0 ] [ x ] = [ x ] [ 0 1 0 ] [ y ] = [ y ] [ 0 0 1 ] [ 1 ] = [ 1 ]
The detailed calculation:
1*x + 0*y + 0*1 = x 0*x + 1*y + 0*1 = y 0*x + 0*y + 1*1 = 1
Swap X and Y
A second test. What is this matrix responsible for:
[ 0 1 0 ] [ 1 0 0 ] [ 0 0 1 ]
This matrix swaps x and y:
[ 0 1 0 ] [ x ] = [ y ] [ 1 0 0 ] [ y ] = [ x ] [ 0 0 1 ] [ 1 ] = [ 1 ]
The detailed calculation:
0*x + 1*y + 0*1 = y 1*x + 0*y + 0*1 = x 0*x + 0*y + 1*1 = 1
Calculation¶
We need three steps for getting pixel x/y
from world x/y
.
Translate
Scale
Mirror
These are discussed below.
Translate Operation¶
We have to shift the origin of the world rectangle to 0,0. This is easy. The
LLC
has values 2000,3000, we only need to subtract 2000 fromx
and 3000 fromy
. We use theURC
with values 10000,12000 to demonstrate the calculation.Java Code
AffineTransform translate= AffineTransform.getTranslateInstance(2000, 3000); System.out.println("Translate:" + translate.toString()); Point2D p = new Point2D.Double(2000,3000); System.out.println(translate.transform(p, null));
Output:
Translate:AffineTransform[[1.0, 0.0, 2000.0], [0.0, 1.0, 3000.0]] Point2D.Double[0.0, 0.0]
The
toString()
method of theAffineTransform
class only shows the first two rows of the matrix. The static methodgetTranslateInstance
is a convenience method, otherwise you have to call a constructor with 6 values.The matrix used:
[ 1.00 0.00 2000.00 ] [ 0.00 1.00 3000.00 ] [ 0.00 0.00 1.00 ]
The detailed calculation:
1 * 2000 + 0 * 3000 + 1 * 2000 = 0 0 * 2000 + 1 * 3000 + 1 * 3000 = 0 0 * 2000 + 0 * 3000 + 1 * 1 = 1
The result of all four corner points is:
Point
before
after
LLC
2000, 3000
0, 0
ULC
2000,12000
0, 9000
URC
10000,12000
8000, 9000
LRC
10000, 3000
8000, 0
Scale Operation¶
The world rectangle has a width of 8000 units and a height of 9000 units, the pixel dimension has a width of 400 pixels and a height of 300 pixels. We need to scale with 400/8000.0 and 300 / 9000.0.
Let us use the point in the middle of the world rectangle after the translate operation, having its
LLC
at 0,0.Java Code
AffineTransform scale= AffineTransform.getScaleInstance(400/8000.0, 300 / 9000.0); System.out.println("Scale:" + scale.toString()); p = new Point2D.Double(4000,4500); System.out.println(scale.transform(p, null));
Output:
Scale:AffineTransform[[0.05, 0.0, 0.0], [0.0, 0.033333333333333, 0.0]] Point2D.Double[200.0, 150.0]
The detailed calculation (omitting the last one, the result is always 1)
0.05 * 4000 + 0 * 5000 + 1 * 0 = 200 0 * 4000 + 0.03.. * 5000 + 1 * 0 = 150
The used matrix is:
[ 0.05 0.00 0.00 ] [ 0.00 0.03.. 0.00 ] [ 0.00 0.00 1.00 ]
Using the output of the translation operation as the input for the mirror operation, the result of all four corner points is:
Point
before
after
LLC
0, 0
0, 0
ULC
0, 9000
0, 300
URC
8000, 9000
400, 300
LRC
8000, 0
400, 0
Mirror Operation¶
Remember: The world rectangle has its origin in the LLC
and the pixel rectangle has its origin
in the ULC
!
There is a need for a mirroring operation. After the scale operation, we have already pixel values, but we must mirror the y value. The x value should not change. For mirroring, we must calculate:
y_new = 300  y
Let us create the appropriate affine transform.
Java Code
AffineTransform mirror_y = new AffineTransform(1, 0, 0, 1, 0, 300); System.out.println("Mirror:" + mirror_y.toString()); p = new Point2D.Double(100,50); System.out.println(mirror_y.transform(p, null));
Output:
Mirror:AffineTransform[[1.0, 0.0, 0.0], [0.0, 1.0, 300.0]] Point2D.Double[100.0, 250.0]
The x value is unchanged, but the y value is mirrored.
The matrix used is:
[ 1.00 0.00 0.00 ] [ 0.00 1.00 300.00 ] [ 0.00 0.00 1.00 ]
The detailed calculation:
1 * 100 + 0 * 50 + 1 * 0 = 100 0 * 100 + 1 * 50 + 1 * 300 = 250
Using the output of the scale operation as the input for the scale operation, the result of all four corner points is:
Point
before
after
LLC
0, 0
0, 300
ULC
0, 300
0, 0
URC
400, 300
400, 0
LRC
400, 0
400, 300
Matrix Magic¶
Concatenation¶
Until now, most of you will say that it is easier to write this calculations without the use of the AffineTransform
class, be patient.
We have created 3 AffineTransform
objects, now we combine them. There is a method
AffineTransform.concatenate(AffineTransform other)
Which we will be introducing in this section. The only important thing to know is that you have
to START with the LAST AffineTransform
object, NOT with the first.
Java Code
AffineTransform world2pixel = new AffineTransform(mirror_y); world2pixel.concatenate(scale); world2pixel.concatenate(translate); System.out.println("World2Pixel:" + world2pixel.toString()); p = new Point2D.Double(2000,3000); System.out.println("LLC: " + world2pixel.transform(p,null)); p = new Point2D.Double(2000,12000); System.out.println("ULC: " + world2pixel.transform(p,null)); p = new Point2D.Double(10000,12000); System.out.println("URC: " + world2pixel.transform(p,null)); p = new Point2D.Double(10000,3000); System.out.println("LRC: " + world2pixel.transform(p,null));
Output:
LLC: Point2D.Double[ 0.0, 300.0] ULC: Point2D.Double[ 0.0, 0.0] URC: Point2D.Double[400.0, 0.0] LRC: Point2D.Double[400.0, 300.0]
The combined matrix is:
[ 0.05 0.00 100.00 ] [ 0.00 0.03 400.00 ] [ 0.00 0.00 1.00 ]
Lets use
LRC
(10000,3000) to show a detailed calculation:0.05 * 10000 + 0 * 3000 + 1 * 100 = 400 0 * 10000 + 0.03.. 3000 + 1 * 400 = 300
At the end of the day, you have exactly one
AffineTransform
object doing the job.
Graphics2D¶
As an example of the power of Affinetransformation
, the
java.awt.Graphics2D
class has a method:
Graphic2d.setTransform(AffineTransform tx)
If you set our transform object in your Graphics2D
object, you can draw and paint with world
coordinates.
Inversion¶
Create an inverse transformation¶
What about calculating world coordinates from pixel coordinates? This is a commonly asked in terms of “what did the user click on?”.
This is easy, get the inverse transform as shown here:
Look at this code segment.
Java Code
AffineTransform pixel2World=null; try { pixel2World = world2pixel.createInverse(); } catch (NoninvertibleTransformException e) { e.printStackTrace(); } System.out.println("Pixel2World:" + pixel2World.toString()); p = new Point2D.Double(200,150); System.out.println("World : " + pixel2World.transform(p,null));
Output:
Pixel2World:AffineTransform[[20.0, 0.0, 2000.0], [0.0, 30.0, 12000.0]] World : Point2D.Double[6000.0, 7500.0]
The inverse matrix is:
[ 20.00 0.00 2000.00 ] [ 0.00 30.00 12000.00 ] [ 0.00 0.00 1.00 ]
Let us use the pixel values 200,150 (representing the center of the pixel rectangle) to show a detailed calculation:
20 * 200 + 0 * 150 + 1 * 2000 = 6000 0 * 200 + 30 * 150 + 1* 12000 = 7500
The point 6000,7500 is indeed the center of our world rectangle.
The inversion result of our pixel corner points is:
Point
before
after
LLC
0, 300
2000, 3000
ULC
0, 0
2000,12000
URC
400, 0
10000,12000
LRC
400, 300
10000, 3000
Hint
As an example, if you want to show the world coordinates while a user moves the mouse over a map, this transform is what you need.
NoninvertibleTransformException¶
It can happen that a matrix is not invertible. This chapter is for the interested reader, if you dislike mathematics, you can skip it. The only import thing you should now is that for this kind of matrices the exception can never occur.
A matrix has a determinant. For creating the inverse matrix, divisions by the determinant are needed. As we know from school, it is not allowed to divide by zero. As a consequence, the determinant with value 0 prevents the creation of an inverse matrix.
For a
2x2
matrix:[ a b ] [ c d ]
the determinant is:
a*d  c*b
For a
3x3
matrix:[ a b c] [ d e f] [ g h i]
the determinant is:
a * ( e*ih*f )  d * (b*i h *c) + g * ( b*f e *c)
Fortunately, our matrices always have g = 0, h = 0 and i = 1.
Setting 0 for g results in:
a * ( e*ih*f )  d * (b*i h *c)
Setting i to 1 results in:
a * ( eh*f )  d * (b h *c)
Finally, we set h to 0:
a * e  d * b
This is in fact the same calculation as for the 2x2
matrix.
Let as construct such a matrix
AffineTransform noInvert = new AffineTransform(5,3,5,3,0,0); System.out.println("NoInvert : "+noInvert.toString()); System.out.println("Determinant : "+noInvert.getDeterminant()); try { noInvert.createInverse(); } catch (NoninvertibleTransformException e) { e.printStackTrace(); }
Output:
NoInvert : AffineTransform[[5.0, 5.0, 0.0], [3.0, 3.0, 0.0]] Determinant : 0.0 java.awt.geom.NoninvertibleTransformException: Determinant is 0.0 at java.awt.geom.AffineTransform.createInverse(AffineTransform.java:2666) at at.linux4all.affine.TestAffineTransform.test(TestAffineTransform.java:164) at at.linux4all.affine.TestAffineTransform.main(TestAffineTransform.java:84)
Remember, our matrix for world to pixel transformation was:
[ 0.05 0.00 100.00 ] [ 0.00 0.03 400.00 ] [ 0.00 0.00 1.00 ]
The determinant is:
0.05 * (0.03..)  0 * 0
which is not equal 0 and we can create the inverse matrix.
Conclusion¶
I hope this tutorial helps to demystify affine transforms, once you are used to working with them you will never return to doing coordinate calculations “by hand”.
Take a look at the Java API of the java.awt.geom.AffineTransform
class to see further
possibilities (rotate, shear,…).