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What exactly does "degrees of freedom" refer to?


@ericchan "Degrees of freedom" means that total number of variables that describe the object of interest. For instance, a line segment in the plane has 4 degrees of freedom: the x and y coordinates of the left endpoint, and the x and y coordinates of the right endpoint. You could also say that this segment has two vector-valued degrees of freedom, i.e., it is completely specified by two elements of $\mathbb{R}^2$. Other kinds of curve or surface patches will have different numbers of degrees of freedom.

Can you tell me (i) how many degrees of freedom a cubic Bezier curve in the plane has and (ii) how many degrees of freedom a bicubic Bezier patch in 3D has?


So would a cubic bezier have 8 degrees of freedom? Two for each point's x and y?


@tcl1 Correct. Nth-order Bezier curve => N+1 control points => d(N+1) degrees of freedom in $R^d$. So for N=3, d=2 you get 4*2=8.

What about a bicubic Bezier patch? What about a Bezier patch of order N in d dimensions?