Hey, what's up, today is the day that you're going to understand curvature continuity in a way that's intuitive, and memorable and simple enough to explain to a five-year-old [phonetic 00:10]. Let's get to it.

The different continuity types show up all over the place in different features. For example, here in the 'Loft feature', 'Match tangent and curvature.' Inside of the 'Fillet feature', you have some different options, including a curvature continuous option in the 'Fill feature'. If you grab an Edge, you can see we have Position, Tangency, and Curvature. But to explain the differences between them, I think it's much easier shown than told. The most intuitive way that I know to imagine and understand these curvature continuity types is to imagine trying to drive on your curves with a car.

Conveniently, the steering wheel directly correlates to the curvature at any given point on your curve.

I made some animations to show a couple different examples of different types and what that would be like to drive on with the car.

The most basic example that I can give this is just driving in a straight line, we all can understand that there is no curvature, and the wheel is just straight and does not move.

Similarly, if we're going to drive in a circle, the steering wheel still isn't moving, but it's in a different position. So you can see from the graph on the bottom right that there is more curvature.

Here's where we start getting into the actual continuity types. So we have two straight lines connected by an arc, the arc is tangent to the lines, but it is not curvature continuous. And it becomes pretty obvious when you look at the steering wheel that this is not a smooth curve. In fact, this would be an impossible combination of curves to drive in a moving vehicle, the wheel has to teleport to achieve an arc and then teleport back to achieve a line. So the difference between the surfaces while they are technically smooth, is not as smooth as it could be.

So if we want a smoother transition between the two curves, then we need to go up one level of continuity. And so here you can see that we're drawing more of a triangle on our graph, because the amount of curvature at the point where the two-- the line and the curve meet is the same. So they have the same curvature at the point that they see the thing [phonetic 02:20]. But you can still see that it's not as smooth as it could be. And in fact, even this would be impossible to drive perfectly mathematically correct in real life, because the wheel itself can accelerate instantaneously. That said, this level of smoothness is totally enough for almost everything that I do, and is in fact the highest level of curvature that Onshape provides internally.

Certain applications can require even higher degrees of curvature usually things involving either high aesthetics and very smooth surfaces like automotive surfacing, or even things that require heavy things to move fast like a train. In order to change the direction of something that's moving, you have to ease in and out of your turns. So in this case, you can see that our steering wheel doesn't move robotically, it doesn't go from zero RPM to one RPM immediately. It accelerates and decelerates. Because this is how cars drive in real life.

This is also how roads are designed in real life. If you start paying attention, you'll notice that on roads with a high speed limit, the curves are much smoother, and the easing in and out is much longer. And on slower roads, you can get away with lines and arcs.

So curvature is not just for industrial design, folks like me to make stuff shiny and pretty. It's also really practical. If you design anything that involves moving stuff around, you might consider a curvature continuous path to ease wear and tear in your system.

Okay, let's go look at some examples in Onshape on how the different continuity types play out on a real model.

So this first example is just positional continuity, which just means an Edge, that's not smooth at all. There's a break in[phonetic 04:07] surface. And you can see from our curvature visualization tool that the reflections don't touch, I mean they they're not related to each other. So if we add a tangent continuous object here, we can see that they're going the same direction where they touch but the curvature of the surface jumps to meet this arc, we have zero curvature and then you know exactly a fixed amount of curvature here.

So if we evaluate that with the curvature visualization, you can see that the lines touch but they break and shoot off to the right. And this visualization, by the way, is as if we just had 100% chrome cube inside of a zebra striped environment. So this is what real reflections in real life will do on your part if you use a tangent continuous relation here.

So taking it one level up, I included a conic here because it's included in the Fillet feature. And I actually really liked to use conics a lot. So this one is actually still just tangent continuous. But because the conic itself does ease in and out a little bit more, you can get something that's a little closer to curvature continuous without. So you can see, as I'm cranking up the road, while you're here, it's getting a lot smoother.

So there are kind of different levels like you can have a tangent that's nicer than another one. So conic is kind of almost halfway in between. But if we go with a true curvature, continuous thing, you can see that our lines now transition just flow across the form really nicely. And again, real reflections in real life will do the same thing.

Another way to understand and evaluate your curves, and their continuity is to use the curvature graph. So if you select any two curves or edges, right click them and go show curvature, you are presented with this sort of cryptic looking diagram. But what it really means is, we can hide the minimum radii. And we don't need to see the inflection points. So these are our curvature combs [phonetic 06:28], curvature graph. And what this is showing us is the amount of curvature at any given point on our curve. So these are both parks, so the amount of curvature doesn't change. And so you can see, every line that's coming off is the same, this arc has less curvature, and so the lines are shorter, and this one has more curvature, so the lines are bigger.

In here, right at the connection point, you can see that the lines are pointing different directions. So that means that we have positional continuity right here, this is a sharp edge, who would form a crease if we extruded it, if I deselect this and grabbed this one. This is what a tangent continuity looks like. So again, it's just another arc. But you can see that right here, where they meet, they're going the same direction, the curve normals are the same. I threw another conic in here again to show that it's still going to be tangent, but it's better tangent than this, because the difference between them is smaller, so it's going to be less noticeable. And you can also start to see here the radius changes across the conic. So you can see that here-- right here on the curve is the pointiest part of the curve, let's say and then for good measure, I threw in a curvature continuous spline and you can see that here is how we can tell that it is curvature continuous, it meets at the exact same point, the graph doesn't break now it can have an elbow there and that would show that we have just plain of [phonetic 07:59] curvature continuous and if the graph itself actually flows smoothly across then you have a higher degree of curvature.

So hopefully that helps you to understand curvature in a simple way that you haven't before. If you have any questions about this or if I missed something, please let me know in the comments. If you did learn something, hit like, if you want to see more content like this, subscribe and hit the bell so you know when I launch more content. Thanks for watching!