Curves n Surfaces_1

7/31/2019 Curves n Surfaces_1

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Dr. N.C.Chauhan @ Dept. of Information Technology, ADIT & M.E. Computer Engineering Programme, BVM

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Representing Curves and Surfaces

Advanced Computer Graphics

Representing Curves and Surfaces: Part I

Dr. NARENDRA C. CHAUHAN

Associate Professor

Department of Information Technology

A.D.Patel Institute of Technology


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Contents

Parametric Bicubic Surfaces

Hermite Curves

Polygon Meshes

Bezier Curves

Hermite, Bezier, and B-Spline Surfaces

Parametric Cubic Curves

BSplines


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Introduction

We need smooth curves and surfaces in many

applications:

model real world objects

computer-aided design (CAD) high quality fonts

data plots

artists sketches

To specify the path of camera or object inanimation sequence


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We need to represent curves and surfaces in 2 cases:

Modeling of existing objects (e.g. car, face, mountain) Physical objects available, but mathematicaldescription may not be available Infinitely many points can not be used

object is approximated with pieces of planes,spheres, or other shapes such the points on model areclose to points on actual object. modeling is not precise

Modeling a new object from scratch(CAD) No preexisting object available modeling is precise


5/53Dr. N.C.Chauhan @ Dept. of Information Technology, ADIT & M.E. Computer Engineering Programme, BVM

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Most common representation forsurface modeling:

polygon mesh surfaces

parametric surfaces They are simple generalizations of parametric curves

quadric surfaces

Solid modeling (next chapter)

Representation of volumes surrounded by surfaces

Surface modeling is also used

Introduction



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Polygon mesh:

set of connected planar surfaces bounded by

polygons

good for boxes, cabinets, building exteriors bad for curved surfaces (Fig.11.1, 11.2),

representation is only approximate (piecewise

linear approximation)

errors can be made arbitrarily small at the cost ofspace and execution time

enlarged images show geometric aliasing



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Parametric Polynomial Curves:

Define points on 3D curve using 3 polynomials in

parameter t, - x(t), y(t), z(t)

We use most commonly cubic polynomials.

Parametric bivariate polynomial surfacepatches:

Define points on curved surface using three

bivariate surface patches x(s,t),y(s,t),z(s,t)

Boundary of patches are parametric polynomial

curves



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Quadric Surfaces:

Defined implicitly by equation f(x,y,z)=0, wh. f is a

quadric polynomial in x, y and z.

Convenient for sphere, ellipsoid, and cylinder



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Polygon Meshes



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Polygon mesh is a collection of edges, vertices, and polygons

connected such that each edge is shared by at most twopolygons. Edge connect two vertices and polygon is closed

sequence of edges.

Methods of representing polygon meshes:

Explicit representation

Pointers to vertex list

Pointers to edge list

Each representation has adv. and disadv. Two basic criteria, timeand space, can be used to evaluate different representations.

Operations on polygon mesh: Find all edges incident to a vertex, finding polygon sharing

edge or vertex, finding vertices connected by an edge, findingedges of a polygon, displaying mesh, etc.



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Explicit Representation: Polygon is represented by list of vertex coordinates:

P=((x1,y1,z1), (x2,y2,z2),, (xn,yn,zn)) in order of theirtraversal

Drawbacks: Not space efficient

Coordinates of shared vertices are duplicated

No explicit representation of shared edges and vertices

To drag vertex ?? Find all polygons Edges are drawn twice, cause problem to some devices

Problem in raster display if edges are drawn in opposite

directions (extra pixels may be intensified)



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Pointers to vertex list

Each vertex stored once, in vertex list, V=((x1,y1,z1),

(x2,y2,z2),, (xn,yn,zn))

Polygon is defined as list of indices in the vertex list, P=(2,3,

5,7), e.g. Fig. 11.3

Advantage:

Space is saved, since each vertex is stored once.

Vertex can be changed easily

Drawbacks:

Still difficult to find polygons that share an edge

Shared polygon edges are still drawn twice


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Pointers to Edge list

Each vertex stored once, in vertex list, V=((x1,y1,z1),(x2,y2,z2),, (xn,yn,zn))

Polygon is not list of vertices but list of edges

Edge points to two vertices, and the list of polygons to which

edge belongs. P=(E1, E2, , En), E=(V1, V2, P1, P2), Fig. 11.4

Advantages:

Earlier all drawbacks are overcomed

Drawbacks: In none of 3 representation, it is easy to determine which

edges are incident to a vertex. (solution: winged-edge

representation)


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Parametric Cubic Curves

d d C G hi


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Polylines and polygons are 1st degree piecewise linear

approximations to curves and surfaces. Large no. of points need to be stored to obtain

reasonable accuracy.

Interactive manipulation of data is tedious.

Alternate solution: Use higher degree polynomials

Easy approximation of desired shape

Less storage, Easy interactive manipulation

Ad d C t G hi


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Higher degree approximations are basedon 3 methods:

Explicit functions

Implicit equations

Parametric representations

Ad n ed Comp te G phi


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Explicit form: y=f(x) and z=g(x) [or z=f(x,y)] It is impossible to get different yfor singlex, (circlue, ellipse)

Not a rotation invariant representation Difficult to represent curves with vertical tangents

Implicit form: f(x, y, z) = 0 Given equation may have more solution than we want. (how

to represent semicircle?) Difficult to connect two curves in a smooth manner Not efficient for drawing Useful for testing object inside/outside (in previous both),Normals to curves are also easily computed

Parametric: x=x(t), y=y(t), z=z(t) Overcomes problems by earlier two representations Very common in modeling Curve is approximated by piecewise curve segments



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Explicit functions:

y = f(x), z = g(x)

impossible to get multiple values for a

single x break curves like circles and ellipses into segments

not invariant with rotation rotation might require further segment breaking

problem with curves with vertical tangents infinite slope is difficult to represent



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Implicit equations:

f(x,y,z) = 0

equation may have more solutions than wewant

circle: x + y = 1, half circle: ?

problem to join curve segments together difficult to determine if their tangent directions agree at

their joint point



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Parametric representation:

x = x(t), y = y(t), z = z(t)

overcomes problems with explicit and implicitforms

no geometric slopes (which may be infinite)

parametric tangent vectors instead (never infinite)

a curve is approximated by a piecewisepolynomial curve

Cubic polynomials are generally used.



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Why cubic?

lower-degree polynomials give too little flexibility

in controlling the shape of the curve

higher-degree polynomials can introduce

unwanted wiggles and require more computation

lowest degree that allows curve to interpolateend points

lowest degree that is not planar in 3D

Given a cubic polynomial with it 4 coefficients, 4knowns are used to solve for the unknowncoefficients.



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General form of parametric cubic polynomials

that define curve segment Q(t)=[x(t) y(t) z(t)]:

CTtztytxtQ

ddd

ccc

bbb

aaa

CtttT

dtctbtatz

dtctbtaty

dtctbtatx

zyx

zyx

zyx

zyx

zzzz

yyyy

xxxx

)]()()([)(

,]1[

)(

)(

)(

23

23

23

23

where, t[0,1]



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Kinds of continuity:

Geometric:

G0: two curve segments join together

G1: directions of tangents are equal at the joint,

i.e. geometric slops are equal at join point Parametric:

C1: directions and magnitudes of tangents are

equal at the joint

Cn: directions and magnitudes of n-th

derivative are equal at the joint



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p p

If the tangent vectors of two cubic curve segments

are equal at the join point, the curve has first-

degree continuity, and is said to be C continuous

If the direction and magnitude ofd / dt [Q(t)]through the nth derivative are equal at the join

point, the curve is called C continuous

If the directions (but not necessarily the

magnitudes) of two segments tangent vectors areequal at the join point, the curve has G

continuity

1

n

n n

1



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p p



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p p

The curves Q1, Q2, Q3 join at point P Q1 and Q2 have equal tangent vectors at P and hence C1 and

G1 continuous

Q1 and Q3 have tangent vectors in the same direction but Q3

has twice magnitude, so they are only G1 continuous at P

Example



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Special case:C1 continuity does not imply G1 continuity.When both curves tangent vectors are [0 0 0] at join point.



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General form:

GMTtztytxtQ

GMCLet

CTtztytxtQ

ddd

ccc

bbbaaa

CtttT

dtctbtatz

dtctbtaty

dtctbtatx

zyx

zyx

zyx

zyx

zzzz

yyyy

xxxx

)]()()([)(

)]()()([)(

,]1[

)(

)(

)(

23

23

23

23

Where,T=vector of parameter

M= Basis MatrixG= Geometric Vector



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A curve segment is defined by constraints on endpoints,

tangent vectors, and continuity between curve segments.

Major types of curves:

Hermit defined by two endpoints and two tangent vectors

Bezier defined by two endpoints and two other points that control the

endpoint tangent vectors

B-Spline several kinds, each defined by four control points

uniform B-spline, non-uniform B-spline, other B-splines

Have c1 and c2 continuity at join points,



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How the coefficients of cubic polynomials depend

on four constraints:



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Hermite Curves



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Constraints



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Basis Matrix



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Blending Functions



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')(')32()2()132(

'

'

0001

0010

1323

1212

1)(

1

23

0

23

1

23

0

23

1

0

1

0

23

xttxttxtttxtt

x

x

x

x

ttttX

Hermite Basis (Blending) Functions



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Hermite Basis (Blending) Functions

x0 x1

x0'

X1

The graph shows the shape of thefour basis functions often called

blending functions.

They are labelled with the elementsof the control vector that they

weight.

Note that at each end only position isnon-zero, so the curve must touch

the endpoints

')(')32()2()132()( 123

0

23

1

23

0

23 xttxttxtttxtttX

R i C d S f



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Condition for Continuity

K>0 G1 continuityK=1 C1 continuity

R ti C d S f



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Interactive manipulation of end points and tangentsFor two Hermite cubic curve segments

Rep esenting C es and S faces



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Effect of Changing Magnitudes of Tangent vector at 1st

end point




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Effect of Changing Directions of Tangent vector at 1st

end point




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Main Drawback:

Requires the specification of the tangents

This information is not always available




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Bezier Curves




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Bezier Curves

Indirectly specifies end point tangent vector by specifying twointermediate points that are not on the curve.




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Generation of Basis Matrix

BBBHBHBHBHHH GMTGMMTGMMTGMTtQ )()()(




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Generation of

Basis MatrixAnd

Blending functions

Condition forC0 and C1 continuity




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p g

Blending Functions

Condition forC0 and C1 continuity




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p g

Continuity and Convex Hull Property

Examining BB polynomials, we can note that their sum is everywhereUnity and each polynomial is everywhere nonnegative for 0 t 1

Hence, Q(t) is weighted average of four control points.This means,

each curve segment is contained in the convex hull of 4 control points.Convex hull for 2D is convex polygon formed by four control points.

Convex hull property holds true for all cubic whose sum=1 and fn. are non vConvex hull property is useful for clipping curve segment.




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THINK GOODBE GOOD.

DO GOOD.

HAVE A NICE DAY !!!




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Generalization of parametric cubic curves.

For each value ofsthere is a family of curves in t.

Major kinds of surfaces:

Hermit, Bezier, B-spline

)()(),( tGMTtCTtsQ




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Summary

Polygon meshes

well suited for representing flat-faced objects

seldom satisfactory for curved-faced objects space inefficient

simpler algorithms

hardware support




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Summary

Piecewise cubic curves and bicubic surfaces

permit multiple values for a single x or y

represent infinite slopes

easier to manipulate interactively

can either interpolate or approximate

space efficient

more complex algorithms little hardware support




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Parametric bivariate (two-variable)

polynomial surface patches:

point on 3D surface = (x(u,v), y(u,v), z(u,v))

boundaries of the patches are parametric

polynomial curves

many fewer parametric patches than polynomial

patches are needed to approximate a curved

surface to a given accuracy more complex algorithms though




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Parametric polynomial curves:

point on 3D curve = (x(t), y(t), z(t))

x(t), y(t), and z(t) are polynomials usually cubic: cubic curves

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Curves n Surfaces_1