Windows Phone 7 : Understanding Matrix Transformations (part 2) - Applying Multiple Transformations

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4. Applying Scaling Transformations

The last of the transformations that we will be working with for the time being is for scaling the objects that we render. Scaling matrices can be either uniform, in which case the object scales by the same amount on all three axes; or non-uniform, in which case each axis scales by a different amount.

Figure 5 shows an object in its identity location on the left, with a uniform scale of 2.0 in the middle, and then on the right with a scale of 4.0 on the x axis and 0.5 on the y axis.

Figure 5. Scaling transformations

To obtain a scaling matrix, call the static Matrix.CreateScale function. It has three calling methods: it can be passed a single float to perform a uniform scale, it can be passed three floats to perform a non-uniform scale with the values provided for the x, y, and z axes; or it can be passed a Vector3 containing the values for non-uniform scaling.

Passing a scale value of 0 for any of the axes will squash the shape on that axis so that it is completely flat. It can be easy to accidentally pass this when you had intended to leave the scaling unchanged for an axis; for any axis that you want to leave unchanged when scaling, pass a value of 1.

Negative scale values are also permitted. These will cause the object to flip over so that the vertices appear on the opposite side of the negatively scaled axis.

5. Applying Multiple Transformations

To apply a single transformation to our objects, we can simply obtain the required matrix and set it into the effect's World matrix property. We have already seen several examples of this.

Any practical application of matrix transformations will quickly find that setting the matrix for just a single transformation is insufficient, however. If we need to perform multiple transformations at once (for example, perhaps we need to move the object to another point within the world and then rotate it), we need some way to combine these transformations together.

Fortunately, matrix transformations are perfectly suited to this task. We can combine two or more transformations by simply multiplying the matrices together. The resulting matrix will contain the effects of both of the input matrices.

The different types of translation can have an effect on each other that might not at first be obvious, so let's first look at the effects of applying multiple transformations. We will then come back to look at how they are implemented in code.

5.1. Rotating Objects

When we rotate an object, we actually rotate its entire coordinate system because we are transforming the entire world, not just the object itself. If the identity matrix is loaded, the world coordinates are reset so that the origin point (0, 0, 0) is in the center, and no scaling or rotation is applied. Once we begin to transform this matrix, the coordinate system moves around accordingly.

Objects always move relative to the transformed coordinate system, not to the identity coordinate system. This means that, if we rotate an object by 45 degrees around the z axis and then translate it along the y axis, it will actually move diagonally onscreen rather than vertically. The rotation has changed the direction of the axes within the world coordinate system.

The effects of this rotation can be seen in Figure 6. On the left is the usual unit square at the identity position. In the middle, we rotate it by 45 degrees around the z axis (counterclockwise). The pale lines show the x and y axes in the identity coordinate system, whereas the darker diagonal lines show the x and y axes for the transformed world coordinate system. On the right, we translate it along its y axis. Observe that it has moved diagonally relative to the identity coordinates, though it has followed the transformed world y axis. Also note that the world coordinate system follows the translation, too. The coordinate (0, 0, 0) moves along with the translations that are applied to the world matrix.

Figure 6. Rotation and translation in the world coordinate system

This sequence of updates brings us to another important feature of matrix transformations: the order in which they are applied is significant. In Figure 5, we first rotated and then translated our object. If we instead translate and then rotate it, the coordinate system for the translation would still be aligned with theidentity coordinate system, so the movement on the screen would be vertical. This is shown in Figure 7, which contains exactly the same transformations but performed with the translation before the rotation.

Figure 7. Translation before rotation

As you can see, the object ends up in a different place if we translate it first.

This is actually very easy to visualize. Imagine that you are standing in place of the square object in these diagrams. You are initially standing at the origin of the identity coordinate system, looking along the positive y axis (up the screen).

You then decide to rotate 45 degrees counterclockwise, just as in Figure 5. You are still facing straight ahead of your body, but relative to the identity coordinates, you are now looking diagonally. If you now take a few paces forward, you are walking diagonally in terms of the identity coordinates but straight ahead in terms of your own position within the world.

If you hold your arms out to the sides, they will be pointing along the x axis relative to your position but are once again at a diagonal angle relative to the identity coordinates.

Any time you want to visualize the transformations that you are applying, think of this same scenario and apply each transformation in sequence to yourself in the world. It should then be easy to see the sequence of transformations that you need to apply to get from one place to another. (It gets slightly harder to visualize in three dimensions, but just imagine you have wings or a jet-pack.)

Hopefully, this makes the effects of cumulative transformations clear. Always remember that, when you transform an object, the transformation will be relative to the existing transformed coordinates, not to those of the identity coordinate system.

5.2. Scaling Objects

When we scale an object, the transformation once again has an effect on the world coordinate system. If we scale an object so that its size doubles, a movement of one unit in the x axis in the transformed world coordinate system will correspond to a movement of two units relative to the identity coordinate system.

If you simply want to draw an object at a different size but without affecting its position, remember to perform the scale transformation after all the translations have been completed to avoid affecting the movement distances.

5.3. Applying Multiple Transformations in XNA

So we know that multiple transformations can be combined by multiplying them together, so let's see some sample code to achieve this in XNA.

The first transformation that we want to use can be obtained directly by calling the appropriate static Matrix function. From that point on, subsequent transformations must be obtained and multiplied by the existing calculated matrix.

To translate an object two units along the y axis and then rotate it by a specified angle, we would use the code shown in Listing 2. It causes the object to rotate on the spot a short distance away from the center of the screen.

Example 2. Multiple transformations: translation and then rotation
// First translate...
_effect.World = Matrix.CreateTranslation(0, 2, 0);
//...then rotate
_effect.World = Matrix.CreateRotationZ(_angle) * _effect.World;

Notice the order of multiplication: the new transformation is on the left of the multiplication symbol, and the existing matrix is on the right. Unlike multiplication of simple numbers, matrix multiplication is not commutative, which is why the order of transformations is significant. If we multiply matrix A by matrix B, we will get results different from multiplying matrix B by matrix A.

We can swap the order of these transformations so that we first rotate and then translate along the (rotated) y axis, as shown in Listing 3.

Example 3. Multiple transformations: rotation and then translation
// First rotate...
_effect.World = Matrix.CreateRotationZ(_angle);
//...then rotate
_effect.World = Matrix.CreateTranslation(0, 2, 0) * _effect.World;

Even though we are generating the same matrices with the same parameter, the resulting behavior is different. Instead of spinning on the spot, the object now rotates around a circular path, centered at the identity origin and with a radius of two units (because this is the distance that the object was translated).

Try plugging each of them into the ColoredSquare project in place of the existing matrix code to see their effects.

We are not limited to using transformation types just once within a transformation sequence, of course, and some movement paths will require the same transformation to be applied repeatedly at different stages of the calculation.

For example, let's get the object to trace a circle as in Listing 3, but this time the circle will be away from the identity origin, and the object itself will remain "upright" without rotating at all. This is achieved using the transformations shown in Listing 4.

Example 4. Repeatedly transforming to achieve a more complex movement path
// First translate to the center of the circular path
_effect.World = Matrix.CreateTranslation(0, 3, 0);
// Rotate the object towards the current position on the circular path
_effect.World = Matrix.CreateRotationZ(_angle) * _effect.World;
// Translate to the edge of the circle
_effect.World = Matrix.CreateTranslation(0, 2, 0) * _effect.World;
//...then rotate back to an upright position
_effect.World = Matrix.CreateRotationZ(-_angle) * _effect.World;

This time the circular path is centered at (0, 3, 0) because we translate to here before rotating. Then the object is rotated toward the point on the circle at which it will be rendered. The object is then translated to the edge of the circle; as the translation is two units along the (rotated) y axis, this will be the radius of the circle. Finally, to keep the object upright, it is rotated back by its angle. This cancels out the rotation that was applied in the original rotation. The original rotation therefore results in having an effect on the position of the object but not its final angle.

As you can see, this entire series of events is eventually contained in the single _effect.World matrix. There is no limit to the number of calculations that can be accumulated into a single matrix in this way.

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