AABB: Axis-Aligned Bounding Box Structure

Fundamental 3D Spatial Container Definition

The AABB (Axis-Aligned Bounding Box) structure provides a fundamental container for 3D spatial operations. This geometric primitive defines rectangular volumes aligned with coordinate axes, enabling efficient collision detection, spatial queries, and geometric computations.

The AABB structure contains two primary components:

$$\text{AABB} = \{ \begin{align} \text{minBounds} &: \text{vec3} \\ \text{maxBounds} &: \text{vec3} \end{align} \}$$

where $\text{minBounds}$ represents the minimum coordinate values and $\text{maxBounds}$ represents the maximum coordinate values along each axis.

Geometric Properties

The AABB defines a rectangular parallelepiped with edges parallel to coordinate axes. This axis-alignment constraint simplifies many geometric operations while maintaining sufficient coverage for most spatial applications.

Key mathematical relationships:

• Volume: $V = (x_{\max} - x_{\min}) \times (y_{\max} - y_{\min}) \times (z_{\max} - z_{\min})$
• Surface Area: $S = 2[(x_{\max} - x_{\min})(y_{\max} - y_{\min}) + (y_{\max} - y_{\min})(z_{\max} - z_{\min}) + (z_{\max} - z_{\min})(x_{\max} - x_{\min})]$

ライブエディター

const fragment = () => Scope(() => {
      const t = iTime.mul(0.5)
      const p = uv.mul(4).sub(2)
      const box = AABB({
              minBounds: vec3(t.sin().mul(0.8).sub(0.5), t.mul(1.1).cos().mul(0.6).sub(0.3), 0),
              maxBounds: vec3(t.mul(1.2).sin().mul(0.9).add(0.7), t.mul(0.9).cos().mul(0.8).add(0.5), 0)
      })
      const pos = vec3(p, 0)
      const inside = aabbContain(box, pos)
      const edge = p.x.sub(box.minBounds.x).abs()
              .min(p.x.sub(box.maxBounds.x).abs())
              .min(p.y.sub(box.minBounds.y).abs()
              .min(p.y.sub(box.maxBounds.y).abs()))
      const wireframe = smoothstep(0.02, 0.05, edge).oneMinus()
      const color = vec3(0.2, 0.4, 0.8).select(vec3(0.8, 0.3, 0.1), inside)
      return vec4(color.add(wireframe.mul(0.5)), 1)
})