I'm trying to make a curve longer, while satisfying these conditions:

1. It has to remain smooth.
2. Its shape shouldn't change. In other words, all points on the original curve should be on the new curve too.

I'm not sure if it mathematically makes sense, but the gut feeling tells me it should be possible.

Any idea?

Method 1: The points at either end of the curve take reference position from their adjacent point, then extrude in the same direction to extend the curve in 2 straight lines.

Method 2: Calculate change in curvature of multiple "previous" points, using dot product, then extrapolating extension points' position based on some function of that curvature (average, min, max, derivative, etc.)