lineSDF: Linear Segment Distance Field

Point-to-Line Distance Calculation in 2D and 3D Space

The lineSDF function family computes the shortest distance from any point to a line segment defined by two endpoints. This primitive enables line rendering, path tracing, and geometric analysis in both 2D and 3D coordinate systems.

Mathematical Foundation

2D Line Segment Distance

The 2D line distance uses orthogonal projection onto the line segment:

$$d_{\text{line2D}}(st, a, b) = \|st - a - h \cdot (b - a)\|$$

where:

$$h = \text{clamp}\left(\frac{(st - a) \cdot (b - a)}{(b - a) \cdot (b - a)}, 0, 1\right)$$

3D Line Distance

The 3D version uses the cross product for perpendicular distance:

$$d_{\text{line3D}}(p, a, b) = \frac{\|(p - a) \times (p - b)\|}{\|b - a\|}$$

Function Signatures

lineSDF2D (2D Line Segment)

Parameter	Type	Description
st	vec2	Sample point position
a	vec2	Line segment start point
b	vec2	Line segment end point

lineSDF3D (3D Line)

Parameter	Type	Description
p	vec3	Sample point position
a	vec3	Line start point
b	vec3	Line end point

Implementation Demonstrations

ライブエディター

const fragment = () => {
      const a = vec2(0.2, 0.3).add(iTime.sin().mul(0.1))
      const b = vec2(0.8, 0.7).add(iTime.mul(1.3).cos().mul(0.1))
      const dist = lineSDF2D(uv, a, b)
      const line = float(0.02).smoothstep(0.005, dist)
      const endpoints = distance(uv, a).step(0.03).max(distance(uv, b).step(0.03))
      return vec4(line.mul(vec3(0.8, 0.4, 1)).add(endpoints.mul(vec3(1, 0.2, 0.3))), 1)
}

結果

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