Can B-splines cause oscillations in second derivatives?

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In summary, the conversation discusses the use of B-splines and the results obtained when using them to approximate the first and second derivatives of a parametrized ellipse. The first derivative was found to be well approximated, but the second derivative showed poor agreement with lots of oscillations. The question is raised whether this is due to a poor choice of numerical derivative, Runge's phenomena, or an inherent quality of B-splines. It is noted that as a general rule, each order of derivative will differ and be worse when approximating a function.
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hunt_mat
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Hi,

I have been playing with B-splines recently and I have had some curious results. As a rest, I had a set of points on the ellipse; [tex]\frac{x^{2}}{9}+y^{2}=1[/tex]. I used the paramatrisation [tex]x=3\cos\theta ,\quad y=\sin\theta[/tex] and I computed the derivatives numerically (I had to come up with a numerical scheme for differentiation using non-uniform grids. It works okay for some simple functions but I am not very familiar with how these things are done)

I found that the first derivative (w.r.t. [itex]\theta[/itex]) was very well approximated (to within 10^-3), plotting it on matlab, you could just about see where they were different. When I calculated the second derivative (w.r.t. [itex]\theta[/itex]) (by applying my derivative formula again), the curves were a poor agreement. There were lots of oscillations in the calculation of the second derivative.

So my question is this:

Are the oscillations just a sign of my poor choice of numerical derivative, this phenomena an example of Runge's phenomena or something which is inherent within B-splines? I was using a degree 5 B-spline, the result was worse for a degree 7 curve.

Any suggestions?
 
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  • #2
As a general rule, when approximating a function, the derivatives will differ, and each order of derivative is worse.
 
  • #3
I understand that. The degree of the curve I am using is 5, so that implies [itex]C^{3}[/itex] continuity. What I don't get is HOW they're worse.
 

What are B-spline curves and surfaces?

B-spline curves and surfaces are mathematical representations of smooth curves and surfaces that are commonly used in computer graphics and computer-aided design. They are defined by a series of control points and mathematical equations that determine the shape of the curve or surface.

What are the advantages of using B-spline curves and surfaces?

One of the main advantages of B-spline curves and surfaces is their ability to represent complex and smooth shapes with a relatively small number of control points. They also have local control, meaning that changes to one control point will only affect the nearby portion of the curve or surface. Additionally, B-splines have a compact and efficient mathematical representation, making them useful for computer graphics applications.

What are the types of B-spline curves and surfaces?

There are several types of B-spline curves and surfaces, including uniform B-splines, non-uniform B-splines, and rational B-splines. Uniform B-splines have equally spaced control points and are commonly used for simple shapes. Non-uniform B-splines allow for more flexibility in controlling the shape of the curve or surface. Rational B-splines are a type of non-uniform B-spline that includes a weight component for each control point, allowing for even more control over the shape.

What are the applications of B-spline curves and surfaces?

B-spline curves and surfaces have a wide range of applications, including computer graphics, computer-aided design, and animation. They are also used in industries such as automotive and aerospace engineering for creating complex and smooth surfaces. B-splines are also commonly used in medical imaging and visualization for reconstructing three-dimensional structures from two-dimensional images.

How are B-spline curves and surfaces evaluated and manipulated?

B-spline curves and surfaces are evaluated and manipulated using algorithms that calculate the positions of the curve or surface based on the control points and equations. These algorithms are used to modify the shape of the curve or surface by changing the control points or equations. Additionally, there are various software tools that allow for interactive manipulation of B-splines, making them accessible for a wide range of users.

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