Interpolation Threads a Curve Through the Points
What we actually get is always scattered points: a temperature read once a second, a handful of values in a table, a few keyframes of an animation. When you care about what happens in between, interpolation fills the gaps by raising one smooth function that passes through the points. Where Taylor mimicked a curve from the derivative information at one point, interpolation builds a curve from the position information of many points. N points pin down a unique degree N-1 polynomial. But naively piling on more points has a trap, where the curve goes wild (the Runge phenomenon). So in practice you place points cleverly, or stitch together low-degree pieces as a spline. Interpolation underlies graphics, animation, data analysis, and numerical integration.
Drag the points up and down. However you move them, one smooth curve follows that passes exactly through all five. This is the interpolating polynomial. Two points give one straight line (degree 1), three give one parabola (degree 2), five give one quartic. N points pin down a unique degree N-1 polynomial — no more, no less. Endlessly many curves connect the points, but adding the condition "degree N-1 polynomial" nails the answer down to exactly one. That is where interpolation starts.
How do you write down that unique polynomial? Lagrange's method is elegant. For each node you build one basis function Lᵢ, a polynomial designed to be 1 at its own node and 0 at every other node. Press the segments to inspect them one by one. Then the interpolating polynomial is simply these bases weighted by the heights yᵢ: P(x) = Σ yᵢ Lᵢ(x). Plug in any node xk and only Lk is 1 while the rest are 0, so P(xk) = yk. Passing exactly through every point is guaranteed for free, with no system of equations to solve.
It is tempting to think more points must mean more accuracy. It is a trap. Interpolate the bell-shaped Runge function with equally-spaced points. As you add points the middle improves, but at the ends the curve swings wildly up and down. By around degree 15 the error at the edges is several times larger than the function itself. This is the Runge phenomenon. High-degree polynomials want to bend freely between the nodes, so the more densely you plant points, the more violently the curve oscillates between them. The lesson is clear: in interpolation, simply raising the degree is not the answer.
So what is the answer? Surprisingly, you keep the interpolation method and only change where you put the points. Toggle from equal spacing to the Chebyshev arrangement, which clusters points more densely toward the ends. Same count, same Lagrange interpolation, yet the edge blow-up settles down as if by magic. The intuition: since the polynomial wants to oscillate at the ends, you plant more points there to pin that oscillation down in advance. This is why Chebyshev is the gold standard in telescope-mirror design and function approximation. The problem was never the interpolation but the placement of the points.
Often you cannot choose where the points sit and must take them as given. Then you change strategy entirely. Instead of covering the whole curve with one high-degree polynomial, you stitch together low-degree (cubic) pieces that each look only at two neighboring points, joined smoothly. This is the cubic spline. Add points and the gold polynomial oscillates at the ends while the blue spline hugs the truth. Because each piece is unaffected by far-away points (locality), a wiggle in one place does not spread across the whole. The curves of a font, car-body design (CAD), and animation paths are all splines. Interpolation is more than connecting points; it is choosing how to connect them.