How to store Drag Input(Irregular shapes included).

I am making a mobile game which takes drag input from a user(any shape which can be made by dragging one finger on the screen) and i need to store it for later use...I know that I can store it with a list of points but I need the curves to be smooth...making it smooth will require a lot of memory and will make the game sluggish, that's why i need another method...Also, i need to access specific positions from that line as well-Ex-Placing an object on the line at position 3.5 units from the start.(The shape can be irregular so LookAt or MoveTowards with the end would not work).Any help is appreciated..Thanks