Cubic Spline Interpolation Tutorial

[hotvette]

Attached below are two cubic spline tutorials:

1. Explanation of the classic tri-diagonal cubic spline formulation. Included are 2 example problems.

2. Extension to parametric cubic splines. Included are 2 example problems
.
:smile:

[Drager]

wow,

After a very long search in google i found this tutorial.

And it was the first one, which describes parametric cubic splines in a good way.

Thank you very much :-)

Now i can lay streets through my landscape :-)

[Cyko007]

Sigh....Thank You!

I was missing something very stupid and now I know what it is!

[emer]

Thank you for tutorials. This has helped me very mush.

[harshm]

Hey there -
Thanks for the great tutorials - they really helped me! I'm trying to duplicate your results for cubic interpolation of a circle with 4 points and I got the same solution for the 2nd derivatives in the x and y directions. However, when I solve for the coefficients and plot the cubic polynomials I can't seem to get the same result as you - Heres the code that calculates the coefficients (in MATLAB)

mx - x''
my - y''
Sx - x coefficients
Sy - y coefficients
s - arc length [0 0.25 0.5 0.75 1]

for i = 1:n-1
Sx(i,1) = (mx(i+1) - mx(i))/6*h;
Sx(i,2) = mx(i)/2;
Sx(i,3) = (x(i+1) - x(i))/h - h*(mx(i+1) + 2*mx(i))/6;
Sx(i,4) = x(i);

Sy(i,1) = (my(i+1) - my(i))/6*h;
Sy(i,2) = my(i)/2;
Sy(i,3) = (y(i+1) - y(i))/h - h*(my(i+1) + 2*my(i))/6;
Sy(i,4) = y(i);

end


%Plotting the function
ds = 0.01;
m=1;
i=1;
for i = 1:n-1
for k = s(i):ds:s(i+1)
genpnx(m) = Sx(i,1)*(k - s(i))^3 + Sx(i,2)*(k-s(i))^2 + Sx(i,3)*(k-s(i)) + Sx(i,4);
genpny(m) = Sy(i,1)*(k - s(i))^3 + Sy(i,2)*(k-s(i))^2 + Sy(i,3)*(k-s(i)) + Sy(i,4);
m = m+1;
end
i
end

plot(genpnx,genpny,'r')

Do you have any idea what I might be doing wrong? (Ive attached an image of what the spline looks like for 4 points)

Thanks!

[hotvette]

Though I'm not a MATLAB person, your expressions for calculating the coefficients look correct. Make sure you are using 0.25 for h. Also, for plotting, t should vary from 0 to 0.25 for each segment.

Here are the coefficient values I got for the 1st segment:

ax = 32
bx = -24
cx = 0
dx = 1

ay = -32
by = 0
cy = 6
dy = 0

Actually, this isn't the best way to fit a circle with cubic polynomials. It is better to fit a quarter circle with a single parametric cubic. The tutorial has been updated - check it out.

[harshm]

Thanks for the coefficients - they helped me figure out what the problem was - a syntax problem - the correct expression for the coefficients are

Sx(i,1) = (mx(i+1) - mx(i))/(6*h)

while I had

Sx(i,1) = (mx(i+1) - mx(i))/6*h

Thanks a ton!

[hotvette]

Glad to help. Fyi, the problem can be made even simpler by making h = 1, meaning s = [0 1 2 3 4]. Second deriveratives are x" = [-3 0 3 -3], y" = [0 -3 0 3 0] and plotting can be done for t varying from 0 to 1 for each segment.

I encourage you to review the updated tutorial on circle fitting.

[hotvette]

Attached is an update to the Cubic Spline Tutorial.

To Staff,

Would someone be willing to replace the 1st attachment in this thread with the below? Thanks.

[vimspal]

thanks a lot buddy.

[spacecamel]

Hi hotvette,
I have been looking for a good tutorial on Cubic Splines, nice work. I have found what I think might be an error in one of your examples, however. Example Problem #1 shows c2 = 0, but when I plug in the values:
y3 = 1
y2 = 0.125
h2 = .5
y3" = 0
y2" = 4.5
I come up with c2 = 1, not 0. Plugging 0.5 into the second polynomial with c2 = 1, I get y2 = 1, thus resulting in a discontinuity with the first polynomial. I'm too lazy to try to work out the root of the problem, but I figured I'd flag the discrepancy.
Cheers

[hotvette]

Agree that C2 = 1.0 (typo), but I think the evaluation of the 2nd poly is OK:

y = a2(x - x_2)^3 + b2(x - x_2)^2 + c2(x - x_2) + d2

Update with typo fixed is attached.

[spacecamel]

Yep, my mistake. I used ax^3 + bx^2 + cx + d rather than using (x-x0), etc.
Thanks again for the excellent tutorial.

[goslar]

For the cubic spline interpolation if we have
A=(x2-x)/h; B=(x-x1/h); C=1/6*(A^3-A)*h^2; D=1/6*(B^3-B)*h^2;
y=A y1 + B y2 + C y1'' + D y2''

What is the interpolation error of above?


For example for Hermitian cubic spline it is smaller than 5/384 ||f^{(4)}|| h^4

[cpile]

Thanks very much my friend! The tutorial for the parametric C Splines was very helpful to me!
You see, every book I ve read doesn t say how we can relate the derivatives of the actual curve to the derivatives of the parametric equations of x & y. Ok, its easy to proof with the chain rule but what about the values of these derivatives on bounds!! Simply, x'(t)/y'(t) must be constant on bounds, so give x'(t),y'(t) any value you want to keep this ratio constant! HAHAHAH, i feel stupid...
Very helpful!
Thanks again!

[$am!]

Thanks! Just what i was looking for! :)

[ark5230]

Any reading suggestion to smooth out a curve using spline or similar approach.