The secret to natural hyperbolic stretching is easy to understand if you grasp the concept of hyperbolic geometry. The idea is this: If you took a circle and transformed it so that the diameter of the circle increased by two times the diameter of the circle, that circle would not equal the same circle that we see on a piece of paper.
If you rotated the circle back to its original position, the circle you would find it, however, would look like the one we use in the real world. This is so because we understand a circle as a solid sphere that has no sharp ends or an angled radius. However, imagine we had come up with a special circle that was similar to the solid sphere we know in real life and with no flat end or an angled radius.
You can also think of the new circle as just a body with only one end. Imagine that you and I look at a body that looks just like the real world circle. We think that is exactly the same shape of the sphere we would see on a piece of paper, but our perception is totally different. We just don’t see a center point and we don’t recognize it.
Hyperbolic stretching is the exact same idea of rotating a circle so that it looks like a hyperbolic sphere. This is the same idea as simply letting the floor of a room to get the surface of a sphere. How does it all work?
The way we draw a sphere is by drawing it in an orthographic projection and normalizing it. For Hyperbolic Stretching, the centers of the spheres would lie anywhere on the circle. Just like any sphere, this new circle would have a center point where it would be equal to the center of the sphere it would be taken from.
We can easily determine this by the fact that we can turn our “planar” circles into larger area of a more uniform size. If we drew a small circle of the same diameter as the two previously mentioned circles on a flat piece of paper, it would be perfect as a circle.
However, if we use two circles that are more similar in shape to one another, we will end up with circles that have both smaller and larger diameter than the first. This is how hyperbolic stretching works.
Hyperbolic stretching is actually very similar to stretching our fingers. For example, think of it like our fingers are a curved stick that we can keep hanging and stretched.
If we make a fist, hold it up and imagine that it is stretched like a sphere, we get something similar to a baseball bat, except that the hands can now hang from the wrist like a baseball glove. Using this idea, stretching a child’s finger is the same as stretching a football stick.
The same thing can be done to a person. Imagine that you are stretched out like a beach ball and imagine the last few inches stretched out. Using this same principle, stretching the person can be done like stretching a baseball ball.
If you look at the hyperbolic shape closely, you will find that it actually contains two smaller circles. Just like that person you held up by your hand could hold you up with two different hand positions, stretching a small circle would not only stretch a small circle, but two smaller circles would be forming. Hyperbolic stretching is the perfect way to stretch out a person.