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The introduction section on this page states: "the spline will take a shape that minimizes the bending", while showing the formula for curvature of a real curve.

Question is, what is the formula for the bending that is to be minimized, the curvature is a function rather than a number?! Is it some integral or something?

Coolwater
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1 Answers1

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The stored energy in a spline goes as the inverse square of the radius of curvature - so if you want to minimize the energy, you want to minimize

$$\int \frac{ds}{r(s)^2}$$

where $s$ is taken along the length of the spline, and $r$ is the local radius of curvature.

Floris
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  • How come the integral result in a number with unit $m^{-1}$, shouldn't it be joule? – Coolwater May 11 '14 at 15:04
  • @Coolwater Floris has left off the constant pre-factors; so what he is computing there is harmonic mean of the radius of curvature along the path. To figure out the pre-factor you need examine the simple cases of bending an arc of constant radius of curvature. – dmckee --- ex-moderator kitten May 11 '14 at 18:42
  • @dmckee is right, I did not write an equation for the energy; I said "if you want to minimize the energy, you want to minimize" [this equation]. Minimizing the integral can happen regardless of the units or scale factors. Put differently - as written the integral does not evaluate to the energy, but it scales with the energy stored. Minimize one, and you minimize the other. – Floris May 11 '14 at 18:45