Derivations of the Bézier Curve

On selected polygon of control points we create Bézier curve. This we differentiate using a direct derivation of the polygon and receive hodograph of a new Bézier curve. The process is repeated for the second derivation, and these results are compared with numerical derivation of the parametric specified curve


INTRODUCTION
Bézier curve needs for its definition a set of control points that determine it completely [3].Its application, as well as the use of B-spline curve we addressed our previous works [2].In finding tangents and normals of Bézier curve is necessary to calculate its derivation [1], [3].Similarly, the first and the second derivation is used in determining of the length and curvature of any curve.The article briefly describes method for obtaining the derivations of a Bézier curve and on a simple example this method is compared with the numerical derivation for the parametric specified curve.

Derivations of a Bézier curve
Bézier curve is defined by the polygon of (n+1) control points P 0 , P 1 , ... , P n and follows the equation where Bernstein polynomial is defined ( ) The control points P i are independent on parameter t and therefore is sufficient to differentiate B i n (t) Then the derivation throughout the Bézier curve has the form ( ) If we introduce a set of control points Q 0 = n(P 1 -P 0 ), Q 1 = n(P 2 -P 1 ), Q 2 = n(P 3 -P 2 ), ..., Q n = n(P n -P n-1 ) writing of derivation (4) simplify on When (5) compared to (1) we see that by differentiation of the Bézier curve arises again Bézier curve of degree (n-1), with control points n(P 1 -P 0 ), n(P 2 -P 1 ), n(P 3 -P 2 ), ... , n(P n -P n-1 ).This new differentiated curve is usually called the hodograph of the original Bézier curve.The form n(P i+1 -P i ) represents n-times direction vector from the point P i to the point P i+1 .
For t = 0, P´(0) = n(P 1 -P 0 ), which means that the first tangent of the curve is in the direction of (P 1 -P 0 ) n-times multiplied.Similarly for t = 1, P´(1) = n(P n -P n-1 ), the last tangent of the curve is in the direction of (P n -P n-1 ) n-times enlarged.
In calculating the second derivation we start from equation ( 5) and again analogously we differentiate as for the first derivation The higher derivations can be obtained likewise by applying the same process.

RESULTS AND DISCUSSION
We chose ten points in the plane, n = 9, where x and y-coordinates are given in the Table 1.These control points have been plotted and we constructed the Bézier curve according to [2].
The result is shown in the Figure 1.
By derivation of the Bézier curve (4) we obtained its hodograph drawn in the Figure 2. The control points of the new curve are calculated in the middle two columns of the Table 1.Degree of the new Bézier curve was reduced.
The process was repeated and it was found the second derivation of the original curve according to (6).The graph is shown in the Figure 3 and the coordinates of the control points are given in the For the parameter t = 0.01, in every tenth value, obtained by numerical calculation Since the Bézier curves for a set of control points are defined parametrically, can be differentiated also numerically.The results of these numerical derivations are given in the Table 3.The last line in this table identified Last means for the original curve i = 100, for the first derivation i = 99 and for the second derivation i = 98.The number of points is reduced, which directly follows from numerical differentiation algorithm.Comparing the pairs of the Figures 2 and 4 we see that result of the first derivation found by direct calculation (5) corresponds to the result of numerical derivation.Similar finding belongs to the comparison of the second derivations in the Figures 3 and 5.
Fig. 3 The second derivation of the Bézier curve with its control points For the parameter t = 0.01, obtained by direct calculation Fig. 4 The first derivation of the Bézier curve with its control points For the parameter t = 0.01, obtained by numerical calculation Fig. 5 The second derivation of the Bézier curve with its control points.
For the parameter t = 0.01, obtained by numerical calculation When comparing the values of the first and second derivation in the Tables 2 and 3 there is not already excessive identity and mainly the second derivations are quite different.However, it should be noted scales of the Figures 1 -5.They result that with increasing derivation also scale is increasing, which is a direct consequence of the relationships (4) and (5).Therefore also relatively large differences between direct and numerical derivation significantly do not affect the outcome and shape of the curve remains the same by both methods.

Fig. 1
Fig. 1 Bézier curve and its control points drawn for the parameter t = 0.01

Fig. 2
Fig.2The first derivation of the Bézier curve with its control points For the parameter t = 0.01, obtained by direct calculation Table 1 in the last two columns.Degree of the Bézier curve of second derivation again decreased.The values of both obtained derivations are presented in the Table2for selected parameter step t = 0.01.Each curve is parameterized through t∈<0, 1>.From all 101 values we point out every tenth value.Tab. 1 Coordinates of control points of a Bézier curve and its first and second derivations Points of Bézier curve and curves forming the first and second derivations