[Matrix Algebra?] I need to learn about regression / curve fitting. Resources? Tips?

Hi everyone, I have a great idea for a new type of software, but I need to do curve fitting based on pixels drawn by a person's mouse. By this I mean... well, resolution is finite on a monitor. I want to make software that lets a person draw a line with their mouse by holding the left click button and dragging the mouse. But, unlike Microsoft Paint, I need for this line to be a smooth, continuous curve. The reason is that I need for my software to afterward turn around and digitize at regular intervals over the horizontal scale, perhaps at intervals that wouldn't lie directly on a pixel, or perhaps at intervals that may lie on the same pixel. Such is the nature of drawing graphics on a monitor, which is a grid of discrete pixels.

If it matters, I plan to write my program in C# .NET in VisualStudio. It's honestly going to be my first bit of software, but I think it should be doable.

Anyways, I just need some help with how to regress this data. The line drawn won't necessarily be continuous, though it definitely will never be vertical. It may have extremely sharp turns / peaks as well.

I need to turn the pixel lines drawn by the user into fitted curves that match the input as close as possible, allowing me to sample that fitted curve to output data.