Geometry

Row Major vs Column Major Vector

Earlier in this lesson, we explained that vectors (or points) could be represented as [1x3] matrices (one row, three columns) or as [3x1] matrices (three rows, one column). Both forms are technically valid, and the choice between them is merely a matter of convention.

• Vector represented as a [1x3] matrix:

  $$V = \begin{bmatrix} x & y & z \end{bmatrix}$$

• Vector represented as a [3x1] matrix:

  $$V = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$$

In the [1x3] matrix representation, the vector or point is in row-major order, meaning it is written as a row of three numbers. Conversely, in the [3x1] matrix representation, points or vectors are in column-major order, with the three coordinates of the vector or point written vertically as a column.

We use matrix representation for points and vectors to facilitate multiplication by [3x3] transformation matrices. For simplicity, we focus on [3x3] matrices rather than [4x4] matrices. Matrix multiplication is possible only when the number of columns in the left matrix matches the number of rows in the right matrix. Hence, a [1x3] matrix can be multiplied by a [3x3] matrix, but a [3x1] matrix cannot directly multiply a [3x3] matrix. This principle is demonstrated below, where the inner dimensions that match allow for valid multiplication (highlighted in green), resulting in a transformed point in the form of a [1x3] matrix:

However, when the inner dimensions do not match (highlighted in red), multiplication is not feasible:

To resolve this, instead of trying to multiply the vector or point by the matrix, we multiply the matrix by the vector, placing the point or vector to the right in the multiplication:

This approach aligns with the conventions of matrix multiplication, ensuring that vectors or points can be correctly transformed by [3x3] matrices.

Note that the result of this operation is a transformed point written in the form of a [3x1] matrix. This means we start with a point and end with a transformed point, effectively solving our problem. To summarize, when we choose to represent vectors or points in row-major order ([1x3]), we position the point on the left side of the multiplication, with the [3x3] matrix on the right side within the multiplication sign. In mathematics, this arrangement is referred to as left or pre-multiplication. Conversely, if we opt to express vectors in column-major order ([3x1]), the [3x3] matrix is placed on the left side of the multiplication, with the vector or point on the right side, known as right or post-multiplication.

The differences between the two conventions can be summarized as follows, where P stands for Point, V for Vector, and M for Matrix.

Row-Major Order

• Representation: Points or vectors (P/V) are expressed as [1x3] matrices, such as \(P/V = \begin{bmatrix}x & y & z\end{bmatrix}\).

• Multiplication Convention: Left or pre-multiplication. The point or vector is placed on the left side of the multiplication operation.

• Operation: \(P/V * M\). The point or vector is multiplied by the matrix, indicating that the transformation matrix follows the vector in the operation.

Column-Major Order

• Representation: Points or vectors (P/V) are expressed as [3x1] matrices, such as

  $$P/V = \begin{bmatrix}x \\ y \\ z\end{bmatrix}$$

• Multiplication Convention: Right or post-multiplication. The matrix is placed on the left side of the multiplication operation.

• Operation: \(M * P/V\). The matrix multiplies the point or vector, indicating that the vector or point follows the transformation matrix in the operation.

Key observations include:

• In row-major order, vectors or points are typically represented in a single row (as [1x3] matrices), and multiplication by a transformation matrix (M) is done from the right (P/V * M), known as pre-multiplication.

• In column-major order, vectors or points are represented in a single column (as [3x1] matrices), and the transformation matrix multiplies from the left (M * P/V), referred to as post-multiplication.

These conventions determine how transformations are applied to points or vectors in various graphics and mathematics applications, ensuring the correct application of operations like translation, rotation, and scaling.

Difference Between Row-Major and Column-Major Conventions

Understanding the difference between row-major and column-major conventions is not just a matter of notation but also has practical implications in mathematics and computer graphics. Here's an exploration of how these conventions affect matrix multiplication and transformations, and why one might be preferred over the other in certain contexts.

Row-Major vs. Column-Major in Matrix Multiplication

When calculating the product of two matrices, A and B, the process involves multiplying each element of a row in matrix A by the corresponding element of a column in matrix B and summing the results. The approach varies between the row-major and column-major conventions:

• Row-Major Order: Multiplication is performed as follows:

  $$\begin{bmatrix}x & y & z\end{bmatrix} * \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}$$

  Resulting in:

  $${\begin{array}{l}x' = x * a + y * d + z * g\\y' = x * b + y * e + z * h\\z' = x * c + y * f + z * i\end{array}}$$

• Column-Major Order: The multiplication is:

  $$\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} * \begin{bmatrix}x\\y\\z\end{bmatrix}$$

  Producing:

  $${\begin{array}{l}x' = a * x + b * y + c * z\\y' = d * x + e * y + f * z\\z' = g * x + h * y + i * z\end{array} }$$

Despite the different approaches, both conventions should ideally yield the same result for a vector or point transformation. However, direct application shows differing outcomes due to the inherent mathematical structure of each convention. To align results, the matrix used in column-major multiplication requires transposition: To match outcomes, the column-major matrix must be transposed (for a refresher on matrix transposition, refer to the chapter on Matrix Operations):

Producing:

Ensuring that, regardless of the convention, the transformation results remain consistent.

Practical Application and Preference

The choice between row-major and column-major impacts the sequence of transformations. For instance:

• Row-Major Transformations: Sequential order is preserved, such as:

  $$P'=P * T * R_z * R_y$$

• Column-Major Transformations: Requires reverse order, which might seem counter-intuitive:

  $$P'=R_y * R_z * T * P$$

The distinction between row-major and column-major conventions prompts a discussion on preference. Both approaches are legitimate, with mathematics and physics typically leaning towards column vectors for various reasons, including historical context and the practical explanations provided below. Scratchapixel adopts the row-major order convention, primarily because it aligns more intuitively with the concept of "transforming this point by this matrix" rather than the inverse.

Educational and API Considerations

• Row-Major Convention: Favored for its intuitive sequence of transformations, making it easier to teach. Used in graphics applications and APIs like Maya and DirectX.

• Column-Major Convention: Used by OpenGL and other 3D APIs, aligning more closely with mathematical function evaluation and composition, suggesting a deeper reason for its preference in certain contexts.

In conclusion, while the row-major convention may offer simplicity and directness in teaching and certain applications, the column-major convention aligns closely with mathematical principles, making it the choice for many 3D graphics APIs. Understanding both conventions allows for flexibility in approaching problems in computer graphics and mathematics, ensuring the correct application of transformations regardless of the chosen convention.

Implication in Coding: Does it Impact Performance?

Choosing between row-major and column-major order in coding does not just concern the conventions of writing. It's also about the practicality concerning the computer's architecture and how it processes data. When dealing with [4x4] matrices in programming, particularly in C++, a matrix might typically be implemented as follows:

class Matrix44 { 
    float m[4][4]; 
    // Additional class details
};

In this setup, the 16 coefficients of a [4x4] matrix are stored in a contiguous two-dimensional array. This arrangement means they're laid out in memory in a row-major fashion: c00, c01, c02, c03, ..., c33.

Row-Major Order Access Pattern:

In row-major order vector-matrix multiplication, the elements are accessed non-sequentially:

// Row-major order
x' = x * c00 + y * c10 + z * c20;
y' = x * c01 + y * c11 + z * c21;
z' = x * c02 + y * c12 + z * c22;

This pattern means that for calculating x', elements are accessed with gaps, leading to potential cache misses due to non-sequential memory access, affecting CPU performance.

Column-Major Order Access Pattern:

Conversely, in column-major order, the elements are accessed sequentially:

// Column-major order
x' = c00 * x + c01 * y + c02 * z;
y' = c10 * x + c11 * y + c12 * z;
z' = c20 * x + c21 * y + c22 * z;

Sequential access aligns with the CPU's cache mechanism, reducing cache misses and potentially enhancing performance.

Theoretically, column-major order could offer better performance due to improved cache usage. However, in practice, especially with compiler optimizations (-O, -O2, -O3 flags), the difference might not be significant. Modern compilers are capable of optimizing memory access patterns in multi-dimensional arrays, mitigating the performance gap between row-major and column-major access patterns.

The provided C++ example showcases a matrix and vector class template, illustrating how vector-matrix multiplication can be implemented for both row-major and column-major conventions. The performance test, involving a large number of iterations, aims to measure the execution time difference between these conventions. Yet, it's noted that real-world performance differences might not be significant due to compiler optimizations.

In summary, while column-major order might theoretically offer performance advantages due to sequential memory access patterns, the actual impact on performance may vary depending on specific use cases and compiler optimizations.

template&lttypename T> 
class Vec3 
{ 
public: 
    Vec3(T xx, T yy, T zz) : x(xx), y(yy), z(zz) {} 
    T x, y, z, w; 
}; 
 
template&lttypename T> 
class Matrix44 
{ 
public: 
    T m[4][4]; 
    Vec3&ltT> multVecMatrix(const Vec3&ltT> &v) 
    { 
#ifdef ROWMAJOR 
        return Vec3&ltT>( 
            v.x * m[0][0] + v.y * m[1][0] + v.z * m[2][0], 
            v.x * m[0][1] + v.y * m[1][1] + v.z * m[2][1], 
            v.x * m[0][2] + v.y * m[1][2] + v.z * m[2][2]); 
#else 
        return Vec3&ltT>( 
            v.x * m[0][0] + v.y * m[0][1] + v.z * m[0][2], 
            v.x * m[1][0] + v.y * m[1][1] + v.z * m[1][2], 
            v.x * m[2][0] + v.y * m[2][1] + v.z * m[2][2]); 
#endif 
    } 
}; 
 
#include &ltcmath> 
#include &ltcstdlib> 
#include &ltcstdio> 
#include &ltctime> 
 
#define MAX_ITER 10e8 
 
int main(int argc, char **argv) 
{ 
    clock_t start = clock(); 
    Vec3&ltfloat> v(1, 2, 3); 
    Matrix44&ltfloat> M; 
    float *tmp = &M.m[0][0]; 
    for (int i = 0; i < 16; i++) *(tmp + i) = drand48(); 
    for (int i = 0; i < MAX_ITER; ++i) { 
        Vec3&ltfloat> vt = M.multVecMatrix(v); 
    } 
    fprintf(stderr, "Clock time %f\n", (clock() - start) / float(CLOCKS_PER_SEC)); 
    return 0; 
} 

In computing, the concepts of row-major and column-major order describe the memory layout for multidimensional arrays. This distinction is crucial for understanding data access patterns, which can affect performance, especially in operations like matrix multiplication.

Row-major Order

• Used by languages like C and C++, where elements are stored sequentially from left to right, top to bottom.

• For a matrix

  \(M = \begin{bmatrix}1&2&3\\4&5&6\end{bmatrix}\),

  it's represented in C/C++ as float m[2][3] = {{1, 2, 3}, {4, 5, 6}};, with elements laid out in memory as 1, 2, 3, 4, 5, 6.

Column-major Order

• Adopted by languages such as FORTRAN and MATLAB, this approach stores elements from top to bottom, left to right.

• Using the same matrix \(M\), elements would be stored in memory as 1, 4, 2, 5, 3, 6.

Understanding these layouts is essential for optimizing memory access. In C/C++, we typically deal with row-major order, but it's important to be aware of column-major order, especially when interfacing with software or libraries that use a different convention, such as OpenGL.

The question of whether real-time APIs like OpenGL use row-major or column-major matrix notation in their documentation, and how these matrices are organized in memory, has historically been a source of confusion. While modern APIs, apart from DirectX (leaving examples like Vulkan), predominantly utilize column-order matrices, they also frequently offer the flexibility within shaders to specify the order in which matrix coefficients should be arranged in memory. This level of adaptability essentially grants all the versatility one might need in matrix manipulation. Though as a consequence, even the most experienced programmers may find themselves experimenting with the order of vector and matrix multiplications several times until the visual output aligns with expectations. Sigh!

Summary

Differences Between Row-Major and Column-Major Vector Conventions in Mathematics:

• Vector Representation:

  • Row-Major: \(P/V=\begin{bmatrix}x & y & z\end{bmatrix}\)

  • Column-Major: \(P/V=\begin{bmatrix}x \\ y \\ z\end{bmatrix}\)

• Multiplication Type:

  • Row-Major: Pre-multiplication (vM).

  • Column-Major: Post-multiplication (Mv).

• Transformation Order:

  • Row-Major: Sequential order is maintained (e.g., \(P'=P*T*R_z*R_y\))

  • Column-Major: Reverse order of transformation (e.g., \(P'=R_y*R_z*T*P\))

• API Compatibility:

  • Row-Major: Compatible with Direct X, Maya.

  • Column-Major: Compatible with OpenGL, PBRT, Blender, Vulkan.

• Basis (Axes) Representation in the Matrix:

  • Row-Major: Rows represent the coordinate system's bases (color-coded: red for x-axis, green for y-axis, blue for z-axis).

    $$ {\begin{bmatrix} \color{red}{c_{00}}& \color{red}{c_{01}}&\color{red}{c_{02}}&0\\ \color{green}{c_{10}}& \color{green}{c_{11}}&\color{green}{c_{12}}&0\\ \color{blue}{c_{20}}& \color{blue}{c_{21}}&\color{blue}{c_{22}}&0\\ 0&0&0&1 \end{bmatrix} } $$

  • Column-Major: Columns represent the coordinate system's bases (color-coded: red for x-axis, green for y-axis, blue for z-axis).

    $$ { \begin{bmatrix} \color{red}{c_{00}}& \color{green}{c_{01}}&\color{blue}{c_{02}}&0\\ \color{red}{c_{10}}& \color{green}{c_{11}}&\color{blue}{c_{12}}&0\\ \color{red}{c_{20}}& \color{green}{c_{21}}&\color{blue}{c_{22}}&0\\ 0&0&0&1 \end{bmatrix} } $$

• Translation Matrices:

  • Row-Major:

    $$ {\begin{bmatrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ Tx&Ty&Tz&1 \end{bmatrix} } $$

  • Column-Major:

    $$ {\begin{bmatrix} 1&0&0&Tx\\ 0&1&0&Ty\\ 0&0&1&Tz\\ 0&0&0&1 \end{bmatrix} } $$

• Matrix Transposition for Convention Switch:

  • Row-Major: Transpose the matrix for use in column-major contexts

  • Column-Major: Transpose the matrix for use in row-major contexts

Vector3 vecMatMult(
    Vector3 v,
    float AXx, float AXy, float AXz, float Tx,
    float AYx, float AYy, float AYz, float Ty,
    float AZz, float AZy, float AZz, float Tz) 
{
    return Vector3(
        v.x * AXx + v.y * AYx + v.z * AZx + Tx,
        v.x * AXy + v.y * AYy + v.z * AZy + Ty,
        v.x * AXz + v.y * AZz + v.z * AZz + Tz);
}