tinkerpad | A little browser-based LOGO interpreter | Graphics library

I want to know how you can achieve modifying 3D object's shape (vertices) and creating a new shape, like let's say changing a part of its height or making a hole on it by programming with a user's interaction like a mouse movement. I put the tag WebGl but I want to know general ideas how to achieve it in a 3d world.

If you don't get what I talk about it, please refer to tinkercad. You can modify 3D object's shape with it.

I have no clue how it's achieved, so any info for it would be appreciated.

EDIT:
This question is similar to mine. But the answers cannot achieve to cut a shape or such stuff, I think.

EDIT2:
OK, so I get an idea from the AutoDesk doc below for changing a shape. To change a shape by moving a point or edge, I need to let a user pick either one of them then change its coordinates as the user moves their mouse.

To cut or clip a shape, I guess I need to split or create triangles where intersections happen with other shapes for achieving the same feature which tinkerpad has.