Bezeir Curves
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Transcript of Bezeir Curves
Apr 11, 2023 3D Object Representation 1
3 D Object Representation
Apr 11, 2023 3D Object Representation 2
Spline Representations Spline: In drafting terminology
A flexible strip used to produce a smooth curve through a designated set of points.
Several small weights are distributed along the strip. Spline Curve: In computer graphics
Any composite curve formed with (cubic) polynomial sections satisfying specified continuity conditions at the boundary of the pieces.
Spline surface Described with two sets of orthogonal spline curves.
Apr 11, 2023 3D Object Representation 3
Interpolation and Approximation Splines Control points
A set of coordinate positions which indicates the general shape of the curve.
A spline curve is defined, modified, and manipulated with operations on the control points.
Interpolation curves: the curve passes through each control points Digitize drawings Specify animation path
Approximation curves: fits to the general control-points path without necessarily passing through any control points
Apr 11, 2023 3D Object Representation 4
Interpolation and Approximation Splines
Apr 11, 2023 3D Object Representation 5
Convex hull The convex polygon boundary that encloses a set of
control points. Provide a measure for the deviation of a curve or
surface from the region bounding the control points
A convex hull for 2 sets of control points
Apr 11, 2023 3D Object Representation 6
Control graph A set of connected line segments connecting the
sequence of control points.
Apr 11, 2023 3D Object Representation 7
Parametric Continuity Continuity conditions
To ensure a smooth transition from one section of a piecewise parametric curve to the next.
Zero-order (C0) continuity First-order (C1) continuity Second-order (C2) continuity
Apr 11, 2023 3D Object Representation 8
Zero-order (C0) continuity
Simply the curves meet.
Apr 11, 2023 3D Object Representation 9
First-order (C1) continuity First parametric derivatives
(tangent lines) of the co ordinate functions of 2 successive curves are equal at their joining point.
Parametric coordinate form x =x(u), y=y(u) z=z(u) u1<=u<=u2
Apr 11, 2023 3D Object Representation 10
Second-order (C2) continuity
Both first and second parametric derivatives of the 2 curve sections are the same at intersection.
Apr 11, 2023 3D Object Representation 11
Parametric Continuity
Apr 11, 2023 3D Object Representation 12
Spline Specifications Three methods for specifying a spline
A set of boundary conditions imposed on the spline.
The matrix that characterizes the spline. The set of blending functions (or basis
functions)
Apr 11, 2023 3D Object Representation 13
1.Boundary Conditions
Suppose x(u)=axu3+bxu2+cxu+dx
be the cubic polynomial representation Endpoints coordinates - x(0) and x(1). First derivatives at the endpoints x’(0) and
x’(1). Determine the values of ax, bx, cx, and dx. Form the matrix from the boundary conditions.
Apr 11, 2023 3D Object Representation 14
2.Matrix Representation
U - row matrix of parameter u , C – Coefficient column matrix.
C = Mspline . Mgeom
Mgeom – control point coordinate values(4 element column matrix
Mspline – 4x4 matrix – characterization for the spline curve x(u) = U. Mspline . Mgeom
Apr 11, 2023 3D Object Representation 15
3.Blending Function
gk- constant parameters such as control points, slope of curve at the control points.
BFk(u) – blending func
Apr 11, 2023 3D Object Representation 16
Bezeir Curves and Surfaces
Apr 11, 2023 3D Object Representation 17
“Computers can’t draw curves.”
The more points/line segments that are used, the smoother the curve.
Apr 11, 2023 3D Object Representation 18
Bezier Curves
An alternative to splines. M.Bezeir a French mathematician who
worked for the Renault motor car company. He invented his curves to allow his firm’s
computers to describe the shape of car bodies.
Apr 11, 2023 3D Object Representation 19
? and provide to How 41 R R
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mation Approxi
and
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32
41
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1P4P
2P 3P
Bezier Approximation
Apr 11, 2023 3D Object Representation 20
Bezier Curves Contd
Typically, cubic polynomials Need four points
Two at the ends of a segment Two control tangent vectors
Apr 11, 2023 3D Object Representation 21
Bezier Curves Contd
Control polygon: control points connected to each other
Easy to generalize to higher order Insert more control points
Apr 11, 2023 3D Object Representation 22
De Casteljau Algorithm
Can compute any point on the curve in a few iterations.
No polynomials, pure geometry. Repeated linear interpolation
Repeated order of the curve times The algorithm can be used as definition of the
curve.
Apr 11, 2023 3D Object Representation 23
De Casteljau Algorithm Contd
Apr 11, 2023 3D Object Representation 24
De Casteljau Algorithm Contd
Apr 11, 2023 3D Object Representation 25
De Casteljau Algorithm Contd
Apr 11, 2023 3D Object Representation 26
De Casteljau Algorithm Contd
Apr 11, 2023 3D Object Representation 27
De Casteljau Algorithm Contd
3P
1P 2P
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t1
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t
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)1(
Pt
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)1(3
Apr 11, 2023 3D Object Representation 28
0
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Cubic Bezeir Curve -Matrix Representation
Apr 11, 2023 3D Object Representation 29
Bezier Polynomial Function
iniin tt
ini
ntB
)1.(.
)!(!
!)(
Apr 11, 2023 3D Object Representation 30
Parametric Equations for x(t),y(t)
)(.)(0
tBxtx in
n
ii
)(.)(0
tByty in
n
ii
Apr 11, 2023 3D Object Representation 31
Cubic Bezeir func in Matrix Form
In cubic matrix form
Normal matrix form
Apr 11, 2023 3D Object Representation 32
Cubic Bezier Curves Contd Bezeir matrix is
Apr 11, 2023 3D Object Representation 33
Design Techniques Using Bezier Curves
Closed Bezier curves Specifying the first and last control points
at the same position.
Apr 11, 2023 3D Object Representation 34
Design Techniques Using Bezier Curves Contd
Multiple control points Specify multiple control points at a single
co ordinate position gives more weight to that position.
Apr 11, 2023 3D Object Representation 35
Design Techniques Using Bezier Curves Contd
When complicated curves are to be generated
Formed by piecing several bezeir sections of lower degree together.
Apr 11, 2023 3D Object Representation 36
Bezier Surfaces Two sets of orthogonal Bezeir curves can be used to design an
object surface by specifying an input mesh of control points. The parametric vector function is formed as the Cartesian
product of Bezier blending functions
pi,k specifying the location Each curve of constant u is plotted by varying v over the interval
0 to 1, with u fixed at one of the values in this unit interval.
)()(),( ,0 0
,, uBEZvBEZpvuP nk
m
j
n
kmjkj
Apr 11, 2023 3D Object Representation 37
Bezier Surfaces Contd Control points connected by dashed lines Solid lines show curves of constant u and constant v
Apr 11, 2023 3D Object Representation 38
Thank You…