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

LLC

lower left corner

ULC

upper left corner

URC

upper right corner

LRC

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

LLC

2000, 3000

ULC

2000, 12000

URC

10000, 12000

LRC

10000, 3000
../_images/world.png

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

LLC

2000, 3000

0,300

ULC

2000,12000

0, 0

URC

10000,12000

400, 0

LRC

10000, 3000

400,300
../_images/screen.png

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.

  1. Translate

  2. Scale

  3. Mirror

These are discussed below.

Translate Operation

  1. 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 from x and 3000 from y. We use the URC with values 10000,12000 to demonstrate the calculation.

  2. 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));

  3. Output:

    Translate:AffineTransform[[1.0, 0.0, -2000.0], [0.0, 1.0, -3000.0]]
Point2D.Double[0.0, 0.0]

  4. The toString() method of the AffineTransform class only shows the first two rows of the matrix. The static method getTranslateInstance is a convenience method, otherwise you have to call a constructor with 6 values.

  5. The matrix used:

    [  1.00  0.00 -2000.00 ]
[  0.00  1.00 -3000.00 ]
[  0.00  0.00  1.00 ]

  6. The detailed calculation:

    1 * 2000 + 0 * 3000 + 1 * -2000 = 0
0 * 2000 + 1 * 3000 + 1 * -3000 = 0
0 * 2000 + 0 * 3000 + 1 * 1     = 1

  7. 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.

  1. Let us use the point in the middle of the world rectangle after the translate operation, having its LLC at 0,0.

  2. 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));

  3. Output:

    Scale:AffineTransform[[0.05, 0.0, 0.0], [0.0, 0.033333333333333, 0.0]]
Point2D.Double[200.0, 150.0]

  4. 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

  5. The used matrix is:

    [  0.05  0.00    0.00 ]
[  0.00  0.03..  0.00 ]
[  0.00  0.00    1.00 ]

  6. 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

  1. Let us create the appropriate affine transform.

  2. 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));

  3. Output:

    Mirror:AffineTransform[[1.0, 0.0, 0.0], [0.0, -1.0, 300.0]]
Point2D.Double[100.0, 250.0]

  4. The x value is unchanged, but the y value is mirrored.

  5. The matrix used is:

    [  1.00  0.00   0.00 ]
[  0.00 -1.00 300.00 ]
[  0.00  0.00   1.00 ]

  6. The detailed calculation:

    1 * 100 +  0 * 50  + 1 *   0 = 100
0 * 100 + -1 * 50 +  1 * 300 = 250

  7. 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

Magic Part 1

The ability to concatenate transforms into a single matrix is vital to the performance of computer graphics and GIS.

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.

  1. 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));

  2. 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]

  3. The combined matrix is:

    [  0.05  0.00 -100.00 ]
[  0.00 -0.03  400.00 ]
[  0.00  0.00    1.00 ]

  1. 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

  2. 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

Magic Part II

The ability to invert a matrix and go the other way allows you to determine what a user clicked on.

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:

  1. 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));

  2. Output:

    Pixel2World:AffineTransform[[20.0, 0.0, 2000.0], [0.0, -30.0, 12000.0]]
World : Point2D.Double[6000.0, 7500.0]

  3. The inverse matrix is:

    [ 20.00  0.00 2000.00 ]
[  0.00 -30.00 12000.00 ]
[  0.00  0.00  1.00 ]

  4. 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

  5. The point 6000,7500 is indeed the center of our world rectangle.

  6. 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*i-h*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*i-h*f ) - d * (b*i -h *c)

  • Setting i to 1 results in:

    a * ( e-h*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.

  1. 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();
}

  2. 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)

  3. 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 ]

  4. The determinant is:

    0.05 *  (-0.03..) - 0 * 0

  5. 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,…).

References