An Elastica curve is defined as one that minimises the bending energy, i.e. the curvature squared. Suppose I have two Elastica curves, and I decide to join them end to end. Then what are the conditions at the joint for the final curve (the union of the two curves) to be also an Elastica?
I presume one necessary condition is smoothness, i.e. continuity of the first derivative. Otherwise there will be infinite bending energy at the joint. Do we also need continuity of the second derivative? If yes, could you explain why?
Is there a theorem regarding this? Could someone give me a reference?
P.S. I presume that what I am describing above is actually spline curves.
UPDATE
The two responses below answer most of my query, one in the the applied context of splines/Bézier curves, and the other in the abstract context of the Elastica equation itself.
I have one remaining doubt, which I ask in a separate question: Can an Elastica curve have a discontinuity in the curvature?
Best Answer
Comment
If shape generation from mechanics is important, it is better at first to avoid splines and look directly at ODE integration, looking into what shapes are generated from physical laws. If you desire smooth generation of shapes of geometrically smooth curves satisfying some end constraints, then use NURBS splines etc.
A single Elastica has no points of sudden change of curvature or slope. Their intrinsic equations found by using Euler-Bernoulli laws usually in one setting. ODE integration or numerical integration/ finite difference methods are used.
What exactly is your problem? Do you want to join two Elastica curves with different slopes at the point of joining, like at a hinge? Then there is no smoothness due to the abrupt slope change. In structural mechanics, freedom of rotation can be incorporated in analysis at the joint by satisfying force and moment equations obeying force/moment equilibria with Euler-Bernoulli laws. Cross section variation can be incorporated in formulation.
All elasticas can be expressed as splines, but not all splines can be associated with elasticas because it involves finding forces, moments that defined them.
To have a single continuous Elastica we should have the same forces, moments and boundary conditions for parts making it.