Approximation of total curvature

In summary, the conversation is about finding an interpolating curve with minimal curvature, or as close to a straight line as possible. The document the speaker is reading mentions a formula for calculating curvature and the speaker is wondering why it is applicable. The other speaker suggests fitting a curve with as few "humps" as possible by lowering the degree and reducing the number of roots in the derivative.
  • #1
Hello, I am trying to find an interpolating curve between a few points that has minimal curvature. That means, as close to a straight line as possible.

Reading a document about cubic splines, they say that

[tex]\kappa \left ( x \right )=\frac{|f''\left ( x \right )|}{\left ( 1+\left [ f'\left ( x \right )^{2} \right ] \right )^{\frac{3}{2}}}\approx |f''\left ( x \right )|[/tex]

Why are they able to say that? Is there any proof or explanation? Thank you very much
 
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  • #2
That's just the formula for calculating curvarture.
I'd rather fit a curve with as few 'humps' as possible (a low total curvature doesn't mean that a curve is nearly straight). Lower the degree, fewer are the roots of its derivative in the range & thus, humps.
 

1. What is total curvature?

Total curvature is a geometric measure that describes the amount of bending or curvature in a two-dimensional surface or curve. It is defined as the integral of the absolute value of the curvature over the entire surface or curve.

2. Why is the approximation of total curvature important?

The approximation of total curvature is important in various fields such as computer graphics, physics, and engineering. It allows for the calculation of important physical properties, such as the bending energy and stability, of surfaces and curves.

3. How is total curvature approximated?

Total curvature can be approximated using various methods, such as the discrete Gaussian curvature method and the discrete mean curvature method. These methods involve breaking down the surface or curve into smaller elements and calculating the curvature at each point.

4. What are the limitations of total curvature approximation?

One limitation of total curvature approximation is that it is an estimation and may not provide exact values. It is also dependent on the accuracy of the data and the chosen approximation method. Additionally, it may not be applicable to surfaces with complex geometries or singularities.

5. How is total curvature used in real-world applications?

Total curvature has various real-world applications, such as in computer graphics for creating smooth and realistic surfaces, in materials science for studying the behavior of thin films and membranes, and in medical imaging for analyzing the curvature of blood vessels and organs. It is also utilized in physics and engineering for studying the stability and deformations of structures and materials.

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