# Coefficents of piecewise polynomial in matlab

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Hi I have a simple example of piecewise polynomial from MATLAB using the function 'cscvn'

cscvn( [1 0 -1 0 1;0 1 0 -1 0] )

ans : form: 'pp'

breaks: [0 1.1892 2.3784 3.5676 4.7568]

coefs: [8×4 double]

pieces: 4

order: 4

dim: 2

My question is : How do I read the coeffiecnt matrix as shown above. I can understand from the above coefs matrix that it has 4 columns corresponding to 4 coefficents of a 4th order polynomial. But why this matrix has 8 columns? I only have 4 pieces of polynomial, then why double the number of rows in coefs matrix?

Seocnd question is : How can I use the above coefs matrix to get the equation of piecewise poilynomial

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### Answers (2)

John D'Errico
on 22 Nov 2023

Edited: John D'Errico
on 22 Nov 2023

Do you understand the result will not be some simple polynomial like you are used to seeing?

plot([1 0 -1 0 1],[0 1 0 -1 0],'-o')

You want a curve that follows around this diamond shaped thing. So you CANNOT have a function of the form y(x). PERIOD. You cannot do so.

Instead, cscvn generates a curve as a function of a parameter, I'll call it t, where t is the arclength of the piecewise linear curve that connects those dots. So cscvn creates the pair of functions x(t), and y(t).

READ THE DOCS FOR CSCVN: cscvn

In there this scheme is explained fairly clearly.

Again, there is no simple equation of the form y(x). There cannot be, as if you look at the plot, the result cannot be a single valued function.

##### 4 Comments

Walter Roberson
on 23 Nov 2023

Bruno Luong
on 24 Nov 2023

@John D'Errico "Instead, cscvn generates a curve as a function of a parameter, I'll call it t, where t is the arclength of the piecewise linear curve that connects those dots"

Actually it's the sum of the square-root of the linear arclength, this thing is called Eugene Lee centripetal parametrization according to cscvn doc

Bruno Luong
on 24 Nov 2023

Edited: Bruno Luong
on 24 Nov 2023

Similar answer as here

points = [1 0 -1 0 1;0 1 0 -1 0];

pp = cscvn(points)

m = pp.dim;

p = pp.pieces;

%D = diff(points,1,2);

%d = sqrt(sqrt(sum(D.^2,1)));

%t = cumsum([0 d]); % Eugene Lee's, equal to pp.breaks, see https://www.mathworks.com/help/curvefit/cscvn.html

t = pp.breaks;

ni = 200;

ti = linspace(t(1),t(end), ni);

loc = discretize(ti, t);

x = zeros(length(ti), m);

coefs = reshape(pp.coefs, m, p, []);

for k=1:p

ink = loc == k;

tik = ti(ink);

dti = tik - t(k);

for d=1:m

P = coefs(d, k, :);

x(ink,d) = polyval(P(:), dti);

end

end

close all

hold on

if m == 2

plot(points(1,:),points(2,:),'or'),

plot(x(:,1), x(:,2),'b.')

elseif m == 3

plot3(points(1,:),points(2,:),points(3,:),'or'),

plot3(x(:,1), x(:,2), x(:,3), 'b.')

else

warning('plotting not supported')

end

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