BEZIER CURVE

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KEY POINTS

• DEFINITION AND HISTORY• PROPERTIES OF BEZIER CURVE• DESIGN TECHNIQUES• REFERENCES

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DEFINITION AND HISTORY

• A BEZIER CURVE IS A PARAMETRIC CURVE THAT USES BERNSTEIN POLYNOMIALS AS BASIS.

• IT EMPLOYS AT LEAST TWO POINTS TO DEFINE A CURVE. THE TWO END POINTS ARE CALLED ANCHOR POINTS AND THE OTHER POINTS, WHICH DEFINE THE SHAPE OF THE CURVE ARE CALLED HANDLES, TANGENT POINTS OR NODES.

• BEZIER CURVE IS FREQUENTLY USED IN COMPUTER GRAPHICS AND RELATED FIELDS.

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DEFINITION AND HISTORY

IN 1962, FRENCH ENGINEER PIERRE BEZIER PUBLICIZED BEZIER CURVE WIDELY, USED THEM TO DESIGN AUTOMOBILE BODIES FOR THE CAR MANUFACTURING COMPANY RENAULT. HOWEVER THE STUDY OF THESE CURVES WAS FIRST DEVELOPED IN 1959 BY MATHEMATICIAN PAUL DE CASTELJAU, USING DE CASTELJAU’S ALGORITHM.

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PROPERTIES OF BEZIER CURVE

• ENDPOINTS INTERPOLATION PROPERTY: THE CURVE BEGINS AT P0 AND ENDS IN PN THAT IS THE FIRST AND LAST CONTROL POINTS ARE THE ENDPOINTS OF THE CURVE. THE START AND END OF THE CURVE IS TANGENT TO THE FIRST AND LAST SECTION OF THE BEZIER POLYGON, RESPECTIVELY.

• CONVEX HULL PROPERTY: A BEZIER CURVE WILL ALWAYS BE COMPLETELY CONTAINED INSIDE OF THE CONVEX HULL OF THE CONTROL POINTS.

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PROPERTIES OF BEZIER CURVE

• SYMMETRY PROPERTY : THE SAME BEZIER CURVE SHAPE IS OBTAINED IF THE CONTROL POINTS ARE SPECIFIED IN THE OPPOSITE ORDER. THE ONLY DIFFERENCE WILL BE THE PARAMETRIC DIRECTION OF THE CURVE.

• MULTIPLE CONTROL POINTS AT A SINGLE COORDINATE POSITION GIVES MORE WEIGHT TO THAT POSITION.

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PROPERTIES OF BEZIER CURVE

• THERE IS NO LOCAL CONTROL OF THIS SHAPE MODIFICATION. EVERY POINT ON THE CURVE (WITH THE EXCEPTION OF THE FIRST AND LAST) MOVE WHENEVER ANY INTERIOR CONTROL POINT IS MOVED.

• CLOSED BEZIER CURVES ARE GENERATED BY SPECIFYING THE FIRST AND THE LAST CONTROL POINTS AT THE SAME POSITION.

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DESIGN TECHNIQUES

A BEZIER CURVE IS DEFINED BY A SET OF CONTROL POINTS P0 THROUGH PN ,WHERE N IS CALLED ITS ORDER AND N-1 IS THE DEGREE OF THE BEZIER CURVE.

• LINEAR BEZIER CURVES: FOR THE GIVEN TWO POINTS P0 AND P1, A BEZIER CURVE IS SIMPLY A STRAIGHT LINE BETWEEN THE TWO POINTS. THE CURVE IS GIVEN BY-

B(T) = P0+T(P1-P0) = (1-T)P0+TP1, 0≤ T ≤1

P0

P1P0 P1

1-T TB(T)

degree 1, order 2

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DESIGN TECHNIQUES

• QUADRATIC BEZIER CURVE:

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P0

P1

P2

P0 P1 P21-T 1-TT T

1-T T

B(T) =(1-T)[(1-T)P0+TP1]+T[(1-T)P1+TP2] = (1-T)2 P0 + 2T(1-T) P1 + T2 P2 , 0≤ T≤1

degree 2, order 3

DESIGN TECHNIQUESCUBIC BEZIER CURVES:

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P0

P1 P2

P3

P0 P1 P2 P31-T

1-T

1-T

T

T

T T

T

T

1-T

1-T

B(T) = (1-T)3 P0 +3T(1-T)2 P1+3T2(1-T) P2 + T3 P3 , 0≤ T≤1

1-T

degree 3, order 4

DESIGN TECHNIQUES

THE GENERALIZE EQUATION OF BEZIER CURVE FOR ANY ORDER N IS –

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DESIGN TECHNIQUES

• WHEN COMPLICATED CURVES ARE TO BE GENERATED, THEY CAN BE FORMED BY PIECING SEVERAL BEZIER SECTIONS OF LOWER DEGREE TOGETHER.

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REFERENCES

• HTTPS://EN.WIKIPEDIA.ORG/WIKI/B%C3%A9ZIER_CURVE• HTTP://WEB.MIT.EDU/HYPERBOOK/PATRIKALAKIS-MAEKAWA-CHO/NODE12.HT

ML• HTTP://POMAX.GITHUB.IO/BEZIERINFO/• HTTP://PEOPLE.EECS.KU.EDU/~MILLER/COURSES/INTROTOCURVESANDSURFA

CES/BEZIERCURVEPROPERTIES.HTML• CURVES AND SURFACES FOR COMPUTER-AIDED GEOMETRIC DESIGN: A

PRACTICAL GUIDE BY GERALD FARIN

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THANK YOU

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