Bezier curve computer graphics

Bezier curve computer graphics

1. Definition of Bezier Curve Properties Design technique Using Bezier Curve Application Conclusion Content 1

2. A Bezier curve is a mathematically defined curve used in two-dimensional graphic applications. The curve is defined by four points: the initial position and the terminating position (which are called "anchors") and two separate middle points (which are called "handles"). The shape of a Bezier curve can be altered by moving the handles. The mathematical method for drawing curves was created by Pierre Bezier in the late 1960's for the manufacturing of automobiles at Renault. Definition 2

3. 1.The degree of a Bézier curve defined by n+1 control points is n: Properties 3

4. 2. The curve passes though the first and the last control point C(u) passes through P and P n . Properties 4

5. 3.Bézier curves are tangent to their first and last edges of control polyline . Properties 5

6. 4.The Bézier curve lines completely in the convex hull of the given control points. Properties 6

7. 5 . Moving control points: Properties 7

8. 6 . Multiple control points at a single coordinate position gives more weight to that position. Properties 8

9. ‍‍ 7. Closed Bézier curves are generated by specifying the first and the last control points at the same position 1 2 3 4 5 6 7 8 Properties 9

10. When complicated curves are to be generated, they can be formed by piecing several Bézier sections of lower degree together. When complicated curves are to be generated, they can be formed by piecing several Bézier sections of lower degree together. Design Technique 10

11. Since Bézier curves pass through endpoints; it is easy to match curve sections (C continuity) Zero order continuity: P´ = P 2 Design Technique 11

12. Since the tangent to the curve at an endpoint is along the line joining that endpoint to the adjacent control point; Design Technique 12

13. To obtain C 1 continuity between curve sections, we can pick control points P´ and P´ 1 of a new section to be along the same straight line as control points P n-1 and P n of the previous section First order continuity: P 1 , P 2 , and P´ 1 collinear . Design Technique 13

14. This relation states that to achieve C 1 continuity at the joining point the ratio of the length of the last leg of the first curve ( i.e. , | p m - p m-1 |) and the length of the first leg of the second curve ( i.e. , | q 1 - q |) must be n/m . Since the degrees m and n are fixed, we can adjust the positions of p m -1 or q 1 on the same line so that the above relation is satisfied Design Technique 14

15. The left curve is of degree 4, while the right curve is of degree7. But, the ratio of the last leg of the left curve and the first leg of the second curve seems near 1 rather than 7/4=1.75. To achieve C 1 continuity, we should increase ( resp. , decrease) the length of the last ( resp. first) leg of the left ( resp. , right). However, they are G 1 continuous Design Technique 15

16. Computer graphics : Bezier curves are widely used in computer graphics to model smooth curves Animation: In animation application ,such as Adobe Flash and synfig,Bezier curves are used to Outline ,for example movement Fon t: TrueType fonts use Bezier splines composed of quadratic Bezier curves Application 16

17. Dynamic Bezier curve is a efficient method to fit geographical curves. It make advantages of GAIT Recognition using Bezier curve that it solve problem of geographical curves. Conclusion 17

18. Thank you all 18