2. Some properties of the cubic Bernstein polynomials

• Values at 0 and 1:
  $$\begin{array}{cc} J_{3,0}(0) = 1& J_{3,0}(1) = 0\\ J_{3,1}(... ...& J_{3,2}(1) = 0\\ J_{3,3}(0) = 0& J_{3,3}(1) = 1 \end{array}$$

• Unit sum property: $J_{3,0} (t) + J_{3,1} (t) + J_{3,2} (t) + J_{3,3} (t) = 1$ for all $t$.

• Nonnegativity: $J_{3,k} \geq 0$ for $0 \leq t \leq 1$.

• Graphs: See Figure .

  Figure: Graphs of Bernstein polynomials of degree 3

  
  

  

  
  

• First derivatives at 0 and 1:
  $$\begin{array}{ll} J'_{3,0} (0) = -3& J'_{3,0} (1) = 0\\ J'_... ...,2} (1) = -3\\ J'_{3,3} (0) = 0& J'_{3,3} (1) = 3 \end{array}$$

• Second derivatives at 0 and 1:
  $$\begin{array}{ll} J''_{3,0} (0) = 6& J''_{3,0} (1) = 0\\ J'... ...(1) = -12\\ J''_{3,3} (0) = 0 & J''_{3,3} (1) = 6 \end{array}$$

• Maximum property: The maximum value of $J_{3,k} (t)$ occurs at $t = {\frac{\displaystyle k}{\displaystyle 3}}$ for $k = 0, 1, 2, 3$.

• Linear sum property: If $x_0, x_1, x_2, x_3$ are evenly spaced (i.e., form an arithmetic progression), then for all $t$, $x_0 J_{3,0} (t) + x_1 J_{3,1} (t) + x_2 J_{3,2} (t) + x_3 J_{3,3} (t)$ $=$ $x_0 + t (x_3 - x_0)$, the linear function that runs from $x_0$ to $x_3$ as $t$ runs from $0$ to $1$.

• Symmetry: For $k = 0, 1, 2, 3$, $J_{3,k} (1-t) = J_{3,3-k} (t)$.

• Basis: $J_{3,0} (t)$, $J_{3,1} (t)$, $J_{3,2} (t)$, $J_{3,3} (t)$ form a basis for the vector space of all polynomials of degree at most three. (In contrast, the standard basis is $1,t,t^2, t^3$.)