The Script of Outside In (part 5):
How to turn any curve into any other of the same turning number

Y: Still not allowing sharp corners, right?

X: Of course. Now can the circle be turned into any curve of turning number 1, say this one?

Y: Let's see: I'll try to go backwards from this curve to the circle... I think I got it. There.

X: Excellent. Now try this one.

Y: I'll undo this loop first... and push this fold back... Now here... Here we go.

X: Very good! And this one?

Y: Whoa! You're not going to ask me to do every single curve of turning number one, are you?

X: Of course not. What we need is a general method. Do you remember the simple way to transform one curve to another when sharp bends are allowed?

Y: Yes. You just go straight from one to the other.

X: That's the one. When the curves have the same turning number, this method can be adapted to work without sharp bends. The trick is to add waves to the curve.

Y: Can we do it on a simpler one?

X: Sure. We start by marking small pieces of the curve that will serve as guides for the transformation. We'll concentrate on these segments now. We move the centers of the guide segments straight toward their final destinations on the circle, without any rotation. Next, we rotate the guides so that they are lined up with the circle.

Y: OK, what about the parts in between?

X: That's where the waviness comes in. We make the connecting segments between adjacent guides bulge out into corrugations. This allows the segments to move freely around each other, as long as they remain more or less parallel.

Y: Oh, I see---the guides can move around without creating sharp bends.

X: Correct. Here is the transformation of the whole curve.

Y: The original curve, in blue, develops sharp corners, but the wavy curve is springy enough to remain smooth throughout.

X: We have to keep adjacent guides roughly parallel as we rotate them to align with the circle. This is possible as long as the turning number of the original curve is one.

Y: Why can't we align the guides if the turning number isn't one?

X: Watch what happens when we try to turn a figure eight into a circle. ... And here both the initial and final curve have turning number zero. Using this method, or others, you can always transform one curve into another with the same turning number. This is called the Whitney--Graustein theorem.

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