square: AABB Cubic Conversion Operation

Dimension Equalization for Uniform Bounding Volumes

The AABB square function converts a rectangular bounding box into a cubic volume by equalizing all dimensions. This operation creates a cube centered at the original box's centroid with edge length equal to the maximum dimension.

Given an AABB with diagonal vector $\vec{d}$ , the square operation computes:

\begin{align} \vec{d} &= \text{diagonal}(\text{box}) \times 0.5 \\ \vec{c} &= \vec{b_{\min}} + \vec{d} \\ d_{\max} &= \max(|d_x|, |d_y|, |d_z|) \\ \vec{b'_{\min}} &= \vec{c} - d_{\max} \\ \vec{b'_{\max}} &= \vec{c} + d_{\max} \end{align}

where $\vec{c}$ represents the center point and $d_{\max}$ becomes the half-edge length of the resulting cube.

Geometric Properties

The square operation maintains several important characteristics:

Center Preservation: The centroid remains unchanged
Volume Expansion: Always increases or maintains original volume
Aspect Ratio Normalization: Eliminates dimensional bias

Applications

Cubic bounding volumes serve specific computational purposes:

Spatial Hashing: Uniform grid cell sizing for consistent indexing
Level of Detail: Distance-based quality metrics using sphere-cube approximations
Collision Detection: Simplified intersection testing with symmetric bounds
Texture Mapping: Cubic environment mapping and reflection calculations

ライブエディター

const fragment = () => Scope(() => {
      const t = iTime.mul(0.4)
      const p = uv.mul(5).sub(2.5)
      const originalBox = AABB({
              minBounds: vec3(t.sin().mul(1.8).sub(0.3), t.mul(1.3).cos().mul(0.7).sub(0.2), 0),
              maxBounds: vec3(t.mul(0.7).sin().mul(0.9).add(0.6), t.mul(0.9).cos().mul(1.5).add(0.8), 0)
      })
      const squaredBox = square(originalBox)
      const pos = vec3(p, 0)
      const inOriginal = aabbContain(originalBox, pos)
      const inSquared = aabbContain(squaredBox, pos)
      const inExpanded = inSquared.and(inOriginal.not())
      const originalColor = vec3(0.8, 0.3, 0.1)
      const expandedColor = vec3(0.2, 0.7, 0.9)
      const outsideColor = vec3(0.1, 0.1, 0.2)
      const color = outsideColor.select(expandedColor.select(originalColor, inOriginal), inSquared)
      const origEdge = p.x.sub(originalBox.minBounds.x).abs()
                  .min(p.x.sub(originalBox.maxBounds.x).abs())
                  .min(p.y.sub(originalBox.minBounds.y).abs()
                  .min(p.y.sub(originalBox.maxBounds.y).abs()))
      const sqEdge = p.x.sub(squaredBox.minBounds.x).abs()
                .min(p.x.sub(squaredBox.maxBounds.x).abs())
                .min(p.y.sub(squaredBox.minBounds.y).abs()
                .min(p.y.sub(squaredBox.maxBounds.y).abs()))
      const origWire = smoothstep(0.02, 0.04, origEdge).oneMinus()
      const sqWire = smoothstep(0.01, 0.03, sqEdge).oneMinus().mul(0.5)
      const wireframe = origWire.add(sqWire)
      return vec4(color.add(wireframe.mul(0.4)), 1)
})

Dimension Equalization for Uniform Bounding Volumes​

Dimension Equalization for Uniform Bounding Volumes