• 一 Translation

The following transformation translates the point (x, y, z) to a new point (x', y', z').

You can manually create a translation matrix in managed code. The following C# code example shows the source code for a function that creates a matrix to translate vertices.

						[C#]
						

private Matrix TranslateMatrix(float dx, float dy, float dz)
{
   Matrix ret;

   ret = Matrix.Identity;

   ret.M41 = dx;
   ret.M42 = dy;
   ret.M43 = dz;

   return ret;
}

For convenience, managed the Microsoft Direct3D supplies the Translation method.

• 二 Scaling

The following transformation scales the point (x, y, z) by arbitrary values in the x-, y-, and z-directions to a new point (x', y', z').

?

The transformations described here are for left-handed coordinate systems, and so might be different from transformation matrices that you have seen elsewhere. For more information, see 3-D Coordinate Systems.

The following transformation rotates the point (x, y, z) around the x-axis, producing a new point (x', y', z').

The following transformation rotates the point around the y-axis.

The following transformation rotates the point around the z-axis.

In these example matrices, the Greek letter theta (?) stands for the angle of rotation, in radians. Angles are measured clockwise when looking along the rotation axis toward the origin.

In a managed application, use the Matrix.RotationX, Matrix.RotationY, and Matrix.RotationZ methods to create rotation matrices. The following C# code example demonstrates how the Matrix.RotationX method performs a rotation.

				[C#]
				

private Matrix MatrixRotationX(float angle)
{
   double sin, cos;
   sin = Math.Sin(angle);
   cos = Math.Cos(angle);

   Matrix ret;
   ret.M11 = 1.0f; ret.M12 = 0.0f; ret.M13 = 0.0f; ret.M14 = 0.0f;
   ret.M21 = 0.0f; ret.M22 = (float)cos; ret.M23 = (float)sin; ret.M24 = 0.0f;
   ret.M31 = 0.0f; ret.M32 = (float)-sin; ret.M33 = (float)cos; ret.M34 = 0.0f;
   ret.M41 = 0.0f; ret.M42 = 0.0f; ret.M43 = 0.0f; ret.M44 = 1.0f;
   return ret;
}

One advantage of using matrices is that you can combine the effects of two or more matrices by multiplying them. This means that, to rotate a model and then translate it to some location, you do not need to apply two matrices. Instead, you multiply the rotation and translation matrices to produce a composite matrix that contains all of their effects. This process, called matrix concatenation, can be written with the following formula.

In this formula, C is the composite matrix being created, and M₁ through M_n are the individual transformations that matrix C contains. In most cases, only two or three matrices are concatenated, but there is no limit.

Use the Matrix.Multiply method to perform matrix multiplication.

The order in which the matrix multiplication is performed is crucial. The preceding formula reflects the left-to-right rule of matrix concatenation. That is, the visible effects of the matrices that you use to create a composite matrix occur in left-to-right order. A typical world transformation matrix is shown in the following example. Imagine that you are creating the world transformation matrix for a stereotypical flying saucer. You would probably want to spin the flying saucer around its center - the y-axis of model space - and translate it to some other location in your scene. To accomplish this effect, you first create a rotation matrix, and then multiply it by a translation matrix, as shown in the following formula.

In this formula, R_y is a matrix for rotation about the y-axis, and T_w is a translation to some position in world coordinates.

The order in which you multiply the matrices is important because, unlike multiplying two scalar values, matrix multiplication is not commutative. Multiplying the matrices in the opposite order has the visual effect of translating the flying saucer to its world space position, and then rotating it around the world origin.

No matter what type of matrix you are creating, remember the left-to-right rule to ensure that you achieve the expected effects.

到底什么時候在左邊什么時候在右邊?


總結(jié):

In applications that work with 3-D graphics, geometrical transformations can be used to do the following.

• Express the location of an object relative to another object.
• Rotate and size objects.
• Change viewing positions, directions, and perspectives.

You can transform any point (x,y,z) into another point (x', y', z') using a 4 x 4 matrix.

Perform the following operations on (x, y, z) and the matrix to produce the point (x', y', z').

The most common transformations are translation, rotation, and scaling. You can combine the matrices that produce these effects into a single matrix to calculate several transformations at once.