The order of a curve deals with the number of degrees of freedom that are necessary to uniquely define that curve. In general the order of a curve equals the degrees of freedom of that curve. [Note: The "Degree" of a curve (dealing with the polynomial degree of the functions that define the curve) is NOT related to the "degrees of freedom" of the curve.]
Degree = (Order - 1)
Order Degree Name ----------------- 1 0 Constant 2 1 Linear 3 2 Quadratic 4 3 Cubic 5 4 Quartic
What does the order of a curve really mean?
For B-Splines, the order of the curve specifies the number of control points that affect the parametric point on the curve. This can be seen by looking at the convex hull that surrounds that point. Notice that for an order 2 curve, the convex hull is actually a straight line. For an order 3 curve, the convex hull is a triangle, and for an order 4 curve, the convex hull can be a quadrilateral.
Degrees of Freedom
A "degree of freedom" is one piece of information used to describe a curve. The more degrees of freedom a curve has, the more complex it can be. For example, how much information is needed to describe a straight line (lines are simple curves)? The answer is two pieces of information, i.e.: two points. Therefore a line has two degrees of freedom.