Boundary Conditions

Boundary conditions (BCs) determine how spline interpolants behave at the domain endpoints. They are essential for quadratic and cubic splines, which require constraints to uniquely determine the spline coefficients.

Not Needed for All Methods

Constant and Linear interpolation do not require boundary conditions
Quadratic splines require one BC (single endpoint)
Cubic splines require two BCs (both endpoints)

Type Hierarchy

AbstractBC{T}
├── PointBC{T}              # Single-point BC (shared by quadratic & cubic)
│   ├── Deriv1{T}           # User-specified first derivative
│   ├── Deriv2{T}           # User-specified second derivative
│   ├── Deriv3{T}           # User-specified third derivative
│   └── PolyFit{D,T}        # D-degree polynomial fit (auto-estimated)
│       ├── LinearFit       # = PolyFit{1} (2 points, O(h))
│       ├── QuadraticFit     # = PolyFit{2} (3 points, O(h²))
│       └── CubicFit        # = PolyFit{3} (4 points, O(h³))
├── BCPair{T,L,R}           # Both endpoints (cubic only)
├── PeriodicBC{T}           # Periodic BC (cubic only)
├── NaturalBC{T}            # Zero curvature at both ends (cubic only)
├── ClampedBC{T}            # Zero slope at both ends (cubic only)
├── MinCurvFit{T}           # Minimum curvature (quadratic only)
├── Left{T,B}               # Wrapper: apply BC at left endpoint
└── Right{T,B}              # Wrapper: apply BC at right endpoint

BC Categories

1. PointBC (Shared)

PointBC types can be used with both quadratic and cubic splines. They specify a derivative condition at a single endpoint.

Type	Meaning	Usage
`Deriv1(v)`	S'(endpoint) = v	Known slope
`Deriv2(v)`	S''(endpoint) = v	Known curvature
`Deriv3(v)`	S'''(endpoint) = v	Known third derivative
`PolyFit{D}()`	Estimate derivative from D+1 points	Auto-fit from data

Convenience aliases for PolyFit:

Alias	Equivalent	Points	Accuracy
`LinearFit()`	`PolyFit{1}()`	2	O(h)
`QuadraticFit()`	`PolyFit{2}()`	3	O(h²)
`CubicFit()`	`PolyFit{3}()`	4	O(h³)

👉 See PointBC Details for in-depth explanation and visualizations.

2. Quadratic-Specific BCs

Quadratic splines need exactly one endpoint constraint:

# Wrap PointBC with Left() or Right() to specify which endpoint
Left(QuadraticFit())     # Default: auto-fit at left endpoint
Right(Deriv1(0.5))      # Slope = 0.5 at right endpoint
Left(Deriv2(0.0))       # Zero curvature at left

# Special quadratic-only BC
MinCurvFit()            # Minimize total curvature (globally smooth)

3. Cubic-Specific BCs

Cubic splines need constraints at both endpoints:

# BCPair: independent constraints at each endpoint
BCPair(Deriv2(0), Deriv2(0))      # Natural BC
BCPair(Deriv1(0), Deriv1(0))      # Clamped BC
BCPair(Deriv1(1.0), Deriv2(0))    # Mixed

# Convenience shortcuts
NaturalBC()      # = BCPair(Deriv2(0), Deriv2(0))  [default]
ClampedBC()      # = BCPair(Deriv1(0), Deriv1(0))

# True periodicity (requires y[1] ≈ y[end])
PeriodicBC()     # S(x) = S(x + τ) with C² continuity

Quick Reference

Quadratic Splines

BC	Description	Best For
`Left(QuadraticFit())`	3-point fit at left	Default - exact for quadratics
`Right(QuadraticFit())`	3-point fit at right	Right-anchored data
`Left(Deriv1(v))`	Known slope at left	Physics constraints
`MinCurvFit()`	Minimize curvature	Globally smooth interpolation

Cubic Splines

BC	Description	Best For
`NaturalBC()`	S''=0 at both ends	Default - general data
`ClampedBC()`	S'=0 at both ends	Flat endpoints
`BCPair(...)`	Custom at each end	Known derivatives
`PeriodicBC()`	True periodicity	Cyclic data
`CubicFit()`	4-point polynomial fit	Exact for cubic polynomials