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Interpolation A method of constructing a function that crosses through a discrete set of known data points. .

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**Spline Interpolation Linear, Quadratic, Cubic**

Preferred over other polynomial interpolation More efficient High-degree polynomials are very computationally expensive Smaller error Interpolant is smoother

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**Spline Interpolation Definition**

Given n+1 distinct knots xi such that: with n+1 knot values yi find a spline function with each Si(x) a polynomial of degree at most n.

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**Linear Spline Interpolation**

Simplest form of spline interpolation Points connected by lines Each Si is a linear function constructed as: Must be continuous at each data point: Continuity:

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**Quadratic Spline Interpolation**

The quadratic spline can be constructed as: The coefficients can be found by choosing a z0 and then using the recurrence relation:

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Quadratic Splines 2 Quadratic splines are rarely used for interpolation for practical purposes Ideally quadratic splines are only used to understand cubic splines

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**Quadratic Spline Graph**

t=a:2:b;

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**Quadratic Spline Graph**

t=a:0.5:b;

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**Natural Cubic Spline Interpolation**

SPLINE OF DEGREE k = 3 The domain of S is an interval [a,b]. S, S’, S’’ are all continuous functions on [a,b]. There are points ti (the knots of S) such that a = t0 < t1 < .. tn = b and such that S is a polynomial of degree at most k on each subinterval [ti, ti+1]. yn … y1 y0 y tn t1 t0 x ti are knots

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**Natural Cubic Spline Interpolation**

Si(x) is a cubic polynomial that will be used on the subinterval [ xi, xi+1 ].

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**Natural Cubic Spline Interpolation**

Si(x) = aix3 + bix2 + cix + di 4 Coefficients with n subintervals = 4n equations There are 4n-2 conditions Interpolation conditions Continuity conditions Natural Conditions S’’(x0) = 0 S’’(xn) = 0

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**Natural Cubic Spline Interpolation**

Algorithm Define Zi = S’’(ti) On each [ti, ti+1] S’’ is a linear polynomial with Si’’(ti) = zi, Si’’ (ti+1) = z+1 Then Where hi = ti+1 – ti Integrating twice yields:

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**Natural Cubic Spline Interpolation**

Where hi = xi+1 - xi S’i-1(ti) = S’i(ti) Continuity Solve this by deriving the above equation

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**Natural Cubic Spline Interpolation**

Algorithm: Input: ti, yi hi = ti+1 – ti a ui = 2(hi-1 + hi) vi = 6(bi – bi-1) Solve Az = b

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**Hand spline interpolation**

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**Bezier Spline Interpolation**

A similar but different problem: Controlling the shape of curves. Problem: given some (control) points, produce and modify the shape of a curve passing through the first and last point.

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**Bezier Spline Interpolation**

Practical Application

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**Bezier Spline Interpolation**

Idea: Build functions that are combinations of some basic and simpler functions. Basic functions: B-splines Bernstein polynomials

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**Bernstein Polynomials**

Definition 5.5: Bernstein polynomials of degree N are defined by: For v = 0, 1, 2, …, N, where N over v = N! / v! (N – v)! In general there are N+1 Bernstein Polynomials of degree N. For example, the Bernstein Polynomials of degrees 1, 2, and 3 are: 1. B0,1(t) = 1-t, B1,1(t) = t; 2. B0,2(t) = (1-t)2, B1,2(t) = 2t(1-t), B2,2(t) = t2; 3. B0,3(t) = (1-t)3, B1,3(t) = 3t(1-t)2, B2,3(t)=3t2(1-t), B3,3(t) = t3;

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**Bernstein Polynomials**

Given a set of control points {Pi}Ni=0, where Pi = (xi, yi), Definition 5.6: A Bezier curve of degree N is: P(t) = Ni=0 PiBi,N(t), Where Bi,N(t), for I = 0, 1, …, N, are the Bernstein polynomials of degree N. P(t) is the Bezier curve Since Pi = (xi, yi) x(t) = Ni=0xiBi,N(t) and y(t) = Ni=0yiBi,N(t) Easy to modify curve if points are added.

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**Bernstein Polynomials Example**

Find the Bezier curve which has the control points (2,2), (1,1.5), (3.5,0), (4,1). Substituting the x- and y-coordinates of the control points and N=3 into the x(t) and y(t) formulas on the previous slide yields x(t) = 2B0,3(t) + 1B1,3(t) + 3.5B2,3(t) + 4B3,3(t) y(t) = 2B0,3(t) + 1.5B1,3(t) + 0B2,3(t) + 1B3,3(t)

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