Quadratic & Cubic Spline Interpolation Calculator

Quadratic Splines ($C^1$)

Connects knots with second-degree polynomials $S_i(x) = a_i x^2 + b_i x + c_i$. Guarantees continuous position and continuous slope ($S'_i(x_i) = S'_{i - 1}(x_i)$), but permits step changes in acceleration and curvature.

Natural Cubic Splines ($C^2$)

Constructs third-degree polynomials enforcing both continuous slopes ($S'$) and continuous curvature ($S''$). Natural splines enforce zero second derivative at terminal knots ($M_0 = 0, M_n = 0$), minimizing overall strain energy.

Clamped Cubic Splines ($C^2$)

Replaces free boundary curvature with explicit initial and terminal first-derivative slopes ($S'(x_0) = f'_0$, $S'(x_n) = f'_n$). Essential for robotics pathing, CAD curve matching, and missile trajectory modeling.

Best Practice Workflow

• • Strict Monotonic Sorting: Spline equations divide by interval step sizes ($h_i = x_{i + 1} - x_i$). Always verify that $x_{i + 1}> x_i$. Coincident or unsorted coordinates trigger zero-division singularities.
• • Choose Clamped Splines for Known Physics:If modeling a physical vehicle starting from rest, clamp the initial derivative to zero ($S'(0) = 0$). Natural splines assume zero inflection at the ends, which can induce spurious initial acceleration.
• • Normalize Coordinates for High Dynamic Ranges: When $X$ span thousands of units while $Y$ ranges between decimals, numerical roundoff error accumulates. Rescale domains to $[0, 1]$ before matrix factorization.
• • Sample Derivatives for Smoothness Audits: Review the first and second derivative outputs at knots to verify that no unnatural oscillatory swings occur between sparse nodes.

Critical Numerical Hazards

• • Extrapolating Beyond Boundary Knots: Splines are rigorously defined only within $[x_0, x_n]$. Evaluating cubic polynomials outside their domain produces diverging polynomial runaway.
• • Overshooting on Step Functions: Splines cannot represent discontinuous step transitions without substantial overshooting (Gibbs-like phenomenon). For steep data transitions, use tension splines or PCHIP.
• • Accumulating Error in Quadratic Splines: In quadratic splines, interval $i+1$ directly inherits the slope calculated from interval $i$. An error or disturbance in $z_0$ propagates and amplifies down the chain.
• • Confusing Spline Interpolation with Regression: Interpolation forces the curve through every single knot. If your data contains measurement noise, use B-spline smoothing or least-squares polynomial regression instead.