Matrix Compendium

Matrix Multiplication Order and Transformations in Three-Dimensional Euclidean Space

In computer graphics applications, transformations such as translation, rotation, scaling, and shearing are combined to create realistic virtual worlds. These transformations are represented using 4 \times 4 matrices, and combining them is equivalent to multiplying matrices.
It is crucial to choose and apply the correct order consistently to achieve the desired result.

While matrix multiplication is non-commutative, it remains associative:

(\mathbf{A} * \mathbf{B}) * \mathbf{C} = \mathbf{A} * (\mathbf{B} * \mathbf{C})

In the examples below, parentheses are used solely to indicate the order in which transformations are combined.

Let’s describe the problem of matrix multiplication order for transformation concatenation using the following example:

\begin{array}{c} \mathbf{Z} = \mathbf{A} * \mathbf{B} * \mathbf{C} * \mathbf{D} \end{array}

where A, B, C, D, Z are matrices.

Parentheses are helpful to clearly indicate the sequence (order) of operations. In mathematics, multiplication is typically assumed to proceed from left to right.

The differences between the two conventions are summarized in the table below:

Where:
\vec{V_w} = Transformed vector in another space.
\vec{V_c} = Transformed vector in original space.
\mathbf{M} = Transform matrix.
\mathbf{M_f} = Final transformation matrix.
\color{yellow}\mathbf{M_m}\color{default} = Model transformation matrix (transforms from local model space to global world space).
\color{cyan}\mathbf{M_v}\color{default} = View matrix (transforms from world space to camera space).
\color{magenta}\mathbf{M_p}\color{default} = Projection matrix (transforms from camera space to clipping space).
\mathbf{M_w} = World transformation matrix.
\color{SpringGreen}\mathbf{M_l}\color{default} = Current local transformation matrix.
\color{Violet}\mathbf{M_a}\color{default} = Transformation matrix of the parent node.
\color{RoyalBlue}\mathbf{M_s}\color{default} = Scaling matrix.
\color{green}\mathbf{M_r}\color{default} = Rotation matrix.
\color{red}\mathbf{M_t}\color{default} = Translation matrix.
\color{grey}\mathbf{M_h}\color{default} = Shear matrix.

Three-Dimensional Cartesian Coordinate System: “Up Direction”

Mathematically, no axis is inherently designated as the Up Direction in three-dimensional Euclidean space.
However, in 3D applications, two conventions have emerged over the years:

• Z-Axis as “Up Direction”

  This convention is particularly useful in fields such as architecture. When the working space is a plane representing, for example, the Earth’s surface, the two main working directions are the X-Axis and Y-Axis, while the Z-Axis represents elevation, or the “Up Direction.”

  

• Y-Axis as “Up Direction”

  In this convention, the working plane corresponds to the monitor screen. The X-Axis and Y-Axis represent the width and height of the screen, respectively, while the Z-Axis represents depth. This setup aligns with our everyday perception of the world, such as in first-person shooter (FPS) games, where the Y-Axis represents the “Up Direction.”

  

Cross Product in Three-Dimensional Euclidean Space

In computer graphics, particularly in three-dimensional Euclidean space, the cross product is a highly useful and common operation. The cross product of two non-collinear vectors results in another vector that is perpendicular to both original vectors.
This operation introduces ambiguity because the direction of the resulting vector does not have a unique solution—it could point in two equally valid directions. To resolve this ambiguity, we select an orientation (or handedness) for the three-dimensional Euclidean space. By making this choice, we designate one cross-product direction as the preferred solution.

To get a unique and consistent result, we use either the right-hand rule or the left-hand rule.
It is worth mentioning that in mathematics and physics, the right-hand orientation is the most commonly used. This orientation is applied to determine directions in magnetic fields, electromagnetic fields, and enantiomers in chemistry.

The right-hand rule and left-hand rule can be described as follows:
Let the straightened index finger of the chosen hand (right or left) point in the direction of the first vector (\vec{a}), and let the bent middle finger point in the direction of the second vector (\vec{b}). Finally, the extended thumb will indicate the direction of the cross-product vector \vec{a} \times \vec{b}.

To determine the directions of the X, Y, and Z axes in three-dimensional Euclidean space, replace the first vector with the direction of the X-Axis and the second vector with the direction of the Y-Axis. The thumb will then indicate the direction of the Z-Axis.

Using this method, we can designate a three-dimensional Euclidean space with the desired handedness (orientation). The orientations, right-hand or left-hand, can be summarized in the table below (if applicable).

Importance of Choosing the Handedness

Choosing the handedness is crucial as it determines not only the direction of the cross-product vector but also:

• The orientation of surface normal vectors.

• The direction in which points rotate around an axis.

• The order of vertices in triangle meshes.

Anticommutativity of Cross Product

Another key property of the cross product of two non-collinear vectors is anticommutativity:

\begin{array}{c} \vec{a} \times \vec{b} = - \vec{b} \times \vec{a} \end{array}

If you need to fix a rendering error by reversing the result of a cross product, keep in mind that you are not simply adding a *negative sign* to the vector. Instead, you are altering the handedness of your three-dimensional Euclidean space.

Positive Direction in Two-Dimensional Euclidean Space

The concept of Positive Direction refers to the direction of rotation established by starting from the first chosen basis vector (or axis of the coordinate system) and moving in the direction of the second basis vector.

Visually, this can be easily explained in two-dimensional Euclidean space, as shown in the table below:

This demonstrates why the order of basis vectors matters.

Positive Direction and Rotation Matrices

To describe a rotation in two- or three-dimensional Euclidean space, a rotation matrix is commonly used. This matrix is then applied to rotate a vector (or a point treated as a vector) by a specified angle.
The rotation is achieved by multiplying the vector by the matrix. Matrix multiplication is not commutative, as we’ve seen, so two conventions exist for rotation: pre-multiplication and post-multiplication.

The order of basis vectors plays a nuanced role in defining the rotation. For example, the same rotation matrix:

R = \begin{bmatrix} \cos(\phi) & -\sin(\phi)\\ \sin(\phi) & \cos(\phi)\\ \end{bmatrix}

depending on whether it is used in pre- or post-multiplication, determines the Positive Direction of rotation for different orders of basis vectors.
In the two-dimensional case, this relates to {\vec{e_0},\vec{e_1}} or {\vec{e_1},\vec{e_0}}.

Ensuring Coherence in Basis Vector Order and Rotation

Maintaining coherence in the concept of the order of basis vectors is critical when determining the position of a point in space and the unique Positive Direction of rotation.
To ensure coherence, the specific form of the rotation matrix must correspond to a particular order of basis vectors, along with the chosen matrix multiplication convention.

All of these options can be summarized in the table below:

Positive Direction in Three-Dimensional Euclidean Space

To determine the Positive Rotation, in three-Dimensional Euclidean Space we can use a hand rule similar to the one used for the cross-product.

The right-hand rule and left-hand rule in this context can be described as follows:

Bent fingers point in the direction of the Positive Direction, and your thumb indicates the axis of rotation
or
Point your thumb in the direction of the axis of rotation, and your bent fingers will indicate the Positive Direction.

Let us illustrate these rules using the following table:

In three-dimensional Euclidean space, we encounter several possible permutations of basis vector orders. Fortunately, these permutations can be grouped into two groups of basis vector orders:

• (X, Y, Z) = {\vec{e_0},\vec{e_1},\vec{e_2}}, {\vec{e_1},\vec{e_2},\vec{e_0}}, {\vec{e_2},\vec{e_0},\vec{e_1}}
• (X, Z, Y) = {\vec{e_0},\vec{e_2},\vec{e_1}}, {\vec{e_2},\vec{e_1},\vec{e_0}}, {\vec{e_1},\vec{e_0},\vec{e_2}}

In the table below, I aim to demonstrate why the topic of the order of basis vectors—and, consequently, the Positive Direction of rotation—can appear complex and challenging to comprehend.

The variety of possible use cases increases the likelihood of making mistakes. How can we summarize the table above? After analyzing each case, the hand used to determine the order of the base vectors (and thus the Positive Direction of rotation), as well as the direction of the vector product, varies. In some cases, different hands are required, while in others, the same hand is used. The table below provides a summary of these cases:

Maintaining coherence in basis vector order is essential for determining point positions in space and the Positive Direction of rotations. Only the basis vector order from the group (X, Y, Z) satisfies these criteria.

Finally, the Positive Direction in three-dimensional Euclidean space are illustrated in the table below:

Basis Vector Order and Rotation Systems

Consider this: If you create a model in a Y-Up left-handed coordinate system but render it in a Z-Up left-handed coordinate system, swapping the Y and Z coordinates may seem like a simple fix.
However, this adjustment implicitly alters elements like plane normal vectors, triangle mesh winding orders, and associated animation data due to changes in Positive Direction and basis vector order.

Positive Direction and Quaternions

The order of basis vectors also affects quaternions. When W. R. Hamilton invented quaternions to describe rotations in three-dimensional Euclidean space, he defined them as Q = a + bi + cj + dk and established dependencies between the imaginary components: ij = k, jk = i, ki = j, and ijk = -1. When quaternions are defined using these rules, the Positive Direction of rotation is determined by the right-hand rule. In computer graphics, however, there are applications that use the left-hand rule to determine the Positive Direction. This raises the question: how should quaternions defined in this manner be handled?
The convention proposed by M.D. Shuster does not change the fundamental definition of the quaternion itself but modifies its dependencies on the imaginary components. This approach differs slightly from the quaternion definitions typically found in mathematics textbooks. Notably, Shuster’s definition is employed by NASA’s Jet Propulsion Laboratory and, in computer graphics, within the Microsoft DirectX Math Library.

The differences between Hamilton’s and Shuster’s quaternion definitions are summarized in the following table:

Handedness in Computer Graphics Applications

To summarize Positive Direction concepts, here’s a table of applications (game engines or software tools for animations or models) that utilize different three-dimensional space orientations and specify the Up axis:

Special Case: Unreal Engine

Unreal Engine uses both hand rules depending on the axis of rotation. For example:

• X and Y-Axis: Right-hand rule

• Z-Axis: Left-hand rule

Representing Matrices in Computer Memory

How can we represent this matrix in computer memory?
A common solution is to use multidimensional arrays. These arrays can be mapped to computer memory in different ways, each offering distinct advantages and disadvantages.
The simplest and most widely used methods are the row-major and column-major conventions, where matrices are stored row by row or column by column, respectively.

We have two conventions for storing matrices in memory, and each programming language must decide which one to adopt for its multidimensional array representation.
The following table summarizes how some popular programming languages implement these conventions:

The majority of widely used languages favor row-major order, while those focused on numerical or scientific computations often prioritize column-major order.

The Storage of Matrices as Transformations

Data storage conventions are inevitably linked to transformations.
For example, in a translation matrix, the translation vector is stored in either the third row or the third column, depending on the multiplication order.

Two multiplication orders and two storage conventions yield four ways to store a translation matrix:

For performance reasons, it is better to read or write memory elements that are adjacent.
As a result, the translation vector should be stored using the convention that allows for this. This may explain why applications that store matrices in row-major order often prefer the pre-multiplication convention, while those storing matrices in column-major order tend to favor the post-multiplication convention.

But what can we do if, for some reason, we need to use the post-multiplication convention, and we don’t want to store matrices in column-major order?
There is a way—though somewhat hacky—but it is possible.
We can reinterpret the multidimensional array indices in a different order.
By doing this, in C/C++, we can pretend that the matrix is stored in column-major order rather than row-major order.

Both approaches to storing multidimensional arrays are summarized in the following table:

The Storage of Matrices in HLSL and GLSL

HLSL and GLSL shader languages are independent from the perspective of matrix multiplication and matrix storage. Similarly, hardware and Graphics APIs, such as DirectX® and Vulkan®, are also independent of these conventions and support various possibilities. However, their default behavior is defined by older API versions.

This independence is evident in how vectors and matrices interact. When a vector is multiplied by a matrix, \vec{v_p} = \vec{v} * \mathbf{m}, the vector \vec{v} is treated as a row vector. Conversely, when the matrix is multiplied by the vector, \vec{v_p} = \mathbf{m} * \vec{v}, the vector \vec{v} is treated as a column vector.