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L5: Hyperbolic Distance Explorer

Drag the two dots. Feel how distance explodes near the boundary.

dH = arcosh(1 + 2||u-v||2 / ((1-||u||2)(1-||v||2)))
Hyperbolic distance in the Poincare ball. Same Euclidean gap = wildly different hyperbolic distance depending on WHERE in the disk.
0.00
Euclidean Distance
0.00
Hyperbolic Distance
1.0x
Amplification

Point A (blue)

||u|| = 0.00

Point B (green)

||v|| = 0.00
What you're feeling: In flat (Euclidean) space, moving 10cm is always 10cm. In hyperbolic space, the same physical movement near the edge covers exponentially more distance.

Try this: Put both dots near the center and note the hyperbolic distance. Then drag them both to the same relative positions near the edge. The Euclidean distance is identical — but the hyperbolic distance explodes.

Why this matters: An attacker trying to reach "unsafe" territory near the boundary has to traverse exponentially more distance. That's the core of SCBE's security guarantee — adversarial behavior is geometrically expensive, not just rule-prohibited.

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