Math Demonstration: Dome of Viviani
What I love about this demonstration is the simplicity of the shapes. With only a half sphere and a Cylinder it manages to make something that looks really beautiful. First let's look at in 2D...
If it looks boring it's because it is. All this is showing is the top view of a half sphere being intersected by a cylinder. What will happen is that the cylinder will make a pivot point on the origin until it reaches the point where it started.
The gifs above show what happens when the cylinder intersects the sphere 4 times. As we increase the number of intersections the sphere becomes more and more blue. If we look at it based on angles then the angle between each cylinder intersection becomes smaller as the number of interstions becomes higher.
As you can see at 32 intersections my computer begins to struggle.
Challenge
I want to find something out. How would you calculate the area of an intersection between a sphere and 2 cylinders with based on the number of intersections? In other words a General equation. To be honest I'm not sure what kind of formula I would use spherical coordinates or cylindrical coordinates. Good Luck!!!