Geometric curve properties Since curves are geometric objects, they possess a number of properties or characteristics which can be used to describe or analyze them. For example, every curve has a starting coordinate and every curve has an ending coordinate. When the distance between these two coordinates is zero, the curve is closed. Also, every curve has a number of control-points, if all these points are located in the same plane, the curve as a whole is planar. Some properties apply to the curve as a whole, others only apply to specific points on the curve. For example, planarity is a global property while tangent vectors are a local property. Also, some properties only apply to some curve types. So far we've dealt with lines, polylines, circles, ellipses, arcs and nurbs curves:
Line
Polyline
Circle
Ellipse
Arc
Nurbs curve
Poly curve
The last available curve type in Rhino is the polycurve, which is nothing more than an amalgamation of other types. A polycurve can be a series of line curves for example, in which case it behaves similarly to a polyline. But it can also be a combination of lines, arcs and nurbs curves with different degrees. Since all the individual segments have to touch each other (G0 continuity is a requirement for polycurve segments), polycurves cannot contain closed segments. However, no matter how complex the polycurve, it can always be represented by a nurbs curve. All of the above types can be represented by a nurbs curve. The difference between an actual circle and a nurbs-curve-that-looks-like-a-circle is the way it is stored. A nurbs curve doesn't have a Radius property for example, nor a Plane in which it is defined. It is possible to reconstruct these properties by evaluating derivatives and tangent vector and frames and so on and so forth, but the data isn't readily available. In short, nurbs curves lack some global properties that other curve types do have. This is not a big issue, it's easy to remember what properties a nurbs curve does and doesn't have. It is much harder to deal with local properties that are not continuous. For example, imagine a polycurve which has a zero-length line segment embedded somewhere inside. The t-parameter at the line beginning is a different value from the t-parameter at the end, meaning we have a curve subdomain which has zero length. It is impossible to calculate a normal vector inside this domain:
This polycurve consists of five curve segments (a nurbs-curve, a zero-length line-segment, a proper line-segment, a 90째 arc and another nurbs-curve respectively) all of which touch each other at the indicated t-parameters. None of them are tangency continuous, meaning that if you ask for the tangent at parameter {t3}, you might either get the tangent at the end of the purple segment or the tangent at the beginning of the green segment. However, if you ask for the tangent vector halfway between {t1} and {t2}, you get nothing. The curvature data domain has an even bigger hole in it, since both line-segments lack any curvature:
Impossible to evaluate
Impossible to evaluate
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