DirectX Transformations

Understanding and implementing transformations is fundamental to 3D graphics programming with DirectX. Transformations allow you to manipulate objects in 3D space, affecting their position, orientation, and scale. These operations are typically performed using matrices.

Types of Transformations

The three basic types of geometric transformations are:

Translation: Moving an object from one position to another.
Rotation: Turning an object around an axis.
Scaling: Resizing an object uniformly or non-uniformly.

Matrix Representation

In DirectX, transformations are commonly represented by 4x4 matrices. This allows us to combine multiple transformations into a single matrix using matrix multiplication. We use homogeneous coordinates, where a 3D point (x, y, z) is represented as a 4D vector (x, y, z, 1).

Translation Matrix

A translation matrix shifts an object by a vector (tx, ty, tz):

    [ 1  0  0  tx ]
    [ 0  1  0  ty ]
    [ 0  0  1  tz ]
    [ 0  0  0  1  ]

Rotation Matrix

Rotation matrices are more complex and depend on the axis of rotation (X, Y, or Z) and the angle of rotation. For example, a rotation around the Z-axis by an angle θ:

    [ cos(θ)  -sin(θ)  0  0 ]
    [ sin(θ)   cos(θ)  0  0 ]
    [ 0           0          1  0 ]
    [ 0           0          0  1 ]

Scaling Matrix

A scaling matrix resizes an object by factors (sx, sy, sz) along each axis:

    [ sx  0  0  0 ]
    [ 0  sy  0  0 ]
    [ 0  0  sz  0 ]
    [ 0  0  0  1 ]

Combining Transformations

To combine multiple transformations (e.g., rotate then translate), you multiply their corresponding matrices. The order of multiplication is crucial, as matrix multiplication is not commutative. The typical order for applying transformations to a vertex is:

Model Transformation (local object transformations: scale, rotate, translate)
View Transformation (camera position and orientation)
Projection Transformation (perspective or orthographic view)

The combined transformation matrix is often referred to as the World Matrix (Model * View * Projection).

DirectXMath Library

DirectX provides the DirectXMath library, a set of highly optimized math functions and types for graphics programming. It includes convenient functions for creating and manipulating transformation matrices, such as:

XMMatrixTranslation()
XMMatrixRotationX(), XMMatrixRotationY(), XMMatrixRotationZ()
XMMatrixScaling()
XMMatrixMultiply()
XMMatrixLookAtLH() (for view matrix)
XMMatrixPerspectiveFovLH() (for projection matrix)

Using DirectXMath significantly simplifies the implementation and improves performance.

World, View, and Projection Matrices

These three matrices are central to transforming geometry from object space to screen space:

World Matrix: Transforms objects from their local model space into the world space.
View Matrix: Transforms the world space to camera space, effectively positioning and orienting the viewer.
Projection Matrix: Transforms camera space into clip space, defining the viewing frustum (the visible volume of space) and the projection type (perspective or orthographic).

Important Note:

The order of operations when building the combined matrix is vital. For typical camera transformations:

FinalMatrix = ModelMatrix * ViewMatrix * ProjectionMatrix;

When applying this to a vertex in model space: TransformedVertex = Vertex * FinalMatrix;

Performance Tip:

Whenever possible, combine multiple transformations into a single matrix before sending it to the GPU. This reduces the number of GPU state changes and improves rendering performance.