3. Some properties of cubic Bézier curves

Let $P(t)$ be the Bézier curve with four control points $P_0$, $P_1$, $P_2$, $P_3$.

• Degree: $P(t)$ has degree at most $3$ (one less than the number of control points).

• Special values: $P(0) = P_0$, $P(1) = P_3$.

• Tangency: At $t = 0$ the Bézier curve is tangent to the first leg of its control polygon; at $t = 1$ it is tangent to the last leg.

• Convex hull: For $0 \leq t \leq 1$, the Bézier curve lies entirely in the convex hull of its control points.

• First derivative at ends: $P'(0) = 3(P_1 - P_0)$ and $P'(1) = 3 ( P_3 - P_2)$.

• Second derivative at ends: $P''(0) = 6(P_0 - 2 P_1 + P_2)$ and $P''(1) = 6(P_1 - 2 P_2 + P_3)$.

• Affine compatibility: The Bézier construction is compatible with affine transformations $T$. In other words, $T(P(t))$ is the same as the point at time $t$ on the Bézier curve with control points $T(P_0)$, $T(P_1)$, $T(P_2)$, $T(P_3)$.

• Evenly spaced control points: If $P_0$, $P_1$, $P_2$, $P_3$ are evenly spaced along a straight line, then $P(t)$ reduces to the usual parametric form of a line segment, namely $P(t) = P_0 + t (P_3 - P_0)$.

• Maximum influence: For $k = 0, 1, 2, 3$, the control point $P_k$ has its maximum influence (i.e., its coefficient is at a maximum) at time $t = {{\frac{\displaystyle k}{\displaystyle 3}}}$.

• Symmetry: $P(1-t)$ is the same as the Bézier curve with control points in the opposite order: $P_3$, $P_2$, $P_1$, $P_0$.

• Half-way point: $P(\frac{1}2)$ is ${\frac 3 4}$ of the way from the midpoint of the segment $\overline {P_0 P_3}$ to the midpoint of the segment $\overline {P_1 P_2}$.

• Generality, algebraically: For any cubic parametric curve $P(t)$, its portion $0 \leq t \leq 1$ is a Bézier curve, for suitable chosen control points.

• Generality, geometrically: Any segment of a cubic parametric curve has the same points as some Bézier curve (with suitably chosen control points).