error estimation

Since the ellipse curve always has a positive curvature and since the linear Bézier curve is a straight line, the distance between the ellipse and its approximation is null at both arc ends and reaches its maximal value for one point $E(\eta)$ between the endpoints. At this point, the tangent to the ellipse is parallel to the line, as depicted in figure 5.

Figure 5: error of a linear approximation

Using the ellipse parametric equation (3) and its derivative (4), we can find $\eta$:

$$\det\big(E'(\eta),E(\eta_2)-E(\eta_1)\big)=0$$

$$\Leftrightarrow x'_\eta\big(y_2-y_1\big)-y'_\eta\big(x_2-x_1\big)=0$$

$$\Leftrightarrow -ab \big(\cos\eta (\cos\eta_2-\cos\eta_1)+\sin\eta (\sin\eta_2-\sin\eta_1)\big)=0$$

$$\Leftrightarrow \cos(\eta_2-\eta)=\cos(\eta-\eta_1)$$

$$\Leftrightarrow \eta_1=\eta_2 \quad\eta=\frac{\eta_1+\eta_2}{2}$$

where $x'_\eta$ and $y'_\eta$ are the coordinates of $E'(\eta)$ and $x_i$ and $y_i$ are the coordinates of $E(\eta_i)$.

Obviously, the second solution is the one we were looking for. Given this value of $\eta$, we compute the error $\varepsilon$ as the distance between $E(\eta)$ (coordinates $x_\eta$ and $y_\eta$) and the line passing through $E(\eta_1)$ and $E(\eta_2)$.