Point cloud

Consider this object:

It is in fact a triple torus.

But what if we zoom in:

We realize that the "object" is just a bunch of points suspended in space! It is called a point cloud.

Another example:

Where do point clouds come from?

They come from scanning:

They come from data:

Usually a simplicial complex is created to tell what's behind the point cloud.

How? A simplistic approach is to choose a threshold r and then

• add an edge between any two points if they are within r from each other,
• add a face spanning three points if the diameter of the triangle is less then r,

etc.

To capture all r's and to deal with noise, persistence is used.