Quadratic Interpolation

C¹-continuous spline interpolation with smooth first derivatives.

Key Feature: Single-endpoint boundary condition — specify constraint at only one end.

BC	Description
`Left(ParabolaFit())`	3-point parabola fit at left (Default)
`Right(ParabolaFit())`	3-point parabola fit at right
`Left(Deriv1(v))`	S'(left) = v
`Left(Deriv2(v))`	S''(left) = v
`Right(Deriv1(v))`	S'(right) = v
`Right(Deriv2(v))`	S''(right) = v
`MinCurvFit()`	Minimize total curvature (globally smooth)

Default: Left(ParabolaFit()) — exact for quadratic polynomials.

Usage

using FastInterpolations

x = range(0.0, 2π, 15)
y = sin.(x)

# One-shot evaluation (default: ParabolaFit - exact for polynomials)
quadratic_interp(x, y, 1.0)                        # default BC
quadratic_interp(x, y, 1.0; bc=Left(Deriv1(1.0)))  # S'(left) = 1.0
quadratic_interp(x, y, 1.0; bc=MinCurvFit())       # minimize curvature

# In-place evaluation (zero allocation)
xq = range(x[1], x[end], 200)
out = similar(xq)
quadratic_interp!(out, x, y, xq)

# Create reusable interpolant
itp = quadratic_interp(x, y; bc=Left(Deriv2(0.0)))
itp(1.0)    # evaluate at single point
itp(xq)     # evaluate at multiple points

# Derivatives
quadratic_interp(x, y, 1.0; deriv=1)  # continuous first derivative
d1 = deriv1(itp); d1(1.0)             # same via interpolant
d2 = deriv2(itp); d2(1.0)             # piecewise constant

Choosing BC

Default ParabolaFit(): Best for polynomial-like data (exact for quadratic)
MinCurvFit(): Best for globally smooth interpolation
Deriv1(v)/Deriv2(v): When you know the endpoint derivative

Visualization

See Visual Comparison for side-by-side plots of all methods, or Derivatives for detailed derivative documentation.

Boundary Condition Comparison

Using non-uniform data to highlight how different boundary conditions affect the spline shape:

using FastInterpolations
using Plots

# Non-uniform grid with asymmetric data
x = [0.0, 0.3, 0.8, 1.5, 2.5, 3.0, 4.0]
y = [0.0, 0.8, 1.2, 0.9, 0.3, 0.6, 1.0]
xq = range(x[1], x[end], 200)

p = plot(layout=(2,3), size=(900, 600), legend=:topright)

for (i, (bc, label)) in enumerate([
    (Left(ParabolaFit()), "Left(ParabolaFit()) [Default]"),
    (MinCurvFit(), "MinCurvFit()"),
    (Left(Deriv2(0.0)), "Left(Deriv2(0))"),
    (Left(Deriv1(0.0)), "Left(Deriv1(0))"),
    (Right(Deriv1(3.0)), "Right(Deriv1(3))"),
    (Right(Deriv2(-1.0)), "Right(Deriv2(-1))")
])
    yq = quadratic_interp(x, y, xq; bc=bc)
    plot!(p[i], xq, yq, label="spline", linewidth=2)
    scatter!(p[i], x, y, label="data", markersize=6, color=:black)
    title!(p[i], label)
    ylims!(p[i], -0.5, 2.0)
end
p

BC Effect

Left(ParabolaFit()) [Default]: Exact for quadratic polynomials
MinCurvFit(): Minimizes total curvature (globally smooth)
Left(Deriv2(0)): First interval is linear (no curvature at left)
Left(Deriv1(0)): Starts flat (zero slope at left endpoint)
Right(Deriv1(3)): Ends with steep upward slope
Right(Deriv2(-1)): Ends with specified negative curvature

When to Use

C¹ continuity (smooth first derivative) is sufficient
Known derivative at one endpoint
Simpler than cubic, smoother than linear

Need C² continuity? → Cubic Speed over smoothness? → Linear