Opis_eng

Bezier-Curves

There are many algorithms to construct curves. A particulary elegant and practi?al one is based on specifaying four poinst that compeletely determine a curve segment: two endpoints and two control points .
curves constructed in this way are referred to as (cubic) Becier curves. We have the endpoints P₀ and P₃ , the control points P₁, P₂ and the curve constructed constructed on the basis of these four points. Writing a method to draw this ?urve is surprisingly easy.

1. With use of recursion

We compute six midpoints , as picture shows:

A , the midpoint of P0 and P1
B, the midpoint of P2 and P3
C, the midpoint of P1and P2
A1, the midpoint of A and C
B1, the midpoint of B and C
C1, the midpoint of A1and B1

As shows expression:

2. Matrix nota?ion

In parametric notacion we take :

Then the curve is defined as:

And in matrix nota?ion:

What is much more efficient then use of recursion becauseof the reduced number of arithmetics operations.

GLAVNA-stran