Geometry studies transformations of space, and these curves arise as a result. A simple movement in a fixed direction such as driving along a straight road is an example, where the vehicle is being transformed by what is called a translation . In our mathematical imagination we can think of the whole of space being transformed in this way. Another example is rotation about an axis. These are simple examples of path curves.
More complicated transformations give rise to more interesting curves.
They allow more freedom than simple rotations and translations, in particular incorporating expansion and contraction. Apart from the path curves they leave a tetrahedron invariant in the most general case. George Adams studied these curves as he thought they would provide a way of understanding how space and counter space interact. A particular version he looked at was for a transformation which leaves invariant two parallel planes, the line at infinity where they meet, and an axis orthogonal to them. This will be recognised as the type of curve lying in the surface of the egg at the top of the page.