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A question regarding control points in a practice exam asks to find the relationship between Bezier control points and degrees of the curves.

How many control points are required to specify a Bezier curve of degree x?

What is the degree of a Bezier curve?

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  • $\begingroup$ $x+1$ control points are needed. The curve degree is the degree of the $x$/$y$-polynomials. $\endgroup$ Commented May 7, 2017 at 17:13
  • $\begingroup$ If the name "control points" is for all the points that participate to the construction: the answer is $x+1$ (for example degree 3 at most for a cubic Bezier curve defined by 4 points A,B,C,D). $\endgroup$
    – Jean Marie
    Commented May 7, 2017 at 17:16

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Cubic Bezier curve is usually defined as:

$$\mathbf{B}(t)=(1-t)^3\mathbf{P}_0+3(1-t)^2t\mathbf{P}_1+3(1-t)t^2\mathbf{P}_2+t^3\mathbf{P}_3 \mbox{ , } 0 \le t \le 1.$$

When you expand and rearrange the term you will get:

$$(-\mathbf{P}_0 + 3\mathbf{P}_1 - 3\mathbf{P}_2 + \mathbf{P}_3)t^3 + (3\mathbf{P}_0 - 6\mathbf{P}_1 + 3\mathbf{P}_2)t^2 + (-3\mathbf{P}_0 + 3\mathbf{P}_1)t + \mathbf{P}_0$$

Which is a polynomial in $t$ of degree 3. In general, the "degree" of a Bezier curve is the highest exponent of $t$ if written as polynomial.

If you stare at the first equation, you will notice the are four terms near the four points, each with raising power of $t$ (i.e., $t^0, t^1, t^2, t^3$). Counting from zero, there are four points for degree 3. In general, one needs degree + 1 points to define the curve.

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