Spline Schmline, part 1
Splines make some people squeamishly uncomfortable. You can’t put dimensions on them, so how do you manufacture it? How do you make parts fit together if you use splines? How do you run QC to check your parts?
There are a lot of misconceptions about splines. The first is that you can’t put dimensions on them. And the second is that if you put dimensions on them that they become fully defined. The Third turns out to be that if you fully define all the points on a spline, that you can’t drag the points around anymore. Each busted misconception in its own time…
I have good answers for each of these misgivings and misconceptions, but I’ll get to those later. I want to start by addressing why splines are so important, infact indispensible.
Great curvature, baby!
Ok, this is going to sound sexist. That’s because it is. So sue me. If you’re male, there is no way to pretend you don’t notice this kind of stuff. Some types of curvature just draw your eye. That’s all there is to it. Just to keep it kind of cartoony and not too explicit, let’s use Jessica Rabbit from the movie Who Framed Roger Rabbit as an example, because she’s a great example.
Splines are sexy. Jessica just wouldn’t be the same if she were just straight and round, drawn with lines and arcs. Jessica, and just about everything else that we think of as “products”, needs curvature, and in particular, changes in curvature.
The reason why splines are so necessary is that they are the one way we have to make shapes that look organic. Organic shapes are never perfectly straight or perfectly round, they are constantly changing curvature. We as humans are programmed to react to organic shape. Jessica Rabbit’s organic shape provokes a reaction, and product design tries to evoke the same kind of reaction. It’s not a coincidence that they both cry out “empty your wallet”, and we dutifully obey.
If Jessica has an equivalent in the automobile world, it is probably something like a 1960 Corvette. Yow. They both look great in red. Nice headlights! Love those rear fenders. How does it look with the top down? It’s no coincidence that product design is so sexual. It’s no coincidence either that they are both beautifully curvaceous.
Splines or lines and arcs?
Cool shapes in product design is all about curvature that changes smoothly from one radius to another rather than simply a series of tangent arcs that change from one discreet radius immediately to another 
discreet radius. Look at these two surfaces below. Which one looks better to you? The one on the right is less lumpy. I think most people would say the one on the right looks better. Of course the one on the right is made with splines. You can almost see the lines and arcs in the one on the left.
Of course it is possible to make splines look very bad if you are trying to do that or are very inexperienced. I think a lot of people make the mistake of thinking that a spline with more spline points is more controllable, and thus a better spline. The truth is that you DON’T want to try to force a spline. The thing about organic curvature that makes it so appealing is that it looks natural. The natural materials are bending or being formed in a natural way. Splines have a very natural way of bending. I’ve been told that the math governing how splines interpolate shapes between points is very much like the natural bending equations for elastic materials.
Interpolated shapes
I classifify shapes into two camps: analytical and interpolated. Analytical shapes have a specific equation that can be written, for example a line or circle each has a familiar equation that you may have learned in high school geometry. A spline I believe is a series of 2nd or 3rd order polynomials. With an analytical shape, you know exactly where every point along that shape is going to be even before you draw the line or the arc. With a spline, you just put down the control points, and SolidWorks interpolates the curve between the points. This concept follows through to features such as the boundary, loft, fill and sweep. You put down the profile curves, and SolidWorks uses the Boundary feature to interpolate the surface between the curves. The difference between analytical and interpolated curves and surfaces is very important. They are not interchangeable.
Arguments against splines
People who don’t trust splines typically talk about dimensions or repeatability. When I create splines in SolidWorks, I never fully dimension them. Its useless to dimension splines. Different CAD packages evaluate splines differently, even if the spline points are all in the same place. Even SolidWorks between releases evaluates splines differently. Even if you were to use a spline on a drawing and fully dimension all of the spline points, the spline could never be exactly recreated based on those dimensions.
The only way to transfer spline based manufacturing data is to send a completed 3D model, and for the manufacturer to use a computer controlled method. Paper drawings may be useful for notes, and reference dimensions but typically not as a way to specify a complex surface or curve. This is most of my business. I don’t remember the last real drawing I gave to a molder, I just hand off 3D data.
Practice with splines
See the spline here with a lot of points looks lumpy. When I first started working with splines, I thought this was the best way to get great control on a shape, but it turns out that the more you try to control the shape, the lumpier it gets. Notice that the second spline is much smoother. This is because you just let the spline math do the work for you. My general rule of thumb is that you add a spline point for every change in convexity (convex up changes to convex down). So you have the two 
endpoints, which would make a straight spline, a middle spline point, which would make a convex or concave spline, and then a third point which gives the undulating S shape.
In the second set of images, I have applied a curvature comb to the splines. This is a way to visually evaluate the spline. The curvature comb represents the instantaneous curvature at that point along the spline. If the curvature comb changes sides of the spline, that means the spline has changed convexity. Notice that the comb on the top spline confirms that it is lumpy and the changes in curvature are too abrupt. The changes in curvature in the lower spline are much more gradual. Changes in curvature are represented by the change in the height of the comb.
Just for reference, curvature is the inverse of radius, or c=1/r. If you have a large radius, it means small curvature. Small radius (tight corner) has large curvature. Notice that the curvature comb is taller in kinks of the top spline. Notice that in flat areas of the spline (almost linear) the curvature comb is almost zero height.
More to come…
We’ll talk more about this topic later. There is a lot to know about splines.
As an industrial designer, I’ve not been filled with all that calculus stuff that fills engineers during college. But as someone who none-the-less attempted engineering before discovering industrial design, I’ve had a little. What I find interesting is the relationship between sexy curves and the little I understand about the math governing.
A couple of months ago, I suddenly came to an understanding of curvature “continuity”, and perhaps why such a thing has such great sex appeal. This happened while driving home on our rural road–lots of great curves in this road. In the city, many such curves lack this curvature continuity–they are made up of lines and arcs, and attempting to drive such roads smoothly can be difficult (without swinging lazily in and out of prescribed lanes). But it’s quite simple with our rural road (with the exception of sudden swerves to avoid deer). Why is that?
I noticed that if I were to perfectly drive in the city, the transition between straight line and circular arc–though tangent–is a sudden change of acceleration (x^3, right?). To perfectly drive this curve would require an instant change of the steering wheel from straight, to whatever matches the radius of the road–jarring. This same, jarring transition shows up in the connecting of lines and arcs in the examples above, and really blow the whole sex appeal in product design. However, my country road allows smooth driving with smooth (analog) transitions of my steering wheel from one position to another, constantly adjusting for the curve of the road. The change in acceleration is not jarring, but gradual and “continuous”. If graphed, we wouldn’t see stair-steps in the acceleration forces as we would if making a tangent transition from line to arc.
I’ve found it quite interesting to discover relationships between various areas of nature (mathematics, industrial design, “sex appeal”, and driving). Perhaps reality itself has continuity.
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Jeff,
it’s great to see comments as philosophical as my posts. It’s interesting, I’ve heard Kieth Pedersen speak on the similarities between spline curvature and the path of a car before. It really is true. To make a car go in lines and arcs, you have to completely stop at the end of a line or arc and turn the wheel.Splines are the natural product of turning the wheel while driving.
Anyway, with calculus velocity is the first derivative of a position equation. Accelleration is the second derivative, and the third derivative is called jerk, or change in acceleration. Geometrically, the first derivative is slope, second derivative is curvature and the third derivative, well, I’m not sure what it is called, but I suppose you could call it “kink”. SolidWorks doesn’t let us go that far with it.
Thanks for the comment!
GREAT post, Matt. I’ve been wanting to read about this type of stuff for a LONG time. I’ve always asked that same question, “How do you define a spline in a drawing?” So far you’ve given me some really good answers. Unfortunately, one of them seems to be, “Well, you don’t need to define them as long as your suppliers have full 3D CAD/CAM CNC capabilities.” And unfortunately, many industries still are nowhere close to getting there. Still, it’d be interesting to hear what things you CAN do to try and use splines to define your product geometry, and try to make it work on the manufacturing floor.
There have been MANY times where I’ve wanted to use splines to define certain geometry on some of the components I’ve design. Sometimes I’ve been able to get away with it. Other times I’m met with the “what the heck is that” response from suppliers, and have had to redefine it with arcs/lines to the best of my ability.
Again, excellent post. Look forward to reading more.
Thanks, Matt. Nice to have something other than rolling eyes from someone regarding this tragic hang-up in my personality (finding universal connections in life).
I’m also glad you’re tackling this spline issue here. It’s interesting to see how often people want to hold to a rule like “always defining all sketch elements” when confronted with a fluid entity like splines. I found out long ago that some vendors simply cannot cope with the designs I give them. So? I find those who can, rather than compromise my design. I don’t often design beige (or black) boxes, so not everyone will be able to manufacture what I design.
Jessica Rabbit- -man, that was perfect!
Thanks for the great blog post. I always like learning about the why behind the how. My simple rule of thumb regarding splines is: Never use three spline points when two will do.
DanO
Interesting article, I hope is the beginning of other technical items.
One thing I regret is your little presence in the SolidWorks forum.
I want you create topics on every SolidWorks area on improvements you’d like and would like users, as is been done on the Drawing.
For example, the command flex seems interesting, but little documented, as well as the command free shape.
Do not tell me you’re tired of fights … other than real….is the only way to bring in line the SolidWorks Corp..
See you soon,
Alessandro
Matt, I truely love how you express words. You said once before that you didn’t have the full capaibilties of a true writer, I think that is bunk. You have the ability to lead us through a message and come out in the end with an understanding of the topic. I love how you bring in the real life (or cartoon in this case) situations and describe what we normal humans take see but can’t express fully.
I remember taking a “Free-Form” modeling class when I was using UG. UG could express curvature very well but the writer of the book made it sound difficult and hard to do. You make people want to use splines, now teach me a way to make an “L” bracket with splines.
Remember, Jessica wasn’t bad – she was just drawn that way.
Steve
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Steve,
I wish you worked for a publisher. Some real writers don’t think I’m a real writer, and who would I be to contradict them? Thanks for the sentiments. I love to try, even if I’m not a real “pro”.
“L” brackets with splines! You might be surprised. I actually could do that, and have honest to goodness real engineering reasons for doing it.
Yes, I wish I had remembered Jessica’s tag line. That’s a great one.
I don’t understand why you wouldn’t fully dimension a spline, at least a two-point spline. Yes, when you first draw a two-point spline, it is a straight line. If you select it in the sketch, you will see arrows show up at those end point. If you dimension from an external line (say horizontal or vertical) to the “diamond” on the arrow, you can control the angle at the beginning and end of the spline (you could even make it tangent or normal with a relation. Then apply dimensions to the length of the arrows by clicking on the arrow head. Now you can change the curvature at the start and end of the spline. Again, a relation could be added to make it curvature continuous to another spline or arc.
So, you can get the same curvature as the 4 point spline (or similar) with only 2 points.
I am glad to see this topic coming up and being discussed.
-Eric
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Yeah, don’t worry. I’ve got more than just a 600 word essay to say about splines… Read the SolidWorks Surfacing and Complex Shape Modeling Bible to read more about it…
Steve, if you use the Fit Spline feature, you might get much more true-to-life L-bracket models (should you ever need them) since the spline can better fit the true nature of bent metal like that. The tolerance setting will determine how tightly the spline matches your lines and arcs, but otherwise can flow a bit more like I’ve seen many L-brackets in manufactured form. (Not sure WHY you’d want to do that unless you need extremely fine analysis of your model or something…)