Computer  Graphics  (CS  543):  Curves  

Prof  Emmanuel  Agu    

Computer  Science  Dept.  Worcester  Polytechnic  Ins7tute  (WPI)  

So  Far…  

!  Dealt  with  straight  lines  and  flat  surfaces  !  Real  world  objects  include  curves  !  Need  to  develop:  

!  Representa?ons  of  curves  !  Tools  to  render  curves    

Curve  Representa<on:  Explicit  

!  One  variable  expressed  in  terms  of  another  !  Example:  

!  Works  if  one  x-­‐value  for  each  y  value  !  Example:  does  not  work  for  a  sphere  

!  Rarely  used  in  CG  because  of  this  limita?on  

),( yxfz =

22 yxz +=

Curve  Representa<on:  Implicit  

!  Represent  2D  curve  or  3D  surface  as  zeros  of  a  formula  !  Example:  sphere  representa?on  

!  May  limit  classes  of  func?ons  used  !  Polynomial:  func?on  which  can  be  expressed  as  linear  

combina?on  of  integer  powers  of  x,  y,  z  !  Degree  of  algebraic  func?on:  highest  power  in  func?on  !  Example:  mx4  has  degree  of  4  

01222 =−++ zyx

Curve  Representa<on:  Parametric  

!  Represent  2D  curve  as  2  func?ons,  1  parameter  

!  3D  surface  as  3  func?ons,  2  parameters  

!  Example:  parametric  sphere  

))(),(( uyux

)),(),,(),,(( vuzvuyvux

φφθ

θφφθ

θφφθ

sin),(sincos),(coscos),(

=

=

=

zyx

Choosing  Representa<ons  

!  Different  representa?on  suitable  for  different  applica?ons  

!  Implicit  representa?ons  good  for:  !  Compu?ng  ray  intersec?on  with  surface  !  Determing  if  point  is  inside/outside  a  surface  

!  Parametric  representa?on  good  for:  !  Breaking  surface  into  small  polygonal  elements  for  

rendering  !  Subdivide  into  smaller  patches  

!  Some?mes  possible  to  convert  one  representa?on  into  another  

Con<nuity  

!  Consider  parametric  curve  

!  We  would  like  smoothest  curves  possible  !  Mathema?cally  express  smoothness  as  con?nuity  (no  jumps)  

!  Defn:  if  kth  deriva?ves  exist,  and  are  con?nuous,  curve  has  kth  order  parametric  con?nuity  denoted  Ck  

TuzuyuxuP ))(),(),(()( =

Con<nuity  

!  0th  order  means  curve  is  con?nuous  !  1st  order  means  curve  tangent  vectors  vary  con?nuously,  etc  

Interac<ve  Curve  Design  

!  Mathema?cal  formula  unsuitable  for  designers  !  Prefer  to  interac?vely  give  sequence  of  points  (control  points)  

!  Write  procedure:  !  Input:  sequence  of  points  !  Output:  parametric  representa?on  of  curve  

Interac<ve  Curve  Design  !  1  approach:  curves  pass  through  control  points  (interpolate)  !  Example:  Lagrangian  Interpola?ng  Polynomial  !  Difficulty  with  this  approach:    

!  Polynomials  always  have  “wiggles”  !  For  straight  lines  wiggling  is  a  problem  

!  Our  approach:  approximate  control  points  (Bezier,  B-­‐Splines)  

De  Casteljau  Algorithm  

!  Consider  smooth  curve  that  approximates  sequence  of  control  points  [p0,p1,….]  

 

!  Blending  func?ons:  u  and  (1  –  u)  are  non-­‐nega?ve  and  sum  to  one  

10)1()( uppuup +−= 10 ≤≤ u

De  Casteljau  Algorithm  

!  Now  consider  3  points  !  2  line  segments,  P0  to  P1  and  P1  to  P2  

 

1001 )1()( uppuup +−= 2111 )1()( uppuup +−=

De  Casteljau  Algorithm  

 

 

)()1()( 1101 uuppuup +−=

22

102 ))1(2()1( pupuupu +−+−=

202 )1()( uub −=

Blending functions for degree 2 Bezier curve

)1(2)(12 uuub −=2

22 )( uub =

)(02 ub )(12 ub )(22 ub

Substituting known values of and )(01 up )(11 up

Note: blending functions, non-negative, sum to 1

De  Casteljau  Algorithm  

!  Extend  to  4  control  points  P0,  P1,  P2,  P3  

!  Final  result  above  is  Bezier  curve  of  degree  3  

32

21

20

3 ))1(3())1(3()1()( upuupuupuup +−+−+−=

)(23 ub)(03 ub )(13 ub )(33 ub

De  Casteljau  Algorithm  

!  Blending  func?ons  are  polynomial  func?ons  called  Bernstein’s  polynomials  

333

223

213

303

)(

)1(3)(

)1(3)(

)1()(

uubuuubuuub

uub

=

−=

−=

−=

32

21

20

3 ))1(3())1(3()1()( upuupuupuup +−+−+−=

)(23 ub)(03 ub )(13 ub )(33 ub

De  Casteljau  Algorithm  

!  Wri?ng  coefficient  of  blending  func?ons  gives  Pascal’s  triangle  

1

4

1

1

1

1

1

2

4

3

6

1 3

1

1

32

21

20

3 ))1(3())1(3()1()( upuupuupuup +−+−+−=

31 3 1

4 control points

3 control points

5 control points

De  Casteljau  Algorithm  

!  In  general,  blending  func?on  for  k  Bezier  curve  has  form  

!  Example  

 

iikik uu

ik

ub −−⎟⎟⎠

⎞⎜⎜⎝

⎛= )1()(

300303 )1()1(

03

)( uuuub −=−⎟⎟⎠

⎞⎜⎜⎝

⎛= −

De  Casteljau  Algorithm  

!  Can  express  cubic  parametric  curve  in  matrix  form  

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

3

2

1

0

32 ],,,1[)(

pppp

Muuuup B

where

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−

−

−=

1331036300330001

BM

Subdividing  Bezier  Curves  

!  OpenGL  renders  flat  objects    !  To  render  curves,  approximate  with  small  linear  segments  

!  Subdivide  surface  to  polygonal  patches  !  Bezier  curves  useful  for  elegant,  recursive  subdivision  

Subdividing  Bezier  Curves  

!  Let  (P0…  P3)  denote  original  sequence  of  control  points  !  Recursively  interpolate  with  u  =  ½  as  below    !  Sequences  (P00,P01,P02,P03)  and  (P03,P12,P21,30)  define  

Bezier  curves  also  !  Bezier  Curves  can  either  be  straightened  or  curved  recursively  

in  this  way  

Bezier  Surfaces  

!  Bezier  surfaces:  interpolate  in  two  dimensions  !  This  called  Bilinear  interpola?on  !  Example:  4  control  points,  P00,  P01,  P10,  P11,  2  parameters  u  

and  v  !  Interpolate  between    

!  P00  and  P01  using  u  !  P10  and  P11  using  u  !  P00  and  P10  using  v  !  P01  and  P11  using  v  

))1(())1)((1(),( 11100100 uppuvuppuvvup +−++−−=

Bezier  Surfaces  

!  Expressing  in  terms  of  blending  func?ons  

11111101011101000101 )()()()()()(),( pubvbpubbvbpubvbvup ++=

Generalizing

∑∑= =

=3

0

3

0,3,3, )()(),(

i jjiji pubvbvup

Problems  with  Bezier  Curves  !  Bezier  curves  are  elegant  but  too  many  control  points  !  To  achieve  smoother  curve    

!  =  more  control  points    !  =  higher  order  polynomial    !  =  more  calcula?ons  

!  Global  support  problem:  All  blending  func?ons  are  non-­‐zero  for  all  values  of  u  

!  All  control  points  contribute  to  all  parts  of  the  curve  !  Means  afer  modelling  complex  surface  (e.g.,  a  ship),  if  one  control  point  is  moves,  recalculate  everything!  

B-­‐Splines  !  B-­‐splines  designed  to  address  Bezier  shortcomings  !  B-­‐Spline  given  by  blending  control  points  !  Local  support:  Each  spline  contributes  in  limited  range  !  Only  non-­‐zero  splines  contribute  in  a  given  range  of  u  

∑=

=m

iii puBup

0

)()(

B-spline blending functions, order 2

NURBS  

!  Encompasses  both  Bezier  curves/surfaces  and  B-­‐splines  !  Non-­‐uniform  Ra?onal  B-­‐splines  (NURBS)  !  Ra?onal  func?on  is  ra?o  of  two  polynomials  !  Some  curves  can  be  expressed  as  ra?onal  func?ons  but  not  as  

simple  polynomials  !  No  known  exact  polynomial  for  circle  !  Ra?onal  parametriza?on  of  unit  circle  on  xy-­‐plane:  

0)(12)(

11)(

2

2

2

=+

=

+

−=

uzuuuy

uuux

NURBS  

!  We  can  apply  homogeneous  coordinates  to  bring  in  w  

!  Using  w,  we  get  we  cleanly  integrate  ra?onal  parametriza?on  !  Useful  property  of  NURBS:  preserved  under  transforma?on  

2

2

1)(0)(2)(1)(

uuwuz

uuyuux

+=

=

=

−=

Subdivision  Surfaces  

!  hips://3dcrea?veworld.wordpress.com/2015/01/20/subdivision-­‐surface-­‐modelling/  

References  

!  Hill  and  Kelley,  chapter  11  !  Angel  and  Shreiner,  Interac?ve  Computer  Graphics,  6th  edi?on,  Chapter  10  

!  Shreiner,  OpenGL  Programming  Guide,  8th  edi?on