Matlab (Software) Radial basis functions Splines theory Knot vectors Lagrange polynomials

MATLAB implementation of the generation of basis functions based on the knot vector

Below you can find a link to the MATLAB code that calculates and draws B-spline basis functions based on a given node vector.

The code execution is also possible in the free Octave environment.
Download or see Appendix 2.

After installing Octave, open the code.
The running of the code starts with the spline () precedent which sets the example vector of knot_vector = [1,1,1,2,3,3,4,5,6,7,7,7] then calculates the number of basis functions using the compute_nr_basis_functions () procedure, and calculates the degree of basis functions (homogeneous for the whole vector of nodes) using the compute_p procedure, and the vector of x points where basis functions will be counted using the mesh () procedure.
The code generates the values of the underlying functions at these points using compute_spline () and then draws the functions using the plot command in a loop. It is possible to modify the node vector in order to generate different bases.

Exercise 1: 1st order Lagrange polynomials

Content of exercise:

Please generate a vector of nodes that generates a base equivalent to the Lagrange basis of polynomials of the first order on two elements [0,1] and [1,2]. Please run MATLAB code and draw base functions.

Exercise 2: 4th order Lagrange polynomials

Content of exercise:

Please generate a vector of nodes that generates a base equivalent to the Lagrange base of polynomials of the fourth degree on five elements [0,1] and [1,2] and [2,3] and [3,4] and [4,5]. Please run MATLAB code and draw base functions.

Exercise 3: 3rd order B-splines with C2 continuity

Content of exercise:

Please generate a vector of nodes that generates the base of B-spline polynomials of the third order with C2 continuity spread over three elements [0,0.1] and [0.1,0.9] and [0.9 1]. Please run MATLAB code and draw base functions.

Exercise 4: 3rd order B-splines with C1 contiunuity

Content of exercise:

Please generate a vector of nodes that generates a base of B-spline polynomials of the third order with continuity C1 spread over four elements [0,1] and [1,2] and [2,3] and [3,4]. Please run MATLAB code and draw base functions.

Exercise 5: Two-dimensional B-splines

Content of exercise:

Please modify the code so that it draws two-dimensional B-splines on the area resulting from the tensor product of two node vectors. Please use MATLAB's meshgrid or surf function. Run the MATLAB code for the node vectors [0 0 0 0 1 2 3 4 4 4 4] x [0 0 1 2 3 4 4] and draw the base functions.

Informatyka

MATLAB implementation of the generation of basis functions based on the knot vector

Exercise 1: 1st order Lagrange polynomials

Content of exercise:

Solution:

Exercise 2: 4th order Lagrange polynomials

Content of exercise:

Solution:

Exercise 3: 3rd order B-splines with C2 continuity

Content of exercise:

Solution:

Exercise 4: 3rd order B-splines with C1 contiunuity

Content of exercise:

Solution:

Exercise 5: Two-dimensional B-splines

Content of exercise:

Solution:

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