Bezier curve computer graphics

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

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

1.The degree of a Bézier curve defined by n+1 control points is n: 10),()( ,
0
 
uuBu nk
n
k
kpC
Properties
3

2. The curve passes though the first and the last control
point C(u) passes through P0 and Pn.
Properties
4

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

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

5. Moving control points:
Properties
7

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

7.Closed Bézier curves are generated by specifying the first and the last
control points at the same position
0
1
2
3
4
5
6
7
8
Properties
9

 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

Since Bézier curves pass through endpoints;
it is easy to match curve sections (C0 continuity)
Zero order
continuity:
P´0=P2
Design Technique
11

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

To obtain C1 continuity between curve sections, we can pick control
points P´0 and P´1 of a new section to be along the same straight line
as control points Pn-1 and Pn of the previous section
First order continuity:
 P1, P2, and P´1 collinear.
Design Technique
13

This relation states that to achieve C1 continuity at the joining point the
ratio of the length of the last leg of the first curve (i.e., |pm - pm-1|) and
the length of the first leg of the second curve (i.e., |q1 - q0|) must be
n/m. Since the degrees m and n are fixed, we can adjust the positions
of pm-1 or q1 on the same line so that the above relation is satisfied
Design Technique
14

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 C1
continuity, we should increase (resp., decrease) the length of the
last (resp. first) leg of the left (resp., right). However, they are G1
continuous
Design Technique
15

 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
 Font: TrueType fonts use Bezier splines composed of quadratic
Bezier curves
Application
16

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

Bezier curve computer graphics

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Bezier curve computer graphics