The document discusses B-splines, which are polynomial curves used for modeling curves and surfaces in computer graphics. B-splines consist of curve segments whose polynomial coefficients depend only on a few nearby control points, allowing for local control of the curve shape. Uniform cubic B-splines provide the highest, C2 continuity between segments. The shape of B-spline curves is determined by the position of control points and the spacing of knots.
Bezier Curves, properties of Bezier Curves, Derivation for Quadratic Bezier Curve, Blending function specification for Bezier curve:, B-Spline Curves, properties of B-spline Curve?
Evaluators provide a way to specify points on a curve or surface (or part of one) using only the control points. The curve or surface can then be rendered at any precision. In addition, normal vectors can be calculated for surfaces automatically. You can use the points generated by an evaluator in many ways - to draw dots where the surface would be, to draw a wireframe version of the surface, or to draw a fully lighted, shaded, and even textured version.
Bezier Curves, properties of Bezier Curves, Derivation for Quadratic Bezier Curve, Blending function specification for Bezier curve:, B-Spline Curves, properties of B-spline Curve?
Evaluators provide a way to specify points on a curve or surface (or part of one) using only the control points. The curve or surface can then be rendered at any precision. In addition, normal vectors can be calculated for surfaces automatically. You can use the points generated by an evaluator in many ways - to draw dots where the surface would be, to draw a wireframe version of the surface, or to draw a fully lighted, shaded, and even textured version.
A Bézier curve is a parametric curve frequently used in computer graphics and related fields. Generalizations of Bézier curves to higher dimensions are called Bézier surfaces, of which the Bézier triangle is a special case.
A Bézier curve is a parametric curve frequently used in computer graphics and related fields. Generalizations of Bézier curves to higher dimensions are called Bézier surfaces, of which the Bézier triangle is a special case.
This presentation was designed to attempt a lucidity and wholism on the topic CURVES IN ENGINEERING in the course ENGINEERING GRAPHICS for first year engineering students
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Representation of curves using cubic polynomials, Hermite form, Bezier form; Surface modelling representations and Solid modelling using B-Rep and CSG techniques are presented in this slide.
a way to specify points on a curve or surface (or part of one) using only the control points. The curve or surface can then be rendered at any precision. In addition, normal vectors can be calculated for surfaces automatically. You can use the points generated by an evaluator in many ways - to draw dots where the surface would be, to draw a wireframe version of the surface, or to draw a fully lighted, shaded, and even textured version.
Java3D is an Application Programming Interface used for writing 3D graphics applications and applets. This paper gives a short introduction of java3D, analyses the mathematics of Hermite, Bezier, FourPoints, B-Splines curve, and describes implementation of curve creation and curve
operations using Java3D API.
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2. Computer Graphics
10/10/2008 Lecture 5 2
Spline
• A long flexible strips of metal used by
draftspersons to lay out the surfaces of
airplanes, cars and ships
• Ducks weights attached to the splines
were used to pull the spline in different
directions
• The metal splines had second order
continuity
3. Computer Graphics
10/10/2008 Lecture 5 3
B-Splines (for basis splines)
• B-Splines
– Another polynomial curve for modelling curves and
surfaces
– Consists of curve segments whose polynomial
coefficients only depend on just a few control points
• Local control
– Segments joined at knots
4. Computer Graphics
B-splines
• The curve does not necessarily pass through
the control points
• The shape is constrained to the convex hull
made by the control points
• Uniform cubic b-splines has C2 continuity
– Higher than Hermite or Bezier curves
5. Computer Graphics
10/10/2008 Lecture 5
Basis Functions
knots
• We can create a long curve using many knots and B-splines
• The unweighted cubic B-Splines have been shown for clarity.
• These are weighted and summed to produce a curve of the
desired shape
t4 8 12
6. Computer Graphics
10/10/2008 Lecture 5 6
Generating a curve
X(t)
t
t
Opposite we see an
example of a shape to be
generated.
Here we see the curve
again with the weighted
B-Splines which
generated the required
shape.
7. Computer Graphics
10/10/2008 Lecture 5 7
The basic one:
Uniform Cubic B-Splines
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• Cubic B-splines with uniform knot-vector is the
most commonly used form of B-splines
8. Computer Graphics
10/10/2008 Lecture 5 8
Cubic B-Splines
• The unweighted spline set of 10 control points, 10 splines,
14 knots, and but only 7 intervals.
• You can see why t3 to t4 is the first interval with a curve
since it is the first with all four B-Spline functions.
• t9 to t10 is the last interval
43
t86 m
m+1
0
9. Computer Graphics
Domain of the function
• Order k, Degree k-1
• Control points Pi (i=0,…,m)
• Knots : tj, (j=0,…, k + m)
• The domain of the function tk-1 t≦ ≦ tm+1
– Below, k = 4, m = 9, domain, t3 t≦ ≦ t10
3
t
m+1
0
10. Computer Graphics
10/10/2008 Lecture 5 10
Cubic Uniform B-Spline
2D example
• For each i ≥ 4 , there is a knot between Qi-1 and Qi at t = ti.
• Initial points at t3 and tm+1 are also knots. The following
illustrates an example with control points set P0 … P9:
Knot.
Control point.
11. Computer Graphics
10/10/2008 Lecture 5 11
Uniform Non-rational B-Splines.
• First segment Q3 is defined by point P0 through P3 over the
range t3 = 0 to t4 = 1. So m at least 3 for cubic spline.
Knot.
Control point.P1
P2
P3
P0
Q3
12. Computer Graphics
10/10/2008 Lecture 5 12
Uniform Non-rational B-Splines.
• Second segment Q4 is defined by point P1 through P4 over
the range t4 = 1 to t5= 2.
Knot.
Control point.
Q4
P1
P3
P4
P2
13. Computer Graphics
10/10/2008 Lecture 5 13
A Bspline of order k is a parametric curve composed of a linear
combination of basis B-splines Bi,n
Pi (i=0,…,m) the control points
Knots: tj, j=0,…, k + m
The B-splines can be defined by
B-Spline :
A more general definition
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14. Computer Graphics
10/10/2008 Lecture 5 14
The shape of the basis functions
Bi,2 : linear basis functions
Order = 2, degree = 1
C0 continuous
http://www.ibiblio.org/e-notes/Splines/Basis.htm
15. Computer Graphics
10/10/2008 Lecture 5 15
The shape of the basis functions
Bi,3 : Quadratic basis functions
Order = 3, degree = 2
C1 continuous
http://www.ibiblio.org/e-notes/Splines/Basis.htm
16. Computer Graphics
10/10/2008 Lecture 5 16
The shape of the basis functions
Bi,4 : Cubic basis functions
Order = 4, degree = 3
C2 continuous
http://www.ibiblio.org/e-notes/Splines/Basis.htm
17. Computer Graphics
Uniform / non-uniform B-splines
• Uniform B-splines
– The knots are equidistant / non-equidistant
– The previous examples were uniform B-splines
• Parametric interval between knots does not
have to be equal.
Non-uniform B-splines
intervalsamet,equidistanwere,...,,, 210 mtttt
18. Computer Graphics
10/10/2008
Non-uniform B-splines.
• Blending functions no longer the same for each interval.
• Advantages
– Continuity at selected control points can be reduced to
C1 or lower – allows us to interpolate a control point
without side-effects.
– Can interpolate start and end points.
– Easy to add extra knots and control points.
• Good for shape modelling !
19. Computer Graphics
Controlling the shape of the curves
• Can control the shape through
– Control points
• Overlapping the control points to make it pass
through a specific point
– Knots
• Changing the continuity by increasing the
multiplicity at some knot (non-uniform bsplines)
20. Computer Graphics
10/10/2008 Lecture 5 20
Controlling the shape through
control points
First knot shown with 4
control points, and their
convex hull.
P1
P2P0
P3
21. Computer Graphics
10/10/2008 Lecture 5 21
Controlling the shape through
control points
First two curve segments
shown with their respective
convex hulls.
Centre Knot must lie in the
intersection of the 2 convex
hulls.
P1
P2P0
P4
P3
22. Computer Graphics
10/10/2008 Lecture 5 22
Repeated control point.
First two curve segments
shown with their respective
convex hulls.
The curve is forced to lie on
the line that joins the 2
convex hulls.
P1=P2
P0
P3
P4
23. Computer Graphics
10/10/2008 Lecture 5 23
Triple control point.
First two curve segments
shown with their respective
convex hulls.
Both convex hulls collapse
to straight lines – all the
curve must lie on these lines.
P1=P2=P3
P0
P4
24. Computer Graphics
10/10/2008 Lecture 5 24
Controlling the shape through knots
• Smoothness increases with order k in Bi,k
– Quadratic, k = 3, gives up to C1 continuity.
– Cubic, k = 4 gives up to C2 continuity.
• However, we can lower continuity order too with Multiple
Knots, ie. ti = ti+1= ti+2 = … Knots are coincident and so
now we have non-uniform knot intervals.
• A knot with multiplicity p is continuous to the
(k-1-p)th derivative.
• A knot with multiplicity k has no continuity at all, i.e. the
curve is broken at that knot.
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30. Computer Graphics
10/10/2008 Lecture 5 30
Summary of B-Splines.
• Functions that can be manipulated by a series of control
points with C2 continuity and local control.
• Don’t pass through their control points, although can be
forced.
• Uniform
– Knots are equally spaced in t.
• Non-Uniform
– Knots are unequally spaced
– Allows addition of extra control points anywhere in the set.