At the heart of all computer graphics lies linear algebra, and specifically matrix multiplication. One can use matrices to perform all sorts of operations, such as transformations to move from one coordinate system to another, as well as a set known as affine transformations. Affine transformations are those that map lines to lines in the transformed coordinate space, and which preserve the relative distance between points. An affine transformation consists of one or more translation, rotation, scaling, and shearing transformations.
See the following external sites for translation and rotation examples, shearing, scaling.
Java has a class to represent these affine transformations, as well as shorthand methods to apply them to a Graphics2D context.
Rotate about origin
Rotate about a point
Scale x and y axis by given amount
Shear
Translation
If you do any work involving Graphics2D in Java (and if you work with Swing components, you implicitly do), knowing how to use affine transforms is extremely beneficial. With them you can express and code things more succintly, and clearly than is possible without them.
We’ll start with a simple example, once with standard Swing painting code, and once using affine transformations. Finally we will end with a more fully fleshed out example that really illustrates the power of affine transformations, rendering a simplified overhead view of the solar system. This example would be extremely difficult to replicate without affine transformations.
For the simple example, let’s draw dots in a circle pattern. The easiest way to start drawing to the screen is simply to subclass the JComponent class and override the paintComponent(Graphics g) method. Here we go:
/**
* Draw a series of dots in a circular pattern
* @param g
*/
@Override
public void paintComponent(Graphics g) {
// Don't forget to call the super method
super.paintComponent(g);
int radius = getWidth() / 2;
for (int i = 0; i < NUM_DOTS; i++) {
double theta = 2 * Math.PI * ((double) i / NUM_DOTS);
int x = (int) (radius * Math.cos(theta));
int y = (int) (radius * Math.sin(theta));
// these x and y are relative to center of circle; currently origin
// is at upper left corner of window. Add to x and y to
// translate.
x += getWidth() / 2;
y += getHeight() / 2;
g.drawOval(x, y, 1, 1);
}
}
Here’s a picture of the result. 
(Full source)
Now, here’s that same code using the implicit affine transformation of the rotate() method of Graphics2D.
/**
* Draw a series of dots in a circular pattern
* @param g
*/
@Override
public void paintComponent(Graphics g) {
Graphics2D g2 = (Graphics2D) g;
// Don't forget to call the super method
super.paintComponent(g);
int radius = getWidth()/2;
// Translate the origin to the center of window
g2.translate(getWidth() /2, getHeight() /2);
for (int i = 0; i < NUM_DOTS; i++) {
g2.rotate(RADIANS_PER_DOT);
// We have rotated about the origin; draw a ray out along x axis
// of new coordinate system
g2.drawOval(radius, 0, 1, 1);
}
}
(Full Source)
As you can see from the screenshots, they come out functionally the same. In this case there’s not a huge advantage to using the rotation over the standard method. But what if we weren’t drawing dots along the radius of the circle, but instead were drawing rectangles that laid tangent to the circle? Here’s how simple that is to do using the rotations..
// Define the number of pixels wide each box is
private static final int BOX_SIZE = 5;
// Replace the call to drawOval with fillRect
g2.fillRect(radius, 0, BOX_SIZE, BOX_SIZE);
Here is the result 
(Full source)
Think how complicated this would be to accomplish if you were not using affine transforms; you would need to manually calculate the coordinates of each corner of each box, create a polygon from those points, and then call fillShape on the polygon.
The other place where affine transformations shine is when you need to place objects relative to each other. For instance, you might draw a table with a bowl of fruit on it; if your table moves, you would like the bowl to move as well. I will show you how you can render a simplified version of the solar system where the earth revolves around the sun, while at the same time the moon orbits the earth. As you can imagine, implementing this without affine transformations would be absolutely infeasible.
First we separate our model from our view as per the model view controller pattern; the state of the solar system is kept in the model which the view uses to render itself. Since the state of the model will be observed by the view, we make it a subclass of the Java Observable class.
package solarsystem;
import java.util.Observable;
public class SolarSystemModel extends Observable {
public static final int DAYS_PER_EARTH_REVOLUTION_AROUND_SUN = 365;
public static final int HOURS_PER_EARTH_REVOLUTION_AROUND_AXIS = 24;
// http://en.wikipedia.org/wiki/Orbit_of_the_Moon
// "The orbit of the Moon around the Earth is completed in approximately 27.3 days"
public static final float DAYS_PER_MOON_ORBIT_AROUND_EARTH = 27.3f;
private int day;
private int hour;
public int getDay() {
return day;
}
public void setDay(int day) {
int oldDay = this.day;
this.day = clampDay(day);
if (oldDay != this.day) {
setChanged();
notifyObservers();
}
}
public int getHour() {
return hour;
}
public void setHour(int hour) {
int oldHour = this.hour;
this.hour = clampHour(hour);
if (oldHour != this.hour) {
setChanged();
notifyObservers();
}
}
private int clampDay(int day) {
return day % DAYS_PER_EARTH_REVOLUTION_AROUND_SUN;
}
private int clampHour(int hour) {
return hour % HOURS_PER_EARTH_REVOLUTION_AROUND_AXIS;
}
}
(Note that we need to call setChanged() before notifyObservers() or our Observers registered with the model will not be updated.)
Now that we have our model defined, we need to make a view to actually render the solar system. Just as in our previous examples, I make the view extend JComponent for ease of display in a JFrame.
public class SolarSystemView extends JComponent implements Observer
The Observer interface allows classes to be notified when an Observable object changes; since we want to keep our view in sync with the model, this is just what we will do.
Here is the meat of the class:
@Override
public void paintComponent(Graphics g) {
Graphics2D g2 = (Graphics2D) g;
drawSpaceBackdrop(g2);
// Set the origin to be in the center of the screen
g2.translate(getWidth()/2, getHeight()/2);
// Order matters, since the earth placement is dependent upon the sun
// placement, and the moon placement is dependent upon the earth placement
drawSun(g2);
drawEarth(g2);
drawMoon(g2);
}
The graphics context is passed into each drawing method, which may or may not modify the context. The drawSpaceBackdrop method merely draws a few random stars on a black background; see the following screenshot:
The code for that is fairly straightforward:
/**
* Draws a black backdrop with star field
* @param g2
*/
private void drawSpaceBackdrop(Graphics2D g2) {
// Draw background as black
g2.setColor(Color.BLACK);
g2.fillRect(0, 0, getWidth(), getHeight());
g2.setColor(Color.WHITE);
for (int i = 0; i < NUM_STARS; i++) {
g2.fillOval(starX[i], starY[i], starRadius[i], starRadius[i]);
}
}
starX, starY, starRadius are parallel int arrays that are initialized earlier in the program by a random int generator.
/**
* Creates and populates our arrays of star x values, star y values, and
* star radii
* @param width what is max x value we should consider for star
* @param height what is max y value we should consider for star
*/
private void createStarField(int width, int height, int maxRadius) {
// Create the arrays
starX = new int[NUM_STARS];
starY = new int[NUM_STARS];
starRadius = new int[NUM_STARS];
// Fill them in with random values
for (int i = 0; i < NUM_STARS; i++) {
starX[i] = random.nextInt(width);
starY[i] = random.nextInt(height);
starRadius[i] = random.nextInt(maxRadius);
}
}
After initializing the arrays and drawing the stars, we then draw the sun. Note that we translate the origin from the upper left corner to the center of the screen; this allows each of the drawing methods to consider its own local coordinate system and not have to remember to translate from upper left corner of screen. For instance, the center of the sun is at (0,0) in its coordinate system.
/**
*
* @param g2 graphics context with (0,0) in center of screen (where sun will
* be centered)
*/
private void drawSun(Graphics2D g2) {
int sunRadius = (int) (SUN_RADIUS_PROPORTION * getWidth());
GradientPaint sunColor = new GradientPaint(0, 0, Color.YELLOW, 0, sunRadius, Color.RED);
g2.setPaint(sunColor);
g2.fillOval(-sunRadius/2, -sunRadius/2, sunRadius, sunRadius);
}
We apply a gradient just to make it look slightly nicer than a monochrome sun.
After having drawn the sun, it’s time to draw the earth.
/**
* Draws the earth to the screen, whose position is dependent upon the
* day of the year
* @param g2 the graphics context with its origin in the center of the sun
*/
private void drawEarth(Graphics2D g2) {
// Draw the earth
// Calculate what portion along its orbit the earth is, and thus how
// far to rotate about our centerpoint
double earthTheta = map(model.getDay(), 0, SolarSystemModel.DAYS_PER_EARTH_REVOLUTION_AROUND_SUN, 0, TWO_PI);
// Rotate our coordinate system by that much
g2.rotate(earthTheta);
// Translate the earth
int distanceFromEarthToSun = (int) (EARTH_DISTANCE_PROPORTION_SCREEN * getWidth());
g2.translate(distanceFromEarthToSun, 0);
int earthRadius = (int) (EARTH_RADIUS_PROPORTION * getWidth());
GradientPaint earthColor = new GradientPaint(0, 0, Color.BLUE, 0, earthRadius, Color.GREEN.darker(), true);
g2.setPaint(earthColor);
g2.fillOval(-earthRadius/2, -earthRadius/2, earthRadius, earthRadius);
}
If you’ve read my earlier blog post on the map function, you know that it maps a value from one range of numbers to another. We must calculate the number of radians to rotate so that we can position our earth correctly along its orbit.
Note that we first rotate and then translate; if we did it in the opposite direction we would see the earth spin about its axis but it would not revolve around the earth.
The drawMoon method is much the same; the main difference is that we calculate its position along its orbit based on its much smaller time to orbit the earth.
/**
* Draw the moon to the screen, whose position is dependent upon that of
* the earth and the day of the year, which dictates its position along
* its orbit around earth
* @param g2 the graphics context with its origin in the center of the earth
*/
private void drawMoon(Graphics2D g2) {
double moonTheta = map(model.getDay(), 0, SolarSystemModel.DAYS_PER_MOON_ORBIT_AROUND_EARTH, 0, TWO_PI);
int moonRadius = (int) (MOON_RADIUS_PROPORTION * getWidth());
g2.setColor(Color.WHITE);
g2.rotate(moonTheta);
int distanceFromEarthToMoon = (int) (MOON_DISTANCE_PROPORTION_SCREEN * getWidth());
// Translate the earth
g2.translate(distanceFromEarthToMoon, 0);
g2.fillOval(-moonRadius/2, -moonRadius/2, moonRadius, moonRadius);
}
Finally all we have to do is create an instance of the model and view, hook them together, and display them in a JFrame.
public static void main(String[] args) {
JFrame frame = new JFrame("Solar System");
final SolarSystemModel model = new SolarSystemModel();
final SolarSystemView view = new SolarSystemView(model);
model.addObserver(view);
JPanel panel = new JPanel();
panel.add(view);
frame.add(panel);
frame.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
frame.pack();
frame.setVisible(true);
}
If we run it as is, we see the planets aligned, since the model starts off at day zero. It’s a lot more fun to be able to interact with the model. To do that, we add a JSlider that modifies the model. 
final JSlider daySlider = new JSlider(0,SolarSystemModel.DAYS_PER_EARTH_REVOLUTION_AROUND_SUN);
daySlider.setPaintLabels(true);
daySlider.setPaintTicks(true);
daySlider.setMajorTickSpacing(100);
panel.add(daySlider);
daySlider.addChangeListener(new ChangeListener() {
public void stateChanged(ChangeEvent e) {
model.setDay(daySlider.getValue());
}
});
With that addition, we can move the slider and watch the planets move.
That’s it for this time. Can you figure out how to use the hour field of the model with another slider to make the earth rotate about its axis as it revolves around the sun?
Full model source
Full view source
