Splines are a powerful tool in various fields, including computer graphics, data fitting, and numerical analysis. This guide provides a practical understanding of splines, covering their theoretical foundations, practical applications, and implementation details. We will focus on key aspects that make splines a versatile and essential technique.
Introduction to Splines
Splines are piecewise polynomial functions that are used to approximate other functions or represent curves and surfaces. They are particularly useful when dealing with complex shapes or data sets where a single polynomial would be inadequate. The primary advantage of splines is their ability to provide smooth and accurate approximations with relatively low-degree polynomials.
I. Polynomial Interpolation Fundamentals
Understanding polynomial interpolation is crucial for grasping the concept of splines. Polynomial interpolation involves finding a polynomial that passes through a given set of data points.
Polynomial Interpolation: Lagrange Form
The Lagrange form of polynomial interpolation is a direct way to construct a polynomial that interpolates a set of data points. Given points (x₁, y₁), (x₂, y₂), …, (xₙ, yₙ), the Lagrange interpolating polynomial P(x) is:
P(x) = ∑ᵢ yᵢ * Lᵢ(x)
Where Lᵢ(x) are the Lagrange basis polynomials, defined as:
Lᵢ(x) = ∏ⱼ₋(x – xⱼ) / (xᵢ – xⱼ) , for all j ≠ i
Polynomial Interpolation: Divided Differences and Newton Form
The Newton form uses divided differences to represent the interpolating polynomial. The divided differences are calculated recursively.
The Newton interpolating polynomial P(x) is:
P(x) = f[x₁] + f[x₁, x₂](x – x₁) + f[x₁, x₂, x₃](x – x₁)(x – x₂) + … + f[x₁, …, xₙ](x – x₁) … (x – xₙ₋₁)
Where f[x₁, …, xₖ] denotes the divided difference of order k-1.
Example: Osculatory Interpolation to the Logarithm
Osculatory interpolation involves matching not only the function values but also the derivatives at the data points. This technique can provide a higher degree of accuracy.
Evaluation of the Newton Form
Efficient evaluation of the Newton form is essential for practical applications. Horner’s method can be adapted to evaluate the Newton form, providing a computationally efficient approach.
Example: Computing the Derivatives of a Polynomial in Newton Form
Computing derivatives of the polynomial is often necessary. The Newton form facilitates the calculation of derivatives using recursive formulas.
Other Polynomial Forms and Conditions
Besides Lagrange and Newton forms, other polynomial representations, such as Bernstein polynomials, are used in specific contexts. The choice of form depends on the application and the properties desired.
II. Limitations of Polynomial Approximation
While polynomial interpolation is useful, it has limitations. High-degree polynomials can exhibit oscillations, especially when the data points are uniformly spaced.
Uniform Spacing of Data Can Have Bad Consequences
Equally spaced data points can lead to Runge’s phenomenon, where the interpolating polynomial diverges between the data points as the degree increases.
Chebyshev Sites Are Good
Choosing Chebyshev nodes as interpolation points can mitigate Runge’s phenomenon. Chebyshev nodes are the roots of Chebyshev polynomials and are clustered near the ends of the interval.
Illustration of Chebyshev sites for interpolation.
Runge Example with Chebyshev Sites
Using Chebyshev sites significantly improves the approximation accuracy in the Runge example, reducing oscillations and providing a more stable interpolation.
Squareroot Example
Approximating the square root function using polynomial interpolation at Chebyshev sites yields good results, demonstrating the effectiveness of this approach.
Interpolation at Chebyshev Sites Is Nearly Optimal
Interpolation at Chebyshev sites is nearly optimal in the sense that it minimizes the Lebesgue constant, which bounds the error of the interpolation.
The Distance from Polynomials
The distance between a function and its polynomial approximation can be estimated using approximation theory. The error depends on the smoothness of the function and the choice of interpolation points.
III. Piecewise Linear Approximation
Piecewise linear approximation involves approximating a function with a series of straight line segments. This is the simplest form of spline approximation.
Broken Line Interpolation
Broken line interpolation, also known as linear spline interpolation, connects the data points with straight lines. It is easy to implement and provides a reasonable approximation for many functions.
Broken Line Interpolation Is Nearly Optimal
Piecewise linear interpolation is nearly optimal in the sense that it achieves the best possible approximation rate for continuous functions.
Least-Squares Approximation by Broken Lines
Least-squares approximation involves finding the best-fitting broken line that minimizes the sum of the squared errors between the function and the approximation.
Good Meshes
Choosing appropriate breakpoints (knots) is crucial for accurate piecewise linear approximation. Adaptive methods can be used to refine the mesh where the function changes rapidly.
IV. Piecewise Cubic Interpolation
Piecewise cubic interpolation uses cubic polynomials to approximate a function. Cubic splines provide smoother approximations than piecewise linear approximations.
Piecewise Cubic Hermite Interpolation
Piecewise cubic Hermite interpolation matches the function values and derivatives at the data points. This method requires derivative information, which may not always be available.
Runge Example Continued
Applying piecewise cubic Hermite interpolation to the Runge example provides a significant improvement over polynomial interpolation, especially when using appropriate derivative estimates.
Piecewise Cubic Bessel Interpolation
Bessel interpolation is a method for estimating the derivatives at the data points using neighboring points. This approach is useful when derivative information is not directly available.
Akima’s Interpolation
Akima’s interpolation is another method for estimating derivatives. It is designed to produce visually pleasing curves with minimal oscillations.
Cubic Spline Interpolation
Cubic spline interpolation involves finding a piecewise cubic polynomial that satisfies certain smoothness conditions, such as continuity of the first and second derivatives.
Boundary Conditions
Boundary conditions are necessary to uniquely define a cubic spline. Common boundary conditions include natural splines (zero second derivative at the endpoints), clamped splines (specified derivative at the endpoints), and periodic splines (matching function values and derivatives at the endpoints).
V. Best Approximation Properties of Complete Cubic Spline Interpolation and Its Error
Complete cubic splines, with clamped boundary conditions, have optimal approximation properties. They minimize the strain energy and provide the best possible approximation in a certain sense.
VI. Parabolic Spline Interpolation
Parabolic splines use piecewise quadratic polynomials to approximate a function. They offer a compromise between linear and cubic splines, providing a good balance between smoothness and computational cost.
VII. A Representation for Piecewise Polynomial Functions
Representing piecewise polynomial functions efficiently is essential for practical use. The ppform is a common representation that stores the breakpoints and polynomial coefficients.
Piecewise Polynomial Functions
Piecewise polynomial functions are defined by a set of polynomials on different intervals, joined together at the breakpoints.
The Subroutine PPVALU
PPVALU is a subroutine for evaluating a piecewise polynomial function at a given point. It uses the ppform to efficiently compute the function value.
The Subroutine INTERV
INTERV is a subroutine for finding the interval containing a given point. It is used by PPVALU to determine which polynomial piece to evaluate.
VIII. The Spaces ∏k,ξ and ∏k,ξ,ν
The spaces ∏k,ξ and ∏k,ξ,ν define the set of piecewise polynomial functions with certain smoothness conditions. These spaces are fundamental to the theoretical understanding of splines.
IX. B-Splines
B-splines are a basis for the space of splines. They are locally supported, which means that each B-spline is nonzero only on a small interval.
Illustration of B-Spline basis functions.
X. More on B-Splines
B-splines have many useful properties, including the partition of unity property, which means that the sum of the B-splines at any point is equal to one.
XI. Calculating with B-Splines
Efficient algorithms for calculating with B-splines are essential for practical applications. These algorithms include recurrence relations for evaluating B-splines and algorithms for knot insertion and removal.
XII. The Distance from Piecewise Polynomials
The distance between a function and its piecewise polynomial approximation can be estimated using approximation theory. The error depends on the smoothness of the function, the degree of the polynomials, and the choice of breakpoints.
XIII. Spline Interpolation
Spline interpolation involves finding a spline that passes through a given set of data points. It is a powerful technique for data fitting and curve smoothing.
XIV. Smoothing Splines
Smoothing splines are used when the data is noisy. They minimize a combination of the error and the smoothness of the spline, providing a good balance between data fitting and noise reduction.
XV. Solving Differential Equations
Splines can be used to solve differential equations. The solution is approximated by a spline, and the coefficients are determined by satisfying the differential equation at certain points.
XVI. Curve Design
Splines are widely used in curve design. They provide a flexible and intuitive way to create and manipulate curves and surfaces.
XVII. Tensor Product Splines
Tensor product splines are used to represent surfaces. They are defined as the product of two B-spline bases, one for each coordinate direction.
Conclusion
Splines are a versatile and powerful tool for approximation, interpolation, and curve design. Understanding their theoretical foundations and practical implementation is essential for many applications. This guide provides a practical introduction to splines, covering the key concepts and techniques needed to effectively use them in various fields.
