Some basic info on curve parameters:

The curve *domain* is a non-zero-length, increasing numeric interval. An interval is delineated by two extremes and these values could be anything, as long as they aren’t the same and as long as the domain start value is less than the domain end value.

The curve ‘exists’ everywhere inside this domain. The first (i.e. smallest) value represents the curve start point. The last (i.e. largest) value represents the curve end point. Any value (or *parameter*) in between represents a point somewhere on the curve interior.

When Rhino makes curves it typically picks a domain from zero to the length of the curve. There is no stringent need for such a choice, except that many algorithms ‘like’ it when the physical size of a curve is roughly equal to its parametric size. When a curve is scaled, its physical length will change but its domain will remain the same, so it is entirely possible to end up with curves that have a wildly different length and domain.

It is possible to *reparameterize* a curve, which essentially just assigns a new domain to it. This will neither change the curve shape, nor its derivatives or control-point locations, or greville-points or what have you. The domain of a curve can be freely chosen. One common action is to assign the *unit interval* (0,1) to a curve domain. This makes certain mathematical operations easier to write down, and this is what the ‘reparameterize’ option in Grasshopper does.

Certain special types of curves have an intrinsic domain. Lines are always (0,1), circles are always (0, \pi), arcs are always (\alpha_0, \alpha_1), where the \alpha's represent the start and end angles of the arc.

However even lines, circles and arcs can be embedded into curve types whose domain is adjustable.

Parameters are usually the way we choose to address curves, because they are fast and unambiguous. If you were to use 3d-points for example then it wouldn’t be clear exactly *where* on the curve you meant for something to be measured if you happened to pick a point where the curve self-intersects. Or a point far away from the curve. Similarly, using lengths or percentages–albeit unambiguous–is counter to the way curves are mathematically defined and it takes a lot of CPU heavy lifting to figure out exactly where along the curve 25mm really is.

To recap, it is perfectly fine to have negative curve parameters. The curve domain may be *entirely* negative, partially negative, or entirely positive, provided it has a measurable length.