Bézier Curve in Computer Graphics

Cubic Bézier Curve:

Bézier Curve Spline can fit to any number of control points. Without tangent vector specification at any of the control points, a set of characteristics polynomial approximating functions that say Bézier Blending Functions. It blends the control points to produce a Bézier curve segment. However, the degree of a Bézier curve is determined by the number of control points to be fitted with that curve segment.

Given a set of n+1 control points P0, P1,…,Pn a parametric Bezier curve that will fit to those points is mathematically defined by,

formula

Now restricting our attention to Bezier curves of degree 3 (n=3). We find that four (n+1) control points required to specify a cubic Bezier curve segment. Thus for a Parametric cubic Bezier curve,

cubic bezier curve

or, P(t)=P0B0,3(t)+ P1B1,3(t)+ P2B2,3(t)+ P3B3,3(t)

Tangent Bézier Curve:

Although it is not necessary to numerically specify the tangent vectors at the end of a Bezier curve, finding out the expression for them will bring out more interesting facts about the bezier curves.
The parametric tangent vector is obtained by differentiating a cubic Bezier curve segment.

Bézier Curve in Computer Graphics