Curves

Unit 2
Bezier Curves & Surfaces

© Bharati Vidyapeeth’s Institute of Computer Applications and Management, New Delhi-63 1

Learning Objectives
In this unit, we will cover the following topics:
Curves
Problems with complex curve
Bezier Curve
Advantages
Cubic Bezier Curves
B Splines
More About B Splines
Knot Vector
Kinds of Knot Vector
Exercise
Summary

© Bharati Vidyapeeth’s Institute of Computer Applications and Management, New Delhi-63. 2

Curves?

Straight lines are easy to store in a computer’s
memory. The computer only has to remember the first
and the last point. But what about curved lines?


Computers can’t draw curves

Curves are just lots of small straight lines joined up
The more points/line segments that are used the smoother
the curve.


Curve representation

The basic problem we face is to represent the curve EASILY and
EFFICIENTLY.


Curve representation (contd.)
Storing a curve as many small straight line segments

doesn’t work well when scaled
inconvenient to have to specify so many points
need lots of points to make the curve look smooth


Curve representation (contd.)
working out the equation that represents the curve
difficult for complex curves
moving an individual point requires re-calculation of the entire
curve


Complex Curves

SOLUTION ???


Solution For Complex Curves

INTERPOLATION

APPROXIMATION


Interpolation vs. Approximation Curves

Interpolation Approximation
curve must pass through control curve is influenced by control
points points


Review of Interpolation
A method of constructing a function that crosses through a discrete
set of known data points.


Interpolation vs. Approximation Curves


Solution - Interpolation
Define a small number of points
Use a technique called “interpolation” to invent the extra points for
us.
Join the points with a series of (short) straight lines


Interpolation Curves
Curve is constrained to pass through all control points
Given points P0, P1, ... Pn, find lowest degree polynomial which
passes through the points

x(t) = an-1tn-1 + .... + a2t2 + a1t + a0
y(t) = bn-1tn-1 + .... + b2t2 + b1t + b0


Bezier Curves
An alternative to interpolation splines
M. Bezier was 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.


Curves & Control Points
Bezier curves are used in computer graphics to produce curves
which appear reasonably smooth at all scales.
Bezier curves use a construction due to Sergei Bernstein, in which
the interpolating polynomials depend on certain control points.


Example : Curves & Control Points
Each set of four points P0, P1, P2, P3 we associate a curve with
the following properties:
It starts at p0 and ends at p3.
When it starts from p0 it heads directly towards p1, and when it
arrives at p3 it is coming from the direction of p2.
The entire curve is contained in the quadrilateral whose corners
are the four given points


Bezier Curves


Bezier Curves- Properties
Not all of the control points are on the line
– Some just attract it towards themselves
Points have “influence” over the course of the line
“Influence” (attraction) is calculated from a polynomial
expression


Bezier Curves
N+1 control points designated by the vectors
pk = (xk,yk,zk) k=0,…n

Thr order n Bezier Curve P(u) is given by
P(u) = ∑pk Bk,,n(u) k= 0 to n


Problems with Bezier Curves

A similar but different problem:
Controlling the shape of curves.

Problem: given some (control) points, produce and modify the
shape of a curve passing through the first and last point.

http://www.ibiblio.org/e-notes/Splines/Bezier.htm


Solution
Idea: Build functions that are combinations of some basic and
simpler functions.

Basic functions: B-splines
Bernstein polynomials


Bernstein Polynomials
Bernstein polynomials of degree n are defined by:

For v = 0, 1, 2, …, n,
In general there are N+1 Bernstein Polynomials of degree N.
For example, the Bernstein Polynomials of degrees 1, 2, and
3 are:
1. B0,1(t) = 1-t, B1,1(t) = t;
2. B0,2(t) = (1-t)2, B1,2(t) = 2t(1-t), B2,2(t) = t2;
3. B0,3(t) = (1-t)3, B1,3(t) = 3t(1-t)2, B2,3(t)=3t2(1-t),
B3,3(t) = t3;


Deriving the Basis Polynomials

Write 1 = t + (1-t) and raise 1 to the third power for cubic
splines (2 for quadratic, 4 for quartic, etc…)
1^3 = (t + (1-t))^3 = t^3 + 3t^2(1-t) + 3t(1-t)^2 + (1-t)^3
The basis functions are the four terms:
B0(t) = (1-t)^3
B1(t) = 3t(1-t)^2
B2(t) = 3t^2(1-t)
B3(t) = t^3


Using the Basis Functions

To use the Bernstein basis functions to make a spline we need
4 “control” points for a cubic. (3 for quadratic, 5 for quartic,
etc…)
Let (x0,y0), (x1,y1), (x2,y2), (x3,y3) be the “control”points
The x and y coordinates of the cubic Bezier spline are given
by:
x(t) = x0B0(t) + x1B1(t) +x2B2(t) + x3B3(t)
y(t) = y0B0(t) + y1B1(t) + y2B2(t) + y3B3(t)
Where 0<= t <= 1.


Bernstein Polynomials
Given a set of control points {Pi}Ni=0, where Pi = (xi, yi). A
Bezier curve of degree N is:
P(t) = Ni=0 PiBi,N(t),
Where Bi,N(t), for I = 0, 1, …, N, are the Bernstein
polynomials of degree N.
P(t) is the Bezier curve
Since Pi = (xi, yi)
x(t) = Ni=0xiBi,N(t) and y(t) = Ni=0yiBi,N(t)
The spline goes through (x0,y0) and (x3,y3)
The tangent at those points is the line connecting
(x0,y0) to (x1,y1) and (x3,y3) to (x2,y2).
The positions of (x1,y1) and (x2,y2) determine the
shape of the curve.

Bernstein Polynomials Example

Find the Bezier curve which has the control points (2,2),
(1,1.5), (3.5,0), (4,1). Substituting the x- and y-coordinates
of the control points and N=3 into the x(t) and y(t) formulas
on the previous slide yields
x(t) = 2B0,3(t) + 1B1,3(t) + 3.5B2,3(t) + 4B3,3(t)
y(t) = 2B0,3(t) + 1.5B1,3(t) + 0B2,3(t) + 1B3,3(t)


Bernstein Polynomials Example

Substitute the Bernstein polynomials of degree three into these
formulas to get
x(t) = 2(1 – t)3 + 3t(1 – t)2 + 10.5t2(1 – t) + 4t3
y(t) = 2(1-t)3 + 4.5t(1 – t)2 + t3
Simplify these formulas to yield the parametric equation:
P(t) = (2 – 3t + 10.5t2 – 5.5t3, 2 – 1.5t – 3t2 + 3.5t3)
where 0 < t < 1
P(t) is the Bezier curve


Cubic Bézier Curve
4 control points
Curve passes through first & last control point
Curve is tangent at P1 to (P1-P2) and at P4 to (P4-P3)


Cubic Bézier Curve


Cubic Bézier Curve


Higher-Order Bézier Curves
> 4 control points
Bernstein Polynomials as the basis functions
Bin(t) is Blending Function

Every control point affects the entire curve
– Not simply a local effect
- More difficult to control for modeling


Composite Bezier Curves

In practice, shapes are often constructed of multiple Bezier
curves combined together.
Allows for flexibility in constructing shapes. (Consecutive
Bezier curves need not join smoothly.)


Parametric equations for x(t),y(t)


Higher-Order Bézier Curves


Properties of Bezier Curves
The degree of the bezier curve is one less than the number of
control point defining it.
N=control point-1
The curve generally follows the shape of the defining polygon (i.e.
polygon formed by connecting the control points.)


Advantages of Bezier Curves

The Bezier Curve is always contained within the convex hull of
its control points.
This is extremely useful for collision detection, because a
bounding box to the Bezier curve’s control points can be
created very fast.


More About Bezier
If we map every point on the Bezier curve with an affine map, the
resulting points also form a Bezier curve.
This is extremely useful in computer graphics, because if we have
a curve in 3D-space and need to project it on a surface, the only
thing we have to do is to project each control point. The same goes
for rotation, mirroring, skewing etc.


More About Bezier (Cont’d)
If we change a control point, the curve will mostly change around
that control point. The other parts of the curve will change very
little.
This is one reason Bezier curves are used in CAD.
If you change one control point a little, you don’t want the whole
drawing to change.


B SPLINE CURVES


An Introduction to B-Spline Curves

Most shapes are simply too complicated to define using a single
Bezier curve.

A spline curve is a sequence of curve segments that are connected
together to form a single continuous curve.

Example, a piecewise collection of Bezier curves, connected end
to end, can be called a spline curve


History behind Splines

The word “spline” comes from the ship building industry, where it
originally meant a thin strip of wood which draftsmen would use
like a ﬂexible French curve.
Metal weights (called “ducks”) were placed on the drawing
surface and the spline was threaded between the ducks as in Figure
.


Splines
Smooth curve defined by some control points

Moving the control points changes the curve


Definition
A B-spline is defined by an ordered set of control points which
determines what path the curve will follow and consequently
how the curve will look.
A longer B-spline is actually made up of several curve segments,
each one of which is defined by some number of control points in
its vicinity


Spline (general case)

Given n+1 distinct knots xi such that:

with n+1 knot values yi find a spline function

with each Si(x) a polynomial of degree at most n.


Spline Interpolation
Linear, Quadratic, Cubic
Preferred over other polynomial interpolation
More efficient
High-degree polynomials are very computationally
expensive
Smaller error
Interpolant is smoother


Natural Cubic Spline Interpolation

Si(x) = aix3 + bix2 + cix + di
4 Coefficients with n subintervals = 4n equations
There are 4n-2 conditions
Interpolation conditions
Continuity conditions
Natural Conditions
S’’(x0) = 0
S’’(xn) = 0


Natural Conditions


Weighting factor in Splines
A point on a particular curve segment is calculated by simply
adding up the coordinate values of its defining control points
after they have been multiplied by a weighting factor.
The weighting factor is calculated using a set of parametric basis
or blending functions.


Spline Notation
A B-spline curve P(t), is defined by

Where
• the {Pi : i = 0, 1, ..., n} are the control points
• k is the order of the polynomial segments of the B-spline curve.
Order k means that the curve is made up of piecewise polynomial
segments of degree k − 1
• The Ni, k(t) are the “normalized B-spline blending functions”. They
are described by the order k and by a non-decreasing sequence of
real numbers {ti : i = 0, ..., n + k}normally called the “knot
sequence”.

Functions used in B Splines
The Ni,k functions are described as follows

and if k > 1


Properties of B-Spline
For n+1 control points the curve is described with n +1 blending
functions.
Each blending function Bk,d is defined over d subintervals of the
total range of u, starting at knot value uk.
The range of parameter u is divided into n + d ,subintervals by
the n + d + 1 values specified in the knot vector.


Properties (Cont’d..)
Each section of the spline curve ( between 2 successive knot
values) is influenced by d control points.
Any one control point can affect the shape of at most d curve
sections.


CUBIC B SPLINE
Iterative method for constructing BSplines


Knot Vector
A knot vector is a list of parameter values, or knots, that specify
the parameter intervals for the individual Bezier curves that make
up a B-spline.

The knot vector can, by its definition, be any sequence of numbers
provided that each one is greater than or equal to the preceding
one.


Example : Knot Vector
For example, the knot vector in Figure would be [t0, 1, 2, 3, 4, 5, 6,
7, 8, t9], where the values of t0 and t9 have absolutely no effect on
the curve.


Uniform Knot Vector
A B-spline curve whose knot vector is evenly spaced is known
as a uniform B-spline.
These are knot vectors for which

For example:


Open Uniform Knot Vector
If any knot value is repeated,it is referred to as a multiple knot.
Also known as open Uniform knot vector.
For example:


Non-uniform Knot Vector
If the knot vector is not evenly spaced, the curve is called a non-
uniform B-spline . Here

For example:


Example: 5 Control Points
The picture below shows a set of 5 control points (so n+1=5), and
two separate B-Spline curves through them.
The red curve has k=3, while the blue curve has k=4. The knot
vector used was uniform.
In the case with k=3, there are n+k+1=8 knots: 0,1,2,3,4,5,6,7,
while in the case with k=4, there are 9 knots: 0,1,2,3,4,5,6,7,8.


Why use Bsplines
B-spline curves require more information (i.e., the degree of the
curve and a knot vector) and a more complex theory than Bezier
curves. But, it has more advantages:
B-spline curve can be a Bézier curve.
B-spline curves satisfy all important properties that Bézier curves
have.
B-spline curves provide more control flexibility than Bézier curves
can do.
For example, the degree of a B-spline curve is separated from the
number of control points. More precisely, we can use lower degree
curves and still maintain a large number of control points.


Bézier Surfaces


Summary
A Curve is specified by set of points called the control point.
The shape of the curve should not get affected when the control
points are measured.
The control points control the shape interactively and hence a
designer tries to manipulate the shape of the curve by moving the
control points in a neighborhood.
Most shapes are simply too complicated to define using a single
Bezier curve.
A spline curve is a sequence of curve segments that are connected
together to form a single continuous curve.
Blending functions are a set of functions which are used to
represent curve.
Knot Vector specify the parameter intervals for the individual
Bezier curves that make up a B-spline.

Exercise
Q.1 Multiple Choice Questions
a) The following knot vector [0, 0, 0, 0, 1 , 2, 2, 2, 2] represents
(A) Bezier spline (B) open uniform B-spline
(C) Periodic B-spline (D) Non-uniform B-spline

b) A B-spline curve with (n+1) blending function has
(A) n control points. (B) (n+1) control points.
(C) (n-1) control points. (D) n2 control points.

c) For which of the curves it is easier to compute basis functions.
(A) Bezier curves. (B) open B-spline.
(C) Uniform B-spline. (D) periodic B-spline


Exercise (Cont’d..)
d) Which one of the following is one of the blending functions for
Bezier curve (cubic)?
(A) 3u2 (B) u2
(C) 3u3 (D) u (1-u)2
e) State True/False
“For n + 1 control points, the B-spline curve is described with n
blending functions”.
f) For second order parametric continuity in Bezier curves with
control points and where , we must have


Q.2 Attempt all questions:
a.‘Bezier Curve do not allow for local control of the curve shape’.
Justify.
b. Give the blending functions for uniform quadratic B-spline with
knot vectors [0, 1, 2, 3, 4, 5,
6]
c. What are the boundary conditions for periodic cubic splines with
four control points?
d. What do you understand by C1 and C2 continuity between two
cubic Bezier segments? Derive the conditions for the same in
terms of control point of the two cubic Bezier curve segments.
e. Define B-spline. What do you mean by knot vector?


f. In computer graphics, why do we prefer parametric
representation of curves?
g. What do you mean by knot vector? What are various types of
knot vectors?
h. Give the general expression for B-spline curve defined by Cox-
de-Boor. Give also the characterstics of B-spline
curve.
i. Write the blending functions of cubic Bezier curves. Draw
their curves also.
j Draw a Bezier curve by hand with four control points P1, P2,
P3, P4


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