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> BEZIER CURVE CIPHER

bezier difficulty: 2–5 field: computer graphics

The idea in plain English: A Bezier curve is a smooth curve defined by a few control points. You draw a line between point A and point B, then point B and point C, then smoothly blend between those lines. The result is a graceful curve controlled by those 3 points (quadratic) or 4 points (cubic). In this puzzle, the message is hidden in the y-coordinates of points along the curve at specific positions. Given the control points, you can compute where the curve passes at any position t (from 0 to 1).

Why this really exists: Bezier curves are how every modern font works — each letter in a font is defined as a set of Bezier curves. Adobe Illustrator, Figma, and every vector graphics tool use them. Your car's body shape was designed with Bezier curves in CAD software. Animation software uses them for smooth motion paths. The pen tool in Photoshop is just drawing Bezier curves.

▸ Concrete Example

3 control points: P0=(0,72), P1=(50,73), P2=(100,69). Sample at t=0.5:

Quadratic Bezier formula: B(t) = (1-t)²P0 + 2(1-t)tP1 + t²P2

At t = 0.5:
x = 0.25×0 + 0.5×50 + 0.25×100 = 50
y = 0.25×72 + 0.5×73 + 0.25×69 = 72.5 → round to 73 → 'I'

Sample at t=0.0: y=72 → 'H'
Sample at t=0.5: y=73 → 'I' → "HI"

▸ How to Decode (Step by Step)

1. Get control points and the t-values (positions along the curve) from the puzzle

2. For each t, compute the point on the curve using the Bezier formula

3. Round the y-coordinate to the nearest integer

4. Convert each integer to an ASCII character

5. Join all characters → the answer word

def quadratic_bezier(p0, p1, p2, t):
return (1-t)**2 * p0 + 2*(1-t)*t * p1 + t**2 * p2

word = ''.join(chr(int(round(quadratic_bezier(...)))))

# For cubic (4 control points):
def cubic_bezier(p0, p1, p2, p3, t):
return (1-t)**3*p0 + 3*(1-t)**2*t*p1 + 3*(1-t)*t**2*p2 + t**3*p3

▸ Real-World Applications

Vector fonts: TrueType and OpenType fonts use Bezier curves for every character shape
Vector graphics: SVG, Illustrator, Figma — all use Bezier curves as the core drawing primitive
CAD/CAM: Car bodies, phone cases, and manufactured products are designed with Bezier surfaces
Animation: Keyframe interpolation in animation software uses Bezier curves for smooth motion
CSS animations: The "cubic-bezier()" function in web CSS controls easing curves

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