b spline blossoms
play

B-Spline Blossoms CS 418 Interactive Computer Graphics John C. - PowerPoint PPT Presentation

Jan 09, 2023 •149 likes •447 views

B-Spline Blossoms CS 418 Interactive Computer Graphics John C. Hart The Blossoming Game B-Spline Rules for Setting Up the Board t =3 t =4 t =2 t =5 The Blossoming Game B-Spline Rules for Setting Up the Board 1. Create a knot vector (with

  1. B-Spline Blossoms CS 418 Interactive Computer Graphics John C. Hart

  2. The Blossoming Game B-Spline Rules for Setting Up the Board t =3 t =4 t =2 t =5

  3. The Blossoming Game B-Spline Rules for Setting Up the Board 1. Create a knot vector (with two extra values at each end) t =3 t =4 t =2 t =5 knot vector: [0 1 2 3 4 5 6 7]

  4. The Blossoming Game B-Spline Rules for Setting Up the Board 1. Create a knot vector (with two extra values at each end) t =3 t =4 2. Label each control vertex with triples of knots t =2 t =5 knot vector: [0 1 2 3 4 5 6 7]

  5. The Blossoming Game B-Spline Rules for Setting Up the Board 1. Create a knot vector (with two extra values at each end) t =3 t =4 2. Label each control vertex with triples of knots t =2 t =5 p (0,1,2) knot vector: [ 0 1 2 3 4 5 6 7]

  6. The Blossoming Game B-Spline Rules for Setting Up the Board 1. Create a knot vector (with two extra values at each end) t =3 t =4 2. Label each control vertex p (1,2,3) with triples of knots t =2 t =5 p (0,1,2) knot vector: [0 1 2 3 4 5 6 7]

  7. The Blossoming Game B-Spline Rules for Setting Up the Board p (2,3,4) 1. Create a knot vector (with two extra values at each end) t =3 t =4 2. Label each control vertex p (1,2,3) with triples of knots t =2 t =5 p (0,1,2) knot vector: [0 1 2 3 4 5 6 7]

  8. The Blossoming Game B-Spline Rules for Setting Up the Board p (2,3,4) p (3,4,5) 1. Create a knot vector (with two extra values at each end) t =3 t =4 2. Label each control vertex p (1,2,3) with triples of knots t =2 t =5 p (0,1,2) knot vector: [0 1 2 3 4 5 6 7]

  9. The Blossoming Game B-Spline Rules for Setting Up the Board p (2,3,4) p (3,4,5) 1. Create a knot vector (with two extra values at each end) t =3 t =4 2. Label each control vertex p (1,2,3) p (4,5,6) with triples of knots t =2 t =5 p (0,1,2) knot vector: [0 1 2 3 4 5 6 7]

  10. The Blossoming Game B-Spline Rules for Setting Up the Board p (2,3,4) p (3,4,5) 1. Create a knot vector (with two extra values at each end) t =3 t =4 2. Label each control vertex p (1,2,3) p (4,5,6) with triples of knots t =2 t =5 p (0,1,2) p (5,6,7) knot vector: [0 1 2 3 4 5 6 7 ]

  11. The Blossoming Game B-Spline Rules for Setting Up the Board p (2,3,4) p (3,4,5) 1. Create a knot vector (with two extra values at each end) t =3 t =4 2. Label each control vertex p (1,2,3) p (4,5,6) with triples of knots t =2 t =5 p (0,1,2) p (5,6,7) knot vector: [0 1 2 3 4 5 6 7]

  12. Bohm Blossoms B-Spline Rules for Winning the Game: p (2,3,4) p (3,4,5) Convert a B-Spline to a Bezier Curve • Bohm Algorithm t =3 t =4 • Trick: Think of each segment p (1,2,3) p (4,5,6) as a Bezier curve t =2 t =5 p (0,1,2) p (5,6,7) knot vector: [0 1 2 3 4 5 6 7]

  13. Bohm Blossoms B-Spline Rules for Winning the Game: p (2,3,4) p (3,4,5) Convert a B-Spline to a Bezier Curve • Bohm Algorithm p (3,3,3) p (4,4,4) • Trick: Think of each segment p (1,2,3) p (4,5,6) as a Bezier curve p (2,2,2) p (5,5,5) p (0,1,2) p (5,6,7) knot vector: [0 1 2 3 4 5 6 7]

  14. Bohm Blossoms B-Spline Rules for Winning the Game: p (2,3,4) p (3,4,5) Convert a B-Spline to a Bezier Curve • Bohm Algorithm p (3,3,3) • Trick: Think of each segment p (1,2,3) p (4,5,6) as a Bezier curve p (2,2,2) • Where should the other two control points go p (0,1,2) for the [2,3] segment? p (5,6,7) knot vector: [0 1 2 3 4 5 6 7]

  15. Bohm Blossoms B-Spline Rules for Winning the Game: p (2,3,4) p (3,4,5) Convert a B-Spline to a Bezier Curve • Bohm Algorithm p (3,3,3) • Trick: Think of each segment p (1,2,3) p (4,5,6) as a Bezier curve p (2,2,2) • Where should the other two control points go p (0,1,2) for the [2,3] segment? p (5,6,7) • Need to find: p (2,2,3) knot vector: [0 1 2 3 4 5 6 7] p (2,3,3)

  16. Bohm Blossoms B-Spline Rules for Winning the Game: p (4,2,3) p (3,4,5) Convert a B-Spline to a Bezier Curve • Bohm Algorithm p (3,3,3) • Trick: Think of each segment p (1,2,3) p (4,5,6) as a Bezier curve p (2,2,2) • Where should the other two control points go p (0,1,2) for the [2,3] segment? p (5,6,7) • Need to find: p (2,2,3) knot vector: [0 1 2 3 4 5 6 7] p (2,3,3)

  17. Bohm Blossoms B-Spline Rules for Winning the Game: p (4,2,3) p (3,4,5) Convert a B-Spline to a Bezier Curve • Bohm Algorithm p (3,3,3) p (2,2,3) • Trick: Think of each segment p (1,2,3) p (4,5,6) as a Bezier curve p (2,2,2) • Where should the other two control points go p (0,1,2) for the [2,3] segment? p (5,6,7) • Need to find: p (2,2,3) = 2/3 p (1,2,3) + 1/3 p (4,2,3) knot vector: [0 1 2 3 4 5 6 7] p (2,3,3)

  18. Bohm Blossoms B-Spline Rules for Winning the Game: p (4,2,3) p (3,4,5) Convert a B-Spline to a Bezier Curve p (3,2,3) • Bohm Algorithm p (3,3,3) p (2,2,3) • Trick: Think of each segment p (1,2,3) p (4,5,6) as a Bezier curve p (2,2,2) • Where should the other two control points go p (0,1,2) for the [2,3] segment? p (5,6,7) • Need to find: p (2,2,3) = 2/3 p (1,2,3) + 1/3 p (4,2,3) knot vector: [0 1 2 3 4 5 6 7] p (2,3,3) = 1/3 p (1,2,3) + 2/3 p (4,2,3)

  19. Bohm Blossoms • Where are the endpoints located? p (4,2,3) p (3,4,5) p (3,2,3) p (2,2,3) p (1,2,3) p (0,1,2)

  20. Bohm Blossoms • Where are the endpoints located? p (4,2,3) p (3,4,5) • Need to find: p (3,2,3) p (2,2,2) = p (2,2,3) p (3,3,3) = p (1,2,3) p (0,1,2)

  21. Bohm Blossoms • Where are the endpoints located? p (4,2,3) p (3,4,5) • Need to find: p (3,2,3) p (2,1,2) = p (2,2,3) p (3,4,3) = p (1,2,3) p (2,2,2) = p (3,3,3) = p (0,1,2)

  22. Bohm Blossoms • Where are the endpoints located? p (4,2,3) p (3,4,5) • Need to find: p (3,2,3) p (2,1,2) = p (2,2,3) p (3,4,3) = p (3,1,2) p (2,2,2) = p (3,3,3) = p (0,1,2)

  23. Bohm Blossoms • Where are the endpoints located? p (4,2,3) p (3,4,5) • Need to find: p (3,2,3) p (2,1,2) = 1/3 p (0,1,2) + 2/3 p (3,1,2) p (2,2,3) p (3,4,3) = p (3,1,2) p (2,2,2) = p (2,1,2) p (3,3,3) = p (0,1,2)

  24. Bohm Blossoms • Where are the endpoints located? p (3,4,2) p (3,4,5) • Need to find: p (3,2,3) p (2,1,2) = 1/3 p (0,1,2) + 2/3 p (3,1,2) p (2,2,3) p (3,4,3) = p (3,1,2) p (2,2,2) = p (2,1,2) p (3,3,3) = p (0,1,2)

  25. Bohm Blossoms • Where are the endpoints located? p (3,4,2) p (3,4,3) p (3,4,5) • Need to find: p (3,2,3) p (2,1,2) = 1/3 p (0,1,2) + 2/3 p (3,1,2) p (2,2,3) p (3,4,3) = 2/3 p (3,4,2) + 1/3 p (3,4,5) p (3,1,2) p (2,2,2) = p (2,1,2) p (3,3,3) = p (0,1,2)

  26. Bohm Blossoms • Where are the endpoints located? p (3,4,2) p (3,4,3) p (3,4,5) • Need to find: p (3,2,3) p (2,1,2) = 1/3 p (0,1,2) + 2/3 p (3,1,2) p (2,3,2) p (3,4,3) = 2/3 p (3,4,2) + 1/3 p (3,4,5) p (3,1,2) p (2,2,2) = p (2,1,2) p (3,3,3) = p (0,1,2)

  27. Bohm Blossoms • Where are the endpoints located? p (3,4,2) p (3,4,3) p (3,4,5) • Need to find: p (3,2,3) p (2,1,2) = 1/3 p (0,1,2) + 2/3 p (3,1,2) p (2,3,2) p (3,4,3) = 2/3 p (3,4,2) + 1/3 p (3,4,5) p (3,1,2) p (2,2,2) p (2,2,2) = 1/2 p (2,1,2) + 1/2 p (2,3,2) p (2,1,2) p (3,3,3) = p (0,1,2)

  28. Bohm Blossoms • Where are the endpoints located? p (3,4,2) p (3,4,3) p (3,4,5) • Need to find: p (3,2,3) p (3,3,3) p (2,1,2) = 1/3 p (0,1,2) + 2/3 p (3,1,2) p (2,3,2) p (3,4,3) = 2/3 p (3,4,2) + 1/3 p (3,4,5) p (3,1,2) p (2,2,2) p (2,2,2) = 1/2 p (2,1,2) + 1/2 p (2,3,2) p (2,1,2) p (3,3,3) = 1/2 p (3,2,3) + 1/2 p (3,4,3) p (0,1,2)

  29. Bohm Blossoms • Where are the endpoints located? • Need to find: p (2,3,3) p (3,3,3) p (2,1,2) = 1/3 p (0,1,2) + 2/3 p (3,1,2) p (2,2,3) p (3,4,3) = 2/3 p (3,4,2) + 1/3 p (3,4,5) p (2,2,2) p (2,2,2) = 1/2 p (2,1,2) + 1/2 p (2,3,2) p (3,3,3) = 1/2 p (3,2,3) + 1/2 p (3,4,3)

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

Subdivision Surfaces CAGD Ofir Weber 1 Spline Surfaces Spline Surfaces Why use them?

Subdivision Surfaces CAGD Ofir Weber 1 Spline Surfaces Spline Surfaces Why use them?

Subdivision Surfaces CAGD Ofir Weber 1 Spline Surfaces Spline Surfaces Why use them? Smooth Good for modeling - easy to control Compact (complex objects are represented by less numbers) Flexibility (different

677 views • 38 slides
Song of Solomon 2:12 The blossoms appear in the countryside. The time of singing has come, and

Song of Solomon 2:12 The blossoms appear in the countryside. The time of singing has come, and

Song of Solomon 2:12 The blossoms appear in the countryside. The time of singing has come, and the turtledoves cooing is heard in our land. Job 39:13 The wings of the ostrich flap joyfully, but are her feathers and plumage like the

602 views • 22 slides
P -spline ANOVA-type interaction models for spatio-temporal smoothing Dae-Jin Lee and Mar

P -spline ANOVA-type interaction models for spatio-temporal smoothing Dae-Jin Lee and Mar

P -spline ANOVA-type interaction models for spatio-temporal smoothing Dae-Jin Lee and Mar a Durb an Universidad Carlos III de Madrid Department of Statistics IWSM Utrecht 2008 D.-J. Lee and M. Durban (UC3M) P -spline

677 views • 28 slides
Bezier Blossoms CS 418 Interactive Computer Graphics John C. Hart de Casteljau de

Bezier Blossoms CS 418 Interactive Computer Graphics John C. Hart de Casteljau de

Bezier Blossoms CS 418 Interactive Computer Graphics John C. Hart de Casteljau de Casteljau algorithm p 1 evaluates a point on a p 12 Bezier curve by p 2 p 012 1- t scaffolding lerps p 123 p 0123 Blossoming renames the p 01 control

539 views • 17 slides
Blossoms CS 419 Advanced Topics in Computer Graphics John C. Hart Borrowed somewhat from Tom

Blossoms CS 419 Advanced Topics in Computer Graphics John C. Hart Borrowed somewhat from Tom

Blossoms CS 419 Advanced Topics in Computer Graphics John C. Hart Borrowed somewhat from Tom Sederbergs notes p 1 de Casteljau p 12 p 2 p 012 1- t de Casteljau algorithm p 123 p 0123 evaluates a point on a p 01 Bezier curve by t

560 views • 19 slides
Computer Graphics - Spline and Subdivision Surfaces - Hendrik Lensch Computer Graphics WS07/08

Computer Graphics - Spline and Subdivision Surfaces - Hendrik Lensch Computer Graphics WS07/08

Computer Graphics - Spline and Subdivision Surfaces - Hendrik Lensch Computer Graphics WS07/08 Spline & Subdivision Surfaces Overview Last Time Image-Based Rendering Today Parametric Curves Lagrange Interpolation

1.05k views • 101 slides
Polygonal Splines and Their Applications Motivation Polynomial for Numerical Solution of PDE 1

Polygonal Splines and Their Applications Motivation Polynomial for Numerical Solution of PDE 1

Ming-Jun Lai Polygonal Splines and Their Applications Motivation Polynomial for Numerical Solution of PDE 1 Splines Spline Method for PDE GBC BB functions Ming-Jun Lai Polygonal Spline Space Department of Mathematics Hexahedral Spline

440 views • 40 slides
G 2 Tensor Product Splines over Extraordinary Vertices Charles Loop Scott Schaefer Microsoft

G 2 Tensor Product Splines over Extraordinary Vertices Charles Loop Scott Schaefer Microsoft

G 2 Tensor Product Splines over Extraordinary Vertices Charles Loop Scott Schaefer Microsoft Research Texas A&M University Spline Rings Spline Rings Spline Rings Problems with Catmull-Clark Subdivision Composed of an infinite

438 views • 42 slides
Blossoms Bezier Curve Construction Reminder Successive linear interpolations at point t

Blossoms Bezier Curve Construction Reminder Successive linear interpolations at point t

Blossoms Bezier Curve Construction Reminder Successive linear interpolations at point t Each point can be found as a function of the parameter, t , and the control points, b i n n i t n b t = i ,n b i = i

1.01k views • 46 slides
Estimating Criteria for for Fitting Fitting B B- -spline Curves spline Curves: : Estimating

Estimating Criteria for for Fitting Fitting B B- -spline Curves spline Curves: : Estimating

Equipe MGII GraphiCon98 GraphiCon98 Moscow - Russia Moscow - Russia September 7-11, 1998 September 7-11, 1998 Estimating Criteria for for Fitting Fitting B B- -spline Curves spline Curves: : Estimating Criteria Application to

545 views • 25 slides
Dr Sean Simpson Cofounder and Chief Scientist of LanzaTech 1 Consider the Cherry Tree " A

Dr Sean Simpson Cofounder and Chief Scientist of LanzaTech 1 Consider the Cherry Tree " A

Dr Sean Simpson Cofounder and Chief Scientist of LanzaTech 1 Consider the Cherry Tree " A cherry tree produces thousands of blossoms which create fruit for birds, humans and other animals in an effort to grow one tree. The blossoms and

364 views • 18 slides
Splines mod m Nealy Bowden Smith College July 24, 2014 Bowden Splines mod m Spline Basics 1

Splines mod m Nealy Bowden Smith College July 24, 2014 Bowden Splines mod m Spline Basics 1

Splines mod m Nealy Bowden Smith College July 24, 2014 Bowden Splines mod m Spline Basics 1 Special Properties (mod m ) 2 Characterizations and the Role of Primes 3 Further Research and New Ideas 4 Bowden Splines mod m Spline Basics

718 views • 32 slides
Multiscale local multiple orientation estimation using Mathematical Morphology and B-spline

Multiscale local multiple orientation estimation using Mathematical Morphology and B-spline

Multiscale local multiple orientation estimation using Mathematical Morphology and B-spline interpolation Jess Angulo 1 , Rafael Verd 2 , Juan Morales 2 1 Centre de Morphologie Mathmatique (CMM), Ecole des Mines de Paris, Fontainebleau

976 views • 63 slides
Spline approximation of random processes with singularity Konrad Abramowicz Department of

Spline approximation of random processes with singularity Konrad Abramowicz Department of

Spline approximation of random processes with singularity Konrad Abramowicz Department of Mathematics and Mathematical Statistics Ume a university Stockholm, 2nd Northern Triangular Seminar, 2010 Coauthor: Oleg Seleznjev Department of

658 views • 47 slides
SKT: A Computationaly Efficient SUPANOVA: Spline Kernel Based Machine Learning Tool Boleslaw

SKT: A Computationaly Efficient SUPANOVA: Spline Kernel Based Machine Learning Tool Boleslaw

SKT: A Computationaly Efficient SUPANOVA: Spline Kernel Based Machine Learning Tool Boleslaw Szymanski, Lijuan Zhu, Long Han and Mark Embrechts Rensselaer Polytechnic Institute, Troy, NY 12180, USA and Alexander Ross and Karsten Sternickel

306 views • 15 slides
PowellSabin spline based multilevel preconditioners for 4th order elliptic equations on the

PowellSabin spline based multilevel preconditioners for 4th order elliptic equations on the

Introduction PowellSabin splines Numerical simulation with PS splines References PowellSabin spline based multilevel preconditioners for 4th order elliptic equations on the sphere Jan Maes, Adhemar Bultheel Department of Computer

932 views • 72 slides
Generalized B-splines and local refinements Carla Manni Department of Mathematics, University of

Generalized B-splines and local refinements Carla Manni Department of Mathematics, University of

Generalized B-splines and local refinements Carla Manni Department of Mathematics, University of Roma Tor Vergata collaboration with P. Costantini, F. Pelosi, H. Speleers 11-th MAIA Conference September 2530, 2013 Ettore

2.13k views • 199 slides
Cubic spline interpolation High-degree polynomial fitting has strong oscillations. Can we get a

Cubic spline interpolation High-degree polynomial fitting has strong oscillations. Can we get a

Cubic spline interpolation High-degree polynomial fitting has strong oscillations. Can we get a piecewise low degree polynomial interpolation instead? S ( x ) S n 2 S n 1 S j S 1 S j 1 S 0 S j ( x j 1 ) f ( x j 1 )

449 views • 7 slides
1 Example Evaluating Bzier and B-Spline Curves, t in [0,1] f(0,0,0) f(0,0,1) f(0,1,1)

1 Example Evaluating Bzier and B-Spline Curves, t in [0,1] f(0,0,0) f(0,0,1) f(0,1,1)

B-Splines Bzier Curves Joining Curve Segments P 2 P 1 Q 3 P 3 Q 0 P 0 Q 2 Q 1 Can only achieve C 1 -continuity Need to pay attention to it explicitly! Desirable: basis functions with continuity built-in B-Splines Polynomial

339 views • 12 slides
The integration of the data from Telescope ROA Montsec to the Virtual Observatory Fabra ROA

The integration of the data from Telescope ROA Montsec to the Virtual Observatory Fabra ROA

The integration of the data from Telescope Fabra The integration of the data from Telescope ROA Montsec to the Virtual Observatory Fabra ROA Montsec to the Virtual Observatory Data centre overview Type of data to be published in Octavi

1.04k views • 60 slides
B-Splines Properties 1. Convex hull property 2. Curve follows the shape of the defining polygon

B-Splines Properties 1. Convex hull property 2. Curve follows the shape of the defining polygon

B-Splines Properties 1. Convex hull property 2. Curve follows the shape of the defining polygon 3. Maximum order = number of control points 4. Invariance to affine transformation 5. Variation diminishing property B-Splines Control Handles

455 views • 19 slides
r t ss

r t ss

rtt rst r t ss st Ptrtr

359 views • 21 slides
Computer Graphics CS 543 Lecture 12 (Part 1) Curves Prof Emmanuel Agu Computer Science Dept.

Computer Graphics CS 543 Lecture 12 (Part 1) Curves Prof Emmanuel Agu Computer Science Dept.

Computer Graphics CS 543 Lecture 12 (Part 1) Curves Prof Emmanuel Agu Computer Science Dept. Worcester Polytechnic Institute (WPI) So Far Dealt with straight lines and flat surfaces Real world objects include curves Need to develop:

345 views • 34 slides
Review What is Computing? Primitive Shapes Occupations in CS point What can be

Review What is Computing? Primitive Shapes Occupations in CS point What can be

Review What is Computing? Primitive Shapes Occupations in CS point What can be Programmed? line Creative Computing triangle Processing quad Downloading Processing rect Sketchpad ellipse

335 views • 19 slides

More recommend