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• Definition of Bezier Curve • Properties • Design technique Using Bezier Curve• Application • Conclusion
Content
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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
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1.The degree of a Bézier curve defined by n+1 control points is n: 10),()( ,0
uuBu nk
n
kkpC
Properties
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2. The curve passes though the first and the last control point C(u) passes through P0 and Pn.
Properties
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3.Bézier curves are tangent to their first and last edges of control polyline.
Properties
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4.The Bézier curve lines completely in the convex hull of the given control points.
Properties
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5. Moving control points:
Properties
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6.Multiple control points at a single coordinate position gives more weight to that position.
Properties
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7.Closed Bézier curves are generated by specifying the first and the last control points at the same position
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Properties
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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
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Since Bézier curves pass through endpoints; it is easy to match curve sections (C0 continuity)
Zeroordercontinuity:
P´0=P2
Design Technique
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Since the tangent to the curve at an endpoint is along
the line joining that endpoint to the adjacent control point;
Design Technique
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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
Firstordercontinuity:
P1, P2, and P´1 collinear.
Design Technique
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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
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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
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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
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Dynamic Bezier curve is a efficient method to fit geographical curves. It make advantages of GAITRecognition using Bezier curve that it solve problem of geographical curves.
Conclusion
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Thank you all
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