Possible to draw an arc (semicircle) between two arbitrary lines, without "bulging" outside the lines.

Question

So you have two lines that need a connection that is a semicircle, or arc.

A    B

|    |
|    |
|    |
|    |

Easy. Focusing on just the bottom portion, you can do a rounded half-circle to join the two ends of the lines (end of line A joined to end of line B). Then there are perpendicular lines:

|
|
|
|    
          _______

This is also easy, it is just a quarter circle. Just make sure to extend the lines so they have the same distance from the meeting point. So the above would look more like:

|
|
|
|    
. _______

Here is how the arc might look. Notice the arc is tangent to each line.

Then there are diagnoal lines (arbitrary lines):

|   \
|    \
|     \
|      \

Or:

|      /
|     /
|    /
|   /

Or a million other diagonals.

The first part of my question is, if it is possible to draw any portion of a circle (to draw an arc) between any two lines, such that there is a smooth transition. By that I mean, the arc dosn't extend beyond the boundary of both lines, and is tangent to it where it intersects with each line.

For example, it seems like you could make these attach with an arc of some sort, but I am not sure of it. Just extend it a little bit, and maybe it would fit.

|      /
|     /
|    /
|   /
   /

Then:

|      /
|     /
|    /
|   /
 ‿ / (but tilted semicircle)

But that wouldn't be quite right because it would bulge outside the lines.

The second part of the question is how to tell how much to extend the lines in order to connect them with a semicircle. In the perpendicular case, you can say that both line ends need to be equidistant from their meeting point. But for the diagonal versions, I'm not sure what the general pattern is.

Answer 1

The key to the pattern? The angle bisector. Put the center of the circle on the bisector of the angle formed by the two lines, and it'll work out.

(I drew the whole circle, but it's easy enough to cut that down to an arc)

The case with two parallel lines is the only one we need to treat as special; in that case, there is no point of intersection, and instead we put the center on the mutually parallel line halfway between the two.

Answer 2

Yes, for any two lines you can draw a circle that transitions smoothly from one line to the other. In fact, any circle tangent to both lines would do the job.

How to build such a circle?

First, you need to cross the lines. Build the angular bisector of the angle between them. The center of the circle would be on this angular bisector.

Then choose one of the lines, say $l_1$, and pick a point $A$ on it where you want the arc joining the lines would start. Run a perpendicular to $l_1$ though $A$ and intersect it with the angular bisector in point $O$. $O$ will be the center of your circle.

Now that you have the center, $O$, and a point on the circle, $A$, you can draw the arc joining the lines using a compass. It will smoothly connect to the second line $l_2$. The point where it connects to $l_2$ can be found by drawing a perpendicular from $O$ to $l_2$.