Joining two circles/cylinders in openscad 1: Geometry

Warning!

This article has more to do with geometry than IT in general. You might possibly be scared of it, and I think that repulse to math in general and geometry specifically is a valid reaction. and I would suggest you to skip to the next posting. Or you might want to go on if just for the pretty pictures. That suits me fine also; you are the one with decider here.

You have been warned!

Recently I decided to learn how to use openscad. As the name kinda implies, it is a (open source) program to build 3D solids. Main difference with the usual suspects is that it is not interactive: you create a file using your favourite text editor (or sed for that matter) and then compile it. That might sound a bit annoying but it can be quite convenient if you are doing engineering objects as you can define the different object that make it as separate functions, which can even become libraries. However, it does not have some features you expect to find in such programs, like fileting. They are still working on providing a official solution to that, but as of right now it ain't there.

Now, sometimes you can go around these limitations by using a bit of geometry. Let me show what I mean by using a simple example: Let's say I want to build a bracket that will hold two pipes together (think chainlink fence gate) that looks like this sketch:

For now we will ignore the pipe holes and look at the external shape of the object. Since we are going to have two circles internally, it would be nice if the ends had a circly feel to it. So, we shall start with two circles around the pipe holes. To make it look pretty, and easy when we do get to build it, how about if the pipe holes are 90% in diameter of those circles we are using to make the body? You know, something like the picture (done in inkscape without making any justice to its potential) on the right.

So we have the ends of the body, but what about the sides? We need to connect the two ends somehow.

The easiest thing I can think of is to use those lines representing the radius of each circle, which happen to be both vertical. We could just connect them on each side (showing just the upper side to keep figure clean and since the object will be symmetric with respect to the horizontal, henceforth called X, axis).

I don't know about you, but even though it might work around the larger (right) circle, it would look odd on the smaller (left) circle: the surface would become parallel to the X axis just before going vertical and then it would make a sharp turn and start moving away of this axis. It just looks like it was put together in a hurry, and I think we can do better.

And how do you plan on doing that? you might ask. Or not. But this is my article so I will answer that question. Our goal is to have the side to transition gently away from the circle. I think we are looking for some kind of curve that started tangent to one circle and then, without much ado, end tangentially to the other circle. Kinda like the picture on the left, with some simple gentle curve in the middle. That does look pretty, but it requires us to determine where it will move away from each circle. In other words, we might need to do some fiddling.

I am lazy, so I would like to keep any fiddling to a minimal. Therefore, I want a solution that makes more work for the computer and less to me. I therefore need a curve that I will provide as few parameters as possible and it will still end up nice and pretty. Euclidean Geometry has the answer: the easiest way to come up with a curved surface that touches both circles tangentially mathematically with the minimal amount of input is to use a third circle.

Here is what I have in mind in a drawing (most of the green circle has been omitted so we would not focus on it): the green circle touches each of the other two circles tangentially as we want. I know the curve in the drawing does not look as nice as the one in the previous drawing, but where it touches and the shape of the curve it creates between those points is determined solely by its radius. Specifically, the bigger the green circle is the closer the line between the blue circles will be to a straight line. The other required information is determined by the two pipes we mentioned before, namely:

1. Radius of the two blue circles
   
2. Distance between the two blue circles.
   

So far so good, thanks to geometry. Let's see if we can continue relying on it and come up with equations that will make our life lazier. First thing we will need is some lines connecting the centers of the three circles (as I mentioned before, I am ignoring the other side due to the symmetry the object has) together. As to be expected, that defines a triangle of some arbitrary (we did not specify any relationship between the radius of the 3 different circles. We will revisit that later) shape, which in our little drawing is represented by the black triangle.

Before we continue, we must do some labeling. So, the centers of the 3 circles are called A, B, and C, which as we know are the vertexes of the triangle ABC. The sides of the triangle are labelled a, b, and c where b is opposite to B and so on. We can also name each circle after its center, so circle C is the green circle. Distance between C and side c is h.

From the picture we can also figure out the radius of each circle is being named and that a is tangent to circles B and C (and b is tangent to circles A and C). Finally, h divides c in the segments c1 and c2 and creates two right triangles. With those parameters defined, we need to start defining a few equations.

First we restate the relationship between the sides a, b, and c and known quantities:

Well, we really do not know c1 or c2, but can find them by applying Pythagoras on the left triangle:

Since we have c2, we can find c1. So, now we can locate the circle C. Next time we will put all these findings in openscad and then do some doodling.