Splines and Curves of Least Energy

I am erstwhile boat designer/builder with an interest in math and physics.

I've long been interested in the math behind CAD curves and surfaces and have wondered why spline curves in most CAD programs don't match the curves a piece of wood or metal takes through control points on a mold or jig in the shop, without a lot of extra 'tweaking and adjustment.' BTW, I design and build interactive museum exhibits for a living, making good use of CAD and CNC machines alongside manual woodworking and machine tools.

The photo below shows an experiment I did with pultruded fiberglass rods through minimal control points to define a symmetric curve of least energy. The background chart is an Excel spreadsheet depicting numerical approximations of 1) an Euler Spiral and 2) a 'true' curve of least energy derived by BKP Horn. Also shown is 3) A CAD cubic spline through three points, which are coincident with the 3 v-wheels in the photo. The end conditions of the CAD spline are 'naturally tangent,' meaning the starting tangent vector is valued (0,1), the end tangent vector is valued (0,-1), and curvature = 0 at both ends.



The 'Horn' curve precisely follows the physical spline. CAD programs do not compute spline curves using Horn's method. I've attached a paper detailing my experiment, along with a PDF of the 1983 paper "Curves of Least Energy" by BKP Horn. I've also attached a copy of a 1962 monograph by R. Frisch-Fay that dives deeply into the math and physics of Flexible Bars. Horn's use of elliptic integrals mirrors Frisch-Fay's, to the point where I wonder if he could have written his paper without being aware of Frisch-Fay's work.

I have uploaded the Excel spreadsheet with a 'xls' extension and VBA macros enabled. If you start to open it, you will probably get a warning not to do so unless you trust the owner. If this is bad form, I will remove it.

 

Fantastic stuff. Of course in boat structure the minimum energy within the water flowing past the hull is the ultimate arbiter, but they may well be congruent. Well worth discussing.

 

Thanks, DogCavalry. I've mused about possible congruence of minimum energy curves as related to flow resistance too.

My guess is one of the easiest ways to start would be to look at foil sections in 2 dimensional flow, in a uniform fluid (just air or just water). Think about all the methods for developing airfoils over time. The NACA 4-digit series profiles are derived from fitting a 4'th degree polynomial to a series of wind tunnel models, with a pair of G1 continuous parabolic curves defining the camber line. The definitions are not directly related to fluid flow or airfoil theory like the 6-series. At some point the air or water doesn't care right?

 

Great approach and documents, I share with you this interest with the geometrical issues of a hull, especially the curvatures and their smooth evolution, for example that can be helpful for a strip-planking construction.
For the bottom line of a sailing dinghy aiming to go planing easily, I did that kind of approach, here below : in the first pdf attached to this post, I give the maths I used, and in the .ods file also atached (an Open office Calc) you can play with :
About the hull shape of sailing dinghies https://www.boatdesign.net/threads/about-the-hull-shape-of-sailing-dinghies.67577/#post-952220

... with the files also attached here directly :

 

Nice job there tropo, thanks!! Ever since late -nineties I've used the 3d spline function in CadKey and later KeyCreate for shaping critical flow elements, like turbopump inlet ducts. The idea is just that with the curvature of least energy, the flow losses can be reduced, to the benefit of efficiency and cavitation performance. Enclosed you find the initial curves defining the "floor" shape of the inlet channel for a double-suction turbopump. Flow is horizontally from the left into the impeller eye downwards, with offsets both to y (transverse) and z (vertical). The yellow line is the area centroid path.

But I also found that the end restraint "1" for the KeyCreate spline function does not give the correct shape. I often end up with something like ~1,25 or higher for maximum smoothness. The math behind the CAD splines is beyond me, but with your calcs and experiment in the background, I'll spend some time tonight to search for the restraint value needed to come close to the Horn curvature.



The spline restraints are linked to the straight lines at the respective ends of the splines. Each spline only defined by endpoints with their direction and a strength value to the director. Sometimes a geometrical restriction requires an additional restraint, but very seldom more than one. The lower half (dotted) of the suction flange leads to the other side of the impeller (mirrored).

 

Every day an education. I can feel the tensile strain on my geranium.(Australia rhyming slang for cranium)

 

"why spline curves in most CAD programs don't match the curves a piece of wood or metal takes through control points on a mold or jig in the shop"

Systems such as NURBS and subdivision surfaces are quite good in general, but there are some features that are valued highly that restrict or determine the parameters used for CAD. One is the ability to draw a circle with minimum control point count. Another is to handle crease edges and patch-add-delete faces. A third, typically not realized, is to be able to draw a pair of curves that are parallel in the sense a curved train track is parallel. A forth is the ability to handle irregular surface grids in 3D.

Both NURBS and subdivision surfaces can be parameterized to render circles exactly, and if you do so, they can both render exactly any surface that the other can render, which is handy for transportability. There are several problems common to both of these methods, though. Arc lengths and surface areas are ill defined by control points and depend on the subdivision scheme in subdivision systems. There are some fancy systems that dither the control points such that the produced surface lands on the user defined control points. Same with being able to specify slope and curvature constraints at control points (this requires higher order splines and iterative approaches with subdivision surfaces.)

For simple curves like with planking a lapstrake or carvel hull over frames, you can do the math for finding control point density as a function of curvature which approximates a low energy curve.

Related approaches can find the control point spacing along the chine and sheer curves for a developable surface plating.

Slightly different again is the control point spacing arrangement for stitch and glue panels. Here you want to space the holes such that the load at every stitch is identical and the only loads on the panel are edge loads.

 

In the old fairing rooms (sorry, I don´t know its english name), draftmen worked with what, in current CAD systems, are the "fit" points. No work was done with the "control" points. Using the same type of "points" you can bring the current splines closer to the battens curves.
(The energy needed to deform a beam has nothing to do with the energy needed to cut the water)
The draftman also had battens of various stiffnesses to attack one curve or another. A single batten was never used to, for example, draw a complete frame. Different battens had to be used but the two curves that passed through the same point must also have the same tangent. The operator's good eyesight saw whether the curvature was also the same or very similar.
Well, what I want to say is that, like any other activity, if you know how to do the work by hand, it is much easier to do it with a computer. We must force the electronic tools at our disposal to do what we need, not what they want, and to do this we also have to know these tools. In most cases, as the old draftman did, using two splines with the same tangent at their common ends is more practical than trying to draw a single curve or surface.

 

Tansl's last sentence reminds me of the other highly sought-after characteristic - so called "local support" - the idea that you can push a control point around and not change the entire curve, only the area between the adjacent 2 (or 4) control points in each direction. This property is inversely related to the order of the spline or subdivision surface. It's why an iterative approach over low order systems is often preferred to a higher order system. Note that a wooden batten spline does not exhibit local support.

<edit> with local support, Your "naturally tangent" (curvature = 0) end conditions aren't any good. To patch a modified curve segment smoothly into the middle of an existing curve, you want to match position, slope, and curvature at each end point. That is six orders of freedom just for the endpoints, plus however more you need to form the new segment as desired.

 

@tropostudio I'm deleting a few posts that were too great a thread drift. I was only able to work for a few minutes at a time during these posts, and they cut the thread up pretty badly. Some of the responses may want editing as a result. I'll try to get back on topic with future posts.

 

Terrible to confess, but my early exposure to curves in a maritime environment occurred in various dockside bars.

 

@baeckmo - Thumbnails of the KeyCreator 'natural' cubic spline with start tangent vectors (0, 1) and end vector (0,-1) from my PDF; and also with start tangent vector (0,2), and end vector (0, -2). Amazingly close fit in the latter to a Horn Curve...but I sure don't want to have to be software specific . Hope it helps.

 

Piecewise cubic splines are typically interpolating (point fit) splines. B-splines/NURBS are based on control points (a 'net'). Some software lets you create NURBS curves or surfaces via interpolating points on the curve or surface, but this involves 'back-figuring' to establish the control points. Stephen Hollister's ProSurf/Nautilus does this (great software, IMO).

For simplicity, let's stick to 2D/planar curves. I will not fault Horn for being an academic, writing an academic paper in 1983 (sorry, @philSweet). I'm just happy I could find it online! (I remember the days of microfiche and inter-library loans).

Will folks agree: if one springs a beam of uniform EI (like a pultruded GRP rod) through a set of simple constraints ( fixed location, but free to rotate), it will form a curve of least energy as described per BKP Horn criteria of 'curvature varies linearly with distance along the axis of symmetry?' I contend the results will scale. The difference in potential energy between Horn's examples of an Euler Spiral, an ellipse, a semi-circle, and the optimal curve are small. That goes to show how little it takes to change the shape of the curve with additional control points, some extra tension on the rod, a little tweaking of the batten with a fingertip before you draw the curve...

 

@philSweet - Good observation. In my PDF I mention the upper pair of v-wheels are irrelevant, and that the reader must trust me on that. The rod floats in the gap, A nudge up or down would have it explode out of the jig! (Safety first - I make exhibits that the public interact with.) Equilibrium is obtained strictly through opposing horizontal forces at the X-axis.

If the condition at y=o includes an upward component (nudge the rod/spline up) the tangents will converge below y=o. The curve above y=0 will 'close' to a 'lightbulb/inverted teardrop' shape as you mentioned. If the condition at y =o includes a downward component (nudge the cusp down), the curve will 'open' below y=0.

And @philSweet - FWIW, online papers I found that were most relevant to his topic were by undergrads looking at deformation of limbs and string angles in regard to compound bows. Very interesting and practical, if slightly divergent. They probably all have jobs now

 

The only design programme i could use was an old DOS type "Plyboats". If i recall in the manual, the choice of curves in that programme were based on wood splines. Too much interface and options even on the new stuff for me.