Important points about the bezier curves - modeling and rendering

1) Generalizing the idea of Bezier curve of degree at n based upon n+1 control point p0,.....pn

P(0)= P₀

P(1) =p_n

Values of parametric initial derivates of Bezier curve at the ending points can be computed from control point Coordinates are:

P'(0) = -nP₀ +n P₁

P'(1) = -nP_n-1 +  nP_n

Hence, the slope at the starting of the curve is beside the line joining two control points and the slope on the end of the curve is beside the line connecting the last two endpoints.

2) Convex hull: For 0≤ t ≤ 1, the Bezier curve lies completely in the convex hull of its control points. This property for a Bezier curve ensures about polynomial will not contain erratic oscillation.

3) Bezier curves are invariant about affine transformations, although they are not invariant about projective transformations.

4) To the Bezier curve, the vector tangent at the beginning or stop is parallel to the line connecting the first two or last two control points.

5) Bezier curves appear a symmetry property: The similar Bezier curve shape is acquired if the control points are considered in the opposite order. The merely difference will be the parametric way of the curve. The direction of rising parameter reverses while the control points are considered in the reverse sequence.

6) Adjusting the location of a control point changes the shape of the curve in a "predictable manner". Spontaneously, the curve "follows" the control point.