Same curvature in spline but different slope

[armin11]

hi,
we have a spline which consists of two splines,is it possible the two splines have the same curvature but different slopes?

 

[armin11]

Why no one ever answers my questions?!Doesn't a highly educated person visit here?

 

[DickL]

Is this spline a non circular curve, as in a drawing, or is it a spline joint, as in an axially sliding, torque transmitting machine member.

You may want to provide a bit more of a picture of what your talking about. You know the context, but we don't. Without the context, it can be very difficult to provide a answer that addresses your question.

 

[armin11]

It is a non circular simple curve,mathematically it is possible because first derivative of the two functions can be the same and at the same time second derivative of them can be inequal,but how can we describe this graphically?

 

[DickL]

Its been a long while since I've done this type of geometry analytically, so no detail equations. However, where the two curves (splines, parabolic, cubic...) meet they must be tangent. At that point the 1st & 2nd derivatives need to be equal for both curves. The 2nd curve (same basic equation as the 1st) is translated and rotated such that the joining ends meet the tangent criteria.

With a simple curve, it may be easier to figure the translation and rotation analytically than graphically. Depending on how the splines are generated (series of short lines, series of circular curves, higher order curves) it may be easier to do a trail & error fitting (copy, translate, rotate).

 

[krisvillez]

Yes you can in certain cases. To solve this problem, you need to write it in following format:

min f(x1,x2)
s.t. g(x1)=g(x2)

where x1 is the set of coefficients for the first curve, x2 is the set of coefficients for the second curve; f(x1,x2) is the fit objective (for both curves simultaneously) and g(x1), g(x2) are second-order derivatives in the knot locations.

This requires that:
1. The same basis is used for both curves
2. The second order derivatives exist in all knot locations (e.g. no multiplicities and spline order equal or higher than 3)

If you are using an orthogonal basis (e.g. b splines) and a least-squares fit objective, this will lead to a quadratic program with linear equality constraints. You can use several platform to solve the problem (e.g. Matlab (quadprog.m) or GAMS).

Have a look at the FDA book by Ramsay and Silverman if the above does not make sense to you:
http://ego.psych.mcgill.ca/misc/fda/resources.html

Hope this helps,
Kris