translate4d: 4D Translation Matrix Generator

Mathematical Foundation of Translation Transformations

The translate4d function generates 4×4 translation matrices that represent spatial displacement in homogeneous coordinate systems. Translation matrices enable parallel displacement of geometric objects without rotation or scaling, fundamental to computer graphics and 3D transformations.

For a translation vector $\mathbf{t} = (t_x, t_y, t_z)$, the translation matrix is defined as:

$$T(\mathbf{t}) = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ t_x & t_y & t_z & 1 \end{bmatrix}$$

When applied to a homogeneous position vector $\mathbf{p} = (x, y, z, 1)^T$, the transformation yields:

$T(\mathbf{t}) \cdot \mathbf{p} = (x + t_x, y + t_y, z + t_z, 1)^T$

This matrix preserves all geometric properties except position, making it essential for coordinate system transformations and spatial positioning operations.

Function Overloading and Parameter Flexibility

The translate4d function supports two distinct parameter patterns:

• translate4d(vec3) - accepts a 3D vector containing translation components
• translate4d(float, float, float) - accepts individual scalar translation values

Both variants generate identical translation matrices, providing interface flexibility for different coding contexts and data representations.

Dynamic Object Translation

ライブエディター

const fragment = () => {
      const center = vec2(0.5)
      const offset = vec3(iTime.sin().mul(0.3), iTime.mul(0.8).cos().mul(0.2), 0)
      const translationMatrix = translate4d(offset)
      const originalPos = uv.sub(center).mul(4)
      const translatedPos = translationMatrix.mul(vec4(originalPos, 0, 1)).xy
      const circle = smoothstep(0.6, 0.4, translatedPos.length())
      const trail = smoothstep(0.1, 0.05, originalPos.y.sub(offset.y).abs())
      const combined = circle.add(trail.mul(0.3))
      const color = vec3(combined, combined.mul(0.8), combined.mul(0.6))
      return vec4(color, 1)
}