march 1, 2009dr. muhammed al-mulhem1 ics 415 computer graphics bézier splines (chapter 8) dr....
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March 1, 2009 Dr. Muhammed Al-Mulhem 1
ICS 415Computer Graphics
Bézier Splines (Chapter 8)
Dr. Muhammed Al-MulhemDr. Muhammed Al-Mulhem
March 1, 2009March 1, 2009
Dr. Muhammed Al-MulhemDr. Muhammed Al-Mulhem
March 1, 2009March 1, 2009
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March 1, 2009 Dr. Muhammed Al-Mulhem 2
Bézier Curves• This spline approximation method was developed by the
French engineer Pierre Bezier for use in the design of Renault car bodies.
• Bezier curve section can be fitted to any number of control points.
• The degree of the Bezier polynomial is determined by the number of control points to be approximated and their relative position.
• In general, Bezier curve can be specified using blending functions.
• This spline approximation method was developed by the French engineer Pierre Bezier for use in the design of Renault car bodies.
• Bezier curve section can be fitted to any number of control points.
• The degree of the Bezier polynomial is determined by the number of control points to be approximated and their relative position.
• In general, Bezier curve can be specified using blending functions.
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March 1, 2009 Dr. Muhammed Al-Mulhem 3
Bézier Curves Equations
• Although, Bezier curve section can be fitted to any number of control points, some graphics packages limit the number of control points to four.
• Although, Bezier curve section can be fitted to any number of control points, some graphics packages limit the number of control points to four.
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March 1, 2009 Dr. Muhammed Al-Mulhem 4
Bézier Curves Equations
• Consider first the general case of n+1 control points, denoted as:
• Consider first the general case of n+1 control points, denoted as:
10)()(0
,
uuBEZPuPn
knkk
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March 1, 2009 Dr. Muhammed Al-Mulhem 5
Bézier Curves Equations
• The Bézier blending function BEZ k, n (u) are the Bernstein
polynomials
• Where parameters C(n,k) are the binomial coefficients
• The Bézier blending function BEZ k, n (u) are the Bernstein
polynomials
• Where parameters C(n,k) are the binomial coefficients
knknk uuknCuBEZ )1(),()(,
)!(!
!),(
knk
nknC
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March 1, 2009 Dr. Muhammed Al-Mulhem 6
Bézier Curves Equations
• P(u) in slide #4 represent a set of three parametric equations for the individual curve coordinates.
• P(u) in slide #4 represent a set of three parametric equations for the individual curve coordinates.
)()(
)()(
)()(
0,
0,
0,
uBEZzuz
uBEZyuy
uBEZxux
n
knkk
n
knkk
n
knkk
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March 1, 2009 Dr. Muhammed Al-Mulhem 7
Bézier Curves Equations
• In most cases, a Bézier curve is a polynomial of degree one less that the designated number of control points.
• Three points generate a parabola, four points a cubic curve, and so on.
• Next slide demonstrate the appearance of some Bézier curves for various selections of control points in the x-y plane (z=0).
• In most cases, a Bézier curve is a polynomial of degree one less that the designated number of control points.
• Three points generate a parabola, four points a cubic curve, and so on.
• Next slide demonstrate the appearance of some Bézier curves for various selections of control points in the x-y plane (z=0).
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March 1, 2009 Dr. Muhammed Al-Mulhem 8
Bézier Curves Equations
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March 1, 2009 Dr. Muhammed Al-Mulhem 9
Bézier Curves Equations
• With certain control points placements, however, we obtain degenerate Bézier polynomials.
• For example,
• A Bézier curve generated with three collinear control points is a straight-line segment.
• A set of control points that are all at the same coordinate position produce a Bézier curve that is a single point.
• With certain control points placements, however, we obtain degenerate Bézier polynomials.
• For example,
• A Bézier curve generated with three collinear control points is a straight-line segment.
• A set of control points that are all at the same coordinate position produce a Bézier curve that is a single point.
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March 1, 2009 Dr. Muhammed Al-Mulhem 10
Bézier Curves Equations
• A recursive calculations can be used to obtain successive binomial-coefficient values as:
• A recursive calculations can be used to obtain successive binomial-coefficient values as:
knforknCk
knknC
)1,(
1),(
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March 1, 2009 Dr. Muhammed Al-Mulhem 11
Bézier Curves Equations
• Also, the Bézier blending function satisfy the recursive relationship:
• Also, the Bézier blending function satisfy the recursive relationship:
kk
kkk
nknknk
uBEZanduBEZ
with
knuBEZuuBEZuuBEZ
)1(
1)()()1()(
,0,
1,11,,
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March 1, 2009 Dr. Muhammed Al-Mulhem 12
Properties of Bézier Curves
• The curve connects the first and last control points.
• Thus, a basic characteristic of any Bézier curve is that:
P(0) = p0
P(1) = pn
• The curve connects the first and last control points.
• Thus, a basic characteristic of any Bézier curve is that:
P(0) = p0
P(1) = pn
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March 1, 2009 Dr. Muhammed Al-Mulhem 13
Properties of Bézier Curves
• Values for the parametric first derivatives of a Bézier curve at the endpoints can be calculated from control point coordinates as:
P’(0) = -np0 + n p1
P’(1) = -npn-1 + n pn
• From these expressions, we see that the slope at the beginning of the curve is a long the line joining the first two control points, and
• the slope at the end of the curve is a long the line joining the last two control points.
• Values for the parametric first derivatives of a Bézier curve at the endpoints can be calculated from control point coordinates as:
P’(0) = -np0 + n p1
P’(1) = -npn-1 + n pn
• From these expressions, we see that the slope at the beginning of the curve is a long the line joining the first two control points, and
• the slope at the end of the curve is a long the line joining the last two control points.
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March 1, 2009 Dr. Muhammed Al-Mulhem 14
Properties of Bézier Curves
• The parametric second derivatives of a Bézier curve at the endpoints are calculated as:
P’’(0) = n (n – 1) [ (p2 - p1) - (p1 - p0) ]
P’’(1) = n (n – 1) [ (pn-2 - pn-1) - (pn-1 - pn) ]
• The parametric second derivatives of a Bézier curve at the endpoints are calculated as:
P’’(0) = n (n – 1) [ (p2 - p1) - (p1 - p0) ]
P’’(1) = n (n – 1) [ (pn-2 - pn-1) - (pn-1 - pn) ]
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March 1, 2009 Dr. Muhammed Al-Mulhem 15
Properties of Bézier Curves
• Another property of any Bézier curve is that it lies within the convex hull (convex polygon boundary) of the control points.
• This follows from the fact that the Bézier bending functions are all positive and their sum is always 1.
• Another property of any Bézier curve is that it lies within the convex hull (convex polygon boundary) of the control points.
• This follows from the fact that the Bézier bending functions are all positive and their sum is always 1.
1)(0
,
uBEZn
knk
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March 1, 2009 Dr. Muhammed Al-Mulhem 16
Design Techniques using Bézier Curves
• A closed Bézier curve is generated when we set the last control point position to the coordinate position of the first control point.
• A closed Bézier curve is generated when we set the last control point position to the coordinate position of the first control point.
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March 1, 2009 Dr. Muhammed Al-Mulhem 17
Design Techniques using Bézier Curves
• Also, specifying multiple control points at a single coordinate position gives more weight to that position.
• The resulting curve is pulled nearer to this position.
• Also, specifying multiple control points at a single coordinate position gives more weight to that position.
• The resulting curve is pulled nearer to this position.
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March 1, 2009 Dr. Muhammed Al-Mulhem 18
Design Techniques using Bézier Curves
• We can fit Bézier curve to any number of control points, but this requires the calculation of polynomial functions of higher degree.
• When complicated curves are to be generated, they can be formed by piecing together several Bézier curves of lower degrees.
• Generating smaller Bézier curves also gives us better local control over the shape of the curve.
• We can fit Bézier curve to any number of control points, but this requires the calculation of polynomial functions of higher degree.
• When complicated curves are to be generated, they can be formed by piecing together several Bézier curves of lower degrees.
• Generating smaller Bézier curves also gives us better local control over the shape of the curve.
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March 1, 2009 Dr. Muhammed Al-Mulhem 19
Design Techniques using Bézier Curves
• Since Bézier curve connect the first and last control points, it is easy to match curve sections (zero-order continuity).
• Also, Bézier curves have the important property that the tangent to the curve at an endpoint is along the line joining that endpoint to the adjacent control point.
• Since Bézier curve connect the first and last control points, it is easy to match curve sections (zero-order continuity).
• Also, Bézier curves have the important property that the tangent to the curve at an endpoint is along the line joining that endpoint to the adjacent control point.
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March 1, 2009 Dr. Muhammed Al-Mulhem 20
Design Techniques using Bézier Curves
• To obtain first-order continuity between curve sections, we can pick control points p0’ and p1’ for the next curve
section to be along the same straight line as control points pn-1 and pn of the preceding section.
• To obtain first-order continuity between curve sections, we can pick control points p0’ and p1’ for the next curve
section to be along the same straight line as control points pn-1 and pn of the preceding section.
p3
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March 1, 2009 Dr. Muhammed Al-Mulhem 21
Cubic Bézier Curves
• Cubic Bézier curves are generated with four control points.
• The four blending function for cubic Bézier curves, obtained by substituting n =3 into equations in slide # 5. The results are:
• Cubic Bézier curves are generated with four control points.
• The four blending function for cubic Bézier curves, obtained by substituting n =3 into equations in slide # 5. The results are:
33,3
23,2
23,1
33,0
)1(3
)1(3
)1(
uBEZ
uuBEZ
uuBEZ
uBEZ
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March 1, 2009 Dr. Muhammed Al-Mulhem 22
Cubic Bézier Curves
• The plots of the four cubic Bézier blending functions are as follows:
• The plots of the four cubic Bézier blending functions are as follows:
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March 1, 2009 Dr. Muhammed Al-Mulhem 23
Cubic Bézier Curves
• The form of the blending functions determines how the control points influence the shape of the curve for values of parameter u over the range from 0 to 1.
• At u=0, the only nonzero blending function is BEZ0,3,
which has value 1.
• At u=1, the only nonzero blending function is BEZ3,3,
which has value 1.
• The form of the blending functions determines how the control points influence the shape of the curve for values of parameter u over the range from 0 to 1.
• At u=0, the only nonzero blending function is BEZ0,3,
which has value 1.
• At u=1, the only nonzero blending function is BEZ3,3,
which has value 1.
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March 1, 2009 Dr. Muhammed Al-Mulhem 24
Cubic Bézier Curves
• A cubic Bézier curve always begins at control point p0
and ends at p3.
• The other functions, BEZ1,3, and BEZ2,3, influence the
shape of the curve at intermediate values of parameter u, so the resulting curve tends toward the points p1 and p2.
• Blending function BEZ1,3, is maximum at u=1/3, and
BEZ2,3, is maximum at u=2/3.
• A cubic Bézier curve always begins at control point p0
and ends at p3.
• The other functions, BEZ1,3, and BEZ2,3, influence the
shape of the curve at intermediate values of parameter u, so the resulting curve tends toward the points p1 and p2.
• Blending function BEZ1,3, is maximum at u=1/3, and
BEZ2,3, is maximum at u=2/3.
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March 1, 2009 Dr. Muhammed Al-Mulhem 25
Bézier in Matrix Form
• A matrix form for the cubic Bézier curve function is obtained by expanding the polynomial expressions for the blending functions.
• Where the Bézier matrix is
• A matrix form for the cubic Bézier curve function is obtained by expanding the polynomial expressions for the blending functions.
• Where the Bézier matrix is
3
2
1
0
23 ..1)(
p
p
p
p
MuuuuP Bez
0001
0033
0363
1331
BezM