Curves

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

Unit 2Bezier Curves & Surfaces

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

Learning ObjectivesIn this unit, we will cover the following topics:

CurvesProblems with complex curveBezier CurveAdvantagesCubic Bezier CurvesB Splines More About B Splines

Knot VectorKinds of Knot VectorExerciseSummary


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?

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Computers can’t draw curves

Curves are just lots of small straight lines joined upThe 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 scaledinconvenient to have to specify so many pointsneed lots of points to make the curve look smooth

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Curve representation (contd.)working out the equation that represents the curve

difficult for complex curvesmoving an individual point requires re-calculation of the entire

curve


SOLUTION ???

Complex Curves


Solution For Complex Curves

INTERPOLATION

APPROXIMATION

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Interpolation vs. Approximation Curves

Interpolationcurve must pass through control points

Approximationcurve is influenced by control points


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


Interpolation vs. Approximation Curves

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


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

x(t) = an-1tn-1 + .... + a2t2 + a1t + a0y(t) = bn-1tn-1 + .... + b2t2 + b1t + b0


Bezier Curves

An alternative to interpolation splinesM. 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.

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Curves & Control PointsBezier 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

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Bezier Curves- PropertiesNot all of the control points are on the line– Some just attract it towards themselvesPoints have “influence” over the course of the line

“Influence” (attraction) is calculated from a polynomial expression


Bezier CurvesN+1 control points designated by the vectorspk = (xk,yk,zk) k=0,…n

Thr order n Bezier Curve P(u) is given byP(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

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SolutionIdea: Build functions that are combinations of some basic and

simpler functions.

Basic functions: B-splinesBernstein polynomials


Bernstein PolynomialsBernstein 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)^2B2(t) = 3t^2(1-t)

B3(t) = t^3

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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”pointsThe 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 PolynomialsGiven a set of control points {Pi}N

i=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)

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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 < 1P(t) is the Bezier curve


Cubic Bézier Curve4 control pointsCurve passes through first & last control pointCurve is tangent at P1 to (P1-P2) and at P4 to (P4-P3)


Cubic Bézier Curve

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Cubic Bézier Curve


Higher-Order Bézier Curves> 4 control pointsBernstein Polynomials as the basis functionsBin(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.)

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


Higher-Order Bézier Curves


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

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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 BezierIf 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.

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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 .

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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 inits 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.

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Spline InterpolationLinear, Quadratic, Cubic

Preferred over other polynomial interpolationMore efficient

High-degree polynomials are very computationally expensive Smaller errorInterpolant is smoother


Natural Cubic Spline Interpolation

Si(x) = aix3 + bix2 + cix + di

4 Coefficients with n subintervals = 4n equationsThere are 4n-2 conditions

Interpolation conditionsContinuity conditionsNatural Conditions

S’’(x0) = 0S’’(xn) = 0


Natural Conditions

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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 NotationA 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 SplinesThe Ni,k functions are described as follows

and if k > 1

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Properties of B-Spline

For n+1 control points the curve is described with n +1 blendingfunctions.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 SPLINEIterative method for constructing BSplines

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Knot VectorA 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 precedingone.


Example : Knot VectorFor 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 VectorA B-spline curve whose knot vector is evenly spaced is known as a uniform B-spline.These are knot vectors for which

For example:

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Open Uniform Knot VectorIf 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 VectorIf the knot vector is not evenly spaced, the curve is called a non-

uniform B-spline . Here

For example:


Example: 5 Control PointsThe 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.

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Why use BsplinesB-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


SummaryA 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.

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Exercise

Q.1 Multiple Choice Questionsa) 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 withcontrol points and where , we must have


Exercise (Cont’d..)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?

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Exercise (Cont’d..)

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-splinecurve.

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

Curves

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