WelcomeTOOur

Presentation

Introducing my group members

Md. Ilias Bappi

ID: 131-15-2266

Ferdous Ahamad

ID:131-15-2408

Md.Kawsar Hamid

ID:131-15-2223

Md. Jahirul Shahed

ID: 131-15-2479

Saiful IslamID:131-15-2516

Our PresentationTopic

Béziercurve

Hermite curve

The hermite curve is the curve for the data P,Q, v, and w. the four polynomials in figure are called the hermite functions, or hermitebasis functions.

Here,

P and Q are points, and velocity vectors

v and w

Figure: The four Hermite polynomials.

Matrix formulation of Hermite curve

The first factor is the geometry matrix (G)

The middle matrix, called the basis matrix (M)

Thus, in brief, the Hermite curve can be written

γ(t) = GMT(t).

This basis matrix M that lists the coefficients of some polynomials, and the vector T(t).

Bézier curve

• Bézier curve defined by four points P1, . . . , P4.

• The curve starts at P1, finishes at P4, and has initial velocity 3(P2 −P1) and final velocity 3(P4 −P3), as shown in Figure.

Finding a point on a Bézier Curve

• A simple way to find the point C(u) on the curve for a particular u is

1.To plug u into every basis function

2.Compute the product of each basis function and its corresponding control point

3.Add them together.

10),()( ,

0

uuBu nk

n

k

kpC

),,( kykk zyxp nk 0knk

nk uuknCuB )1(),()(,

Derivatives or Design Techniques Using

Bézier Curve

Design Techniques Using Bézier Curve(Weights)

Multiple control points at a single coordinate position

gives more weight to that position.

Design Techniques Using Bézier Curve(Closed Curves)

Closed Bézier curves are generated by specifying

the first and the last control points at the same

position.

Note: Bézier curves are polynomials which cannot represent

circles and ellipses.

0

1

2

3

45

6

7

8

Example Cubic Bézier Curves

knk

nk uuknCuB )1(),()(,

Cubic Bézier curves are generated with four control points.

The four blending functions for cubic Bézier curves (n=3):

3

3,3

2

3,2

2

3,1

3

3,0

)(

)1(3)(

)1(3)(

)1()(

uuB

uuuB

uuuB

uuB

3,1B

Properties

of

Bézier Curves

Properties of a Bézier Curve

10),()( ,

0

uuBu nk

n

k

kpC

1. The degree of a Bézier curve defined by n+1

control points is n:

Parabola Curve Cubic Curve Cubic Curve

Cubic Curve

Properties of a Bézier Curve

2. The curve passes though the first and the last control point

C(u) passes through P0 and Pn.

Properties of a Bézier Curve

3. Bézier curves are tangent to their first and

last edges of control polyline.

1

2

0

3

4

5

8

7

6

109

0

1

2

3

4

5

6

7

8

Properties of a Bézier Curve

4. The Bézier curve lies completely in the convex hull of the

given control points.

Note that not all control points are on the boundary of the convex

hull. For example, control points 3, 4, 5, 6, 8 and 9 are in the

interior. The curve, except for the first two endpoints, lies

completely in the convex hull.

Properties of a Bézier Curve

5. Moving control points:

Thank You