And it's not just circles, either.

Pi

I know how to calculate the area of a circle.

Do it mathematically correct.

Center Of Curvature

Center of Curvature of This Curve

Osculating circle of this curve

Center of Curvature of This Conic

Center of Curvature of This Cubic Curve

The center of the osculating circle to a curve

Of course not.

Let's see if we can learn a thing or two about the hyperbola.

And then I'll do another video where I actually shift the hyperbola.

OK, that's interesting.

My volume is going to be 4 3...And then I don't want to just put a pi there because that might interprit it as 4 divided by 3 pi.

What about bigger numbers?

Pi's just a number.

He's reciting the digits of pi.

In the case of positively or negatively curved ones, there is a crtuch.

Well, this rear tire is actually going to travel along a smaller inner circle while this tire is going to travel along a larger outer circle.

There you go.

OK, this is a neat property of circles.

With that out of the way, let's just apply this radius being 7 centimeters to this formular right over here.

So we're going to have a volume is equal to 4 thirds pi 7. cm 3.

Then they find out that the diameter is roughly 2 centimeters.

I'll just write it like this, the probability of ... that the point is also lies in the smaller circle .. so all of that stuff I'll put it ... the probability of that is going to be equal to the percentage of this larger circle that it is the smaller one so that's going to be... well... the fraction of the larger circle's area that is the smaller circle's area.

The first equation I want to talk about is this one.

And then he takes a wire hanger and twists it around...

Center of Curvature at This Point

And this was a challenge now thrown open to the entire computer vision community.

So the probability that a point that we select from this larger circle is also in the smaller circle is really just a percentage of the larger circle that is the smaller circle so our probability...

And it amazes me that this and it never repeats.

In this painting that I made of the first 20 decimals of Pi,

And when I say we, I really mean the computer program.

Osculating circle of a curve at a point

What I had written before, with a plus, that would have been an ellipse.

But assuming that we're dealing with a circle, an ellipse, or a hyperbola, there will be an x squared term and a y squared term.

So our new ellipse looks something like this.

However, Pi is much more than that.

Since Archimedes, people all over the world have been obsessed with calculating Pi.

In fact, calculating Pi was used to test proper functioning of early computers.

I'm obviously passionate about the number and what we can do with it.

And we'll see five years later what turns out to be a good idea or not. So here is a example of a precision inaudible .

I know what a parabola is.

(Laughter)