S-Bend Generation Architecture

This document describes the internal architecture and construction flow of an S-Bend geometry generator implemented in Python. The S-Bend is constructed using cubic Bézier curves derived directly from their mathematical formulation.

1. Mathematical Foundation

The S-Bend geometry is based on the cubic Bézier curve, defined as:

Each Bézier segment is fully defined by four control points:

• P₀: start point
• P₁, P₂: shape control points
• P₃: end point

2. Geometric Architecture

The S-Bend is composed of two cubic Bézier segments connected continuously:

• Segment 1: Input → Midpoint
• Segment 2: Midpoint → Output

The endpoint of the first segment is reused as the starting point of the second segment to guarantee geometric continuity.

3. Software Architecture

3.1 Control Point Definition

Each Bézier segment is parameterized using NumPy vectors representing the control points in Cartesian space.

P0, P1, P2, P3   → First Bézier segment
Q0, Q1, Q2, Q3   → Second Bézier segment

The midpoint P3 of the first segment is reused as Q0 for the second segment, ensuring a smooth transition.

3.2 Bézier Curve Evaluation

A dedicated function evaluates the cubic Bézier equation by sampling the parameter t uniformly in the range [0, 1].

def cubic_bezier(A, B, C, D, n):
    t = linspace(0, 1, n)
    return B(t)

Vectorized computation is used to efficiently generate curve points.

4. Construction Flow

1. Define control points for both Bézier segments
2. Evaluate each segment independently
3. Merge the two curves while removing duplicate midpoint samples
4. Interpret the resulting polyline as a waveguide centerline
5. Visualize the curve and its control polygons

5. Visualization Layer

The generated curve is plotted using Matplotlib:

• Thick line: waveguide centerline
• Dashed lines: control polygons
• Markers: control points

This separation clearly illustrates the relationship between the mathematical model and the resulting physical geometry.

6. Summary

This architecture demonstrates a clean separation between:

• Mathematical formulation (cubic Bézier curves)
• Geometric modeling (control points and segments)
• Numerical evaluation (sampling and vectorization)
• Visualization (waveguide representation)

Such an approach is well-suited for parametric photonic and semiconductor layout generation.