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Paper is here

enter image description here

I have read this paper. But i don't know what the Invariance means in page 3 and 5. Calculation/estimation of κ is NOT invariant and Invariant to viewpoint. What's the meaning of these sayings? When i read "Invariant to viewpoint", i think it means the k value is same when rotation and translation. But after i read "Calculation/estimation of κ is NOT invariant" ,i have no idea. Then i try the code what i got here,to caculate the k value.

Code is here

And i have found that, the k value is different when shape have rotated which have the same sigma. Then i changed the sigma, k value is also different while the shape is same. When i set the sigma as 0.6667, the curvature of straight line is 1.6 not 0, why? The curvature of straight line should be zero,right? I don't why the value is 1.6. Then I tried to rotate the straight line, the curvature changed!!new value is -6.6. The papge said to find zero crossing to get concave-convex. But here curvature of straight line is not 0 is 1.6? Can somebody help me. Thank you very much!

jack fukey
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  • It says it is not invariant to zooming. If you have the earth, zooming in on an ocean it would seem flat on the horizon, zooming out it would look more and more curved as the earth actually is round. – mathreadler Sep 16 '15 at 05:59
  • I don't get what the dots mean. Is it derivative with respect to $s$? So these are parametrized curves? – mathreadler Sep 16 '15 at 06:02
  • @mathreadler Yes,these are parametrized curves.What does zooming mean here?These are 2d curves. – jack fukey Sep 16 '15 at 06:15
  • Imagine a circle drawn on a piece of paper. If you zoom out it looks like it has a higher curvature. If you zoom in the circle looks almost flat. – mathreadler Sep 16 '15 at 07:44
  • @mathreadler Thank you!got it. But i don't know why the curvature of straight line not be zero, and will change by rotation? – jack fukey Sep 16 '15 at 08:55

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