This document discusses B-spline curves. It defines B-splines as piecewise polynomial curves defined by a set of control points and a knot vector. B-splines provide local, rather than global, control over the curve shape and produce smoother curves than Bezier curves. The document covers properties of B-splines like degree independence and continuity, as well as blending functions, knot vectors, advantages over Bezier curves, and applications in modeling and animation.

1. B-SPLINE CURVE
FAKIR MOHAN UNIVERSITY
DEPT. OF COMPUTER SCIENCE
NAME-SUBHASHREE PRADHAN
ROLLNO:-15101FT222032
GUIDED BY DR.MINATI MISHRA

2. CONTENT:-
SPLINE CURVE
TYPES OF SPLINE CURVE
B-SPLINE CURVE
PROPERTIES OF B-SPLINE CURVE
BLENDING FUNCTION
TYPES OF KNOT VECTOR
BEZIER CURVE VS B-SPLINE CURVE
ADVANTAGE OF B-SPLINE CURVE
APPLICATION OF B-SPLINE CURVE
CONCLUSION

3. Spline Curve:-
 A spline curve is a mathematical function that is defined piecewise by polynomials.
This means that the curve is made up of a series of polynomial segments, each of which
is connected to the next segment smoothly. Each segment, called a spline, is controlled
by a set of control points or knots.
 Spline curves are often used in computer graphics and computer-aided design (CAD)
because they can be used to represent smooth, complex curves.

4. Types Of Spline Curve:-
Some of the types of Spline curves that are commonly used are :-
Spline Curve
Bezier B-Spline

5. B-Spline Curve:-
 Concept of B-spline curve came to resolve the disadvantages having by Bezier curve, as
we all know that both curves are parametric in nature. In Bezier curve we face a problem,
when we change any of the control point respective location the whole curve shape gets
change. But here in B-spline curve, the only a specific segment of the curve-shape gets
changes or affected by the changing of the corresponding location of the control points.
 A B-spline curve is a piecewise polynomial curve that is defined by a set of control
points and a knot vector. The knot vector specifies the location of the knots, which are the
points at which the polynomial segments of the curve are joined. The control points control
the shape of the curve.

7. • We can write a general expression for the calculation of coordinate positions along a B-spline curve in a
blending-function formulation as
• where the 𝑃𝑘are an input set of n + 1 control points.
• The range of parameter ‘u’ now depends on how we choose the B-spline parameters.
• The B-spline blending functions 𝑩𝒌,𝒅 are polynomials of degree (d – 1), where parameter d can be
chosen to be any integer value in the range from 2 up to the number of control points ( n + 1).
• Local control for B-splines is achieved by defining the blending functions over subintervals of the total
range of ‘u’.

8. Properties of B-spline curve:-
1) B-spline basis is non-global(local) effect. In this each control point affects the shape of the curve only
over range of parameter values where its associated basis function is non-zero.
2) The degree of B-spline is independent of No. of control points.
3) We can add/modify any no. of control points to change the shape of the curve without affecting the
degree of polynomial.
4) Local control for the B-spline is achieved by defining a blending function over “d” subintervals over the
total range of “u”. The selected set of subinterval endpoints 𝑢𝑘is referred to as a knot vector.
o for example;
Degree=2, Control points=3, Total Knot=?, Knot Vector =?
A:- Degree=d-1=2 Control Points=n+1=3
=>d=3 =>n=2

9. Here 𝑢𝑖 = 0 ≤ 𝑖 ≤ 𝑛 + 𝑑
= 0 ≤ 𝑖 ≤ 5
Knot Vectors={𝑢0, 𝑢1, 𝑢2, 𝑢3, 𝑢4, 𝑢5}
Knot Values= {0,0,0,1,1,1}
𝑢𝑖 =
0, 𝑖 < 𝑑
𝑖 − 𝑑 + 1, 𝑑 ≤ 𝑖 ≤ 𝑛
𝑛 − 𝑑 + 2, 𝑖 > 𝑛
5) The polynomial curve has degree d-1 & 𝐶𝑑−2continuity over the range of u.
6) The range of parameter “u” is divided into “n+d” subintervals by “n+d+1” values.
7) The curve lies within the convex hull of its defining polygon.
8) Each Blending function 𝐵𝑘,𝑑 is defined over d no. of subinterval over the range of “u”, starting from knot
value 𝑢𝑘.
9) The sum of the B-spline basis function for any parameter value u is 1.
𝑘=0
𝑛
𝐵𝑘,𝑑 𝑢 = 1

10. Blending Function(𝑩𝒌,𝒅):-
 Blending functions for B-spline curves are defined by the Cox-deBoor recursion formulas:

11. Types of Knot Vector:-
1) Uniform Knot:- In a Uniform Knot vector individual knot values are evenly spaced
e.g.[0 1 2 3 4].
2) Open Uniform Knot:- It has multiplicity of knot values at the ends equal to the order k
of B-spline basis function/blending function. Interval knot values are evenly spaced.
e.g.
k=2[0 0 1 2 3 3 ]
k=3[0 0 0 1 2 3 3 3]

12. • The data points in the Bezier Curve are n-1 and the
Bezier Curve deals with polynomials of degree n.
• The data points and the degree of the data points are
dependent in the Bezier Curve modelling.
• The shape of the curve is manipulated globally using
the control points.
• The shape of the curve is can easily be disturbed
because when you try to change the single control
point whole shape of the curve will be changed.
• The Bezier Curve has the global control over the
curve.
• The curves produced by the Bezier Curve are not that
clear when compared to the curves produced by the
B-Spline curve.
• The computation of the Bezier curve is easy when
compared to B-Spline curve.
• The data points in the B-Spline curve are n+1 data
points and the B-Spline curve deals with the
polynomials of any degree from one to n.
• The data points and the degree of the data points
in the B-Spline curve are independent there will
be no relation between them.
• The shape of the curve is locally manipulated
using the control points.
• The shape of the curve is not disturbed easily
because changing one of the control points does
not change the whole shape of the curve.
• The B-Spline curve has the local control over the
curve.
• The B-Spline curve produces clearer and neat
curves than the Bezier Curve.
• The computation of the B-Spline curve is
somewhat tough when compared to Bezier Curve.
BEZIER CURVE:- B-SPLINE CURVE:-

13. ADVANTAGES & DISADVANTAGES OF B-SPLINE CURVE:-
ADVANTAGES:-
The B-Spline curve produces clearer and neat
curves when compared to Bezier Curve.
The B-Spline Curves can be used for the
polynomials of any degree.
The shape of the curve is locally manipulated
or controlled so that there will be fewer
disturbances in the curve.
DISADVANTAGES:-
The B-Spline curve always require more
control points because every control point can
only control some part of the curve.
The B-Spline curve computation is a bit hard
process when compared to the computation of
the Bezier Curve.

14. APPLICATION :-
Modeling animation characters for the purposes of modeling
animated movies and True Type fonts.
In the calculation of optimal orbit and trajectory of aircraft
flight.

15. CONCLUSION:-
B-splines are not used very often in 2D graphics software but are used quite
extensively in 3D modeling software.
They have an advantage over Bezier curves in that they are smoother and
easier to control.
B-splines consist entirely of smooth curves, but sharp corners can be
introduced by joining two spline curve segments.
The continuous curve of a b-spline is defined by control points. While the
curve is shaped by the control points, it generally does not pass through
them.

16. REFERENCES:-
Computer Graphics C Version by Donald Hearn & M Pauline Baker II
Edition.
Fred T. Hofstetter, Multimedia Literacy, Tata McGraw Hill, 1995.
R1. Roy A. Plastock & Zhigang Xiang, Schaum’s Outline of Computer
Graphics, Second Edition, Tata McGraw-Hill.