SDF Transformations & Curves

2 min read•1 page

Unlike boundary meshes, SDFs are transformed by applying the inverse transform to the query point $p$ before evaluating the distance function. This lets us bend, twist, or duplicate geometry infinitely with zero memory overhead.

Deformation Operators:

1. Translation / Rotation: $f(T^-1(p))$. Query point is shifted/rotated opposite to the desired movement. 2. Infinite Repetition: Spatial domain folding using a modulo operator $mod(p, s) - 0.5s$. 3. Twist / Bend: Varying the rotation angle dynamically as a function of the query point's coordinate (e.g. height $y$).

python

1# Modifiers work by transforming the query point 'p' in reverse
2def op_translate(p, offset):
3    return p - offset
4
5def op_repeat(p, size):
6    # Map p to local repeated cell
7    return Vector3(
8        (p.x % size) - 0.5 * size,
9        p.y,
10        (p.z % size) - 0.5 * size
11    )
12
13def op_twist(p, k):
14    # Twist angle proportional to height (y)
15    angle = p.y * k
16    c, s = cos(angle), sin(angle)
17    rx = p.x * c - p.z * s
18    rz = p.x * s + p.z * c
19    return Vector3(rx, p.y, rz)

Modifier (0=Translate/Rotate, 1=Repeat, 2=Twist)

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2 min read•1 page

Calculating distances to curves is essential for outline rendering and wireframe effects. A line segment has a straightforward projection calculation. A quadratic Bezier curve is more complex, requiring us to solve a cubic equation to find the closest point.

Projection Mathematics:

1. Segment Projection: Project query point $p$ onto the infinite line of the segment, then clamp to the boundaries $[0, 1]$. 2. Bezier Closest Point: The vector from the closest point on the Bezier curve $B(t)$ to $p$ must be orthogonal to the tangent vector $B'(t)$, yielding $dot(B(t) - p, B'(t)) = 0$.

python

1def sd_segment(p, a, b):
2    pa = p - a
3    ba = b - a
4    # Clamped projection factor
5    t = clamp(dot(pa, ba) / dot(ba, ba), 0.0, 1.0)
6    return length(pa - ba * t)
7
8def sd_quadratic_bezier(p, a, b, c):
9    # Find parameter t in [0, 1] minimizing:
10    # dist(p, B(t)) where B(t) = (1-t)^2*a + 2(1-t)*t*b + t^2*c
11    # Involves finding roots of a cubic equation: dot(B(t) - p, B'(t)) = 0
12    ...
13    return min_distance

Curve Type (0=Line Segment, 1=Bezier Curve)

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Query Point X

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