juliaSDF: Julia Set Fractal Distance Field

Complex Number Iteration for Fractal Boundary Detection

The juliaSDF function generates a Julia set fractal pattern through iterative complex number calculations. This function evaluates the escape time of points under the iteration $z_{n+1} = z_n^2 + c$, where $c$ is a complex constant.

Mathematical Foundation

The Julia set is defined by the iterative formula:

$$z_{n+1} = z_n^2 + c$$

where:

• $z_0$ represents the initial complex coordinate (transformed input position)
• $c$ is a complex constant that determines the fractal shape
• The iteration continues until $|z_n| > 4$ or maximum iterations are reached

The escape time normalization:

$$n = \frac{\text{escape\_iteration}}{\text{max\_iterations}}$$

produces a continuous field value representing the fractal boundary distance.

Function Variants

Function	Parameters	Description
juliaSDF	st, center, c, r	Full parameterization with center and radius
juliaSDFSimple	st, c, r	Simplified version with default center at (0.5, 0.5)

Implementation Demonstrations

Live Editor

const fragment = () => {
      const pos = uv.sub(0.5)
      const c = vec2(-0.7, 0.27015)
      const fractal = juliaSDF(pos.add(0.5), vec2(0.5), c, 1.2)
      const color = fractal.mul(vec3(1, 0.8, 0.3))
      return vec4(color, 1)
}

Result

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