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Mike Tate Mathematics

Bézier Curves and the Geometry of Smooth Control

Bézier Curves
Control, interpolation, and smooth geometric flow
📐 What is a Bézier Curve?

A Bézier curve is a parametric curve defined by control points. Rather than passing through every point, the curve is pulled by them — a principle used throughout computer graphics, typography, CAD, and animation.

🧮 Mathematical Definition

A cubic Bézier curve with control points P₀, P₁, P₂, P₃ is defined by:

B(t) = (1−t)³P₀ + 3(1−t)²tP₁ + 3(1−t)t²P₂ + t³P₃

As t varies from 0 to 1, the curve smoothly interpolates between endpoints while respecting tangent directions.

🎨 Why Bézier Curves Matter
  • Vector graphics & typography (SVG, fonts)
  • Animation easing & interpolation
  • Physical trajectories & splines
  • Geometric continuity (C¹, C² smoothness)
📐 From Curves to Phase

A Bézier curve interpolates position using weighted control points. The curve does not “push” itself forward — its smoothness is imposed by constraint.

A Josephson junction interpolates phase in an analogous way. The superconducting phase evolves smoothly because it minimizes an energy functional, not because it is forced.

In both cases, smoothness emerges from geometry.

Josephson Junctions · Phase Geometry

Smooth motion governed by phase, energy, and constraint

ℹ️ What is a Josephson Junction?

A Josephson junction consists of two superconductors separated by a thin barrier. Remarkably, current can flow without voltage, governed purely by a phase difference θ between quantum wavefunctions.

The system is described by a geometric rule: 

I = I₀ sin(θ)
dθ/dt ∝ V

Instead of forces pushing particles, the system evolves by minimizing energy along a smooth phase landscape — much like a curve interpolating between control points.

Phase evolution on a sinusoidal energy landscape
Geometric connection: Bézier curves interpolate position · Josephson junctions interpolate phase

SO(2) · Winding Number

Rotation as topology, not motion

ℹ️ What is SO(2)?

SO(2) is the group of rotations in the plane. Each element corresponds to a rotation by an angle θ: 

R(θ) =
[ cosθ  −sinθ ]
[ sinθ   cosθ ]

Although θ and θ + 2π produce the same matrix, the path taken may differ. That difference is captured by the winding number.

Phase angle wrapping around a circle
Same orientation does not imply the same history.

Ampère’s Circuital Law · Circulation Geometry

How loops generate fields

ℹ️ What is Ampère’s Law?

Ampère’s circuital law relates the circulation of a field around a closed loop to what passes through the loop. 

∮ B · dℓ = μ₀ I_enclosed

Geometrically, this says:

  • Fields prefer to circulate
  • Loops measure rotation
  • Enclosed structure generates motion around it

This is the same logic behind phase winding, Josephson junctions, and rotation groups like SO(2).

Circulating field around an enclosed current
Geometric thread:
Bézier → smooth paths
Josephson → phase loops
Ampère → circulation around structure