The tySlice modifier has a cross-section mode you can use to slice up an object similar to your images, but it doesn't rely on intersections.
The Birth Intersections operator just finds intersections and randomly scatters particles on them...you can't really use it to analytically extract intersection splines.
There's not really a straight-forward way to extract intersection splines that I can think of...closest thing would be performing a boolean surface subtract operation between your pieces of geometry using the tyBoolean modifier, then using a tyMeshToSplines modifier to convert the remaining open edges to splines. But that wouldn't only work as intended if the source geometry being sliced is a closed surface.
The Birth Intersections operator just finds intersections and randomly scatters particles on them...you can't really use it to analytically extract intersection splines.
There's not really a straight-forward way to extract intersection splines that I can think of...closest thing would be performing a boolean surface subtract operation between your pieces of geometry using the tyBoolean modifier, then using a tyMeshToSplines modifier to convert the remaining open edges to splines. But that wouldn't only work as intended if the source geometry being sliced is a closed surface.