simple symmetric venn diagrams with 11 and 13 curves
play

Simple Symmetric Venn Diagrams with 11 and 13 Curves Khalegh - PowerPoint PPT Presentation

Nov 22, 2022 •252 likes •648 views

Simple Symmetric Venn Diagrams with 11 and 13 Curves Khalegh Mamakani and Frank Ruskey Department of Computer Science, University of Victoria, Canada 1 Tuesday, 9 July, 13 WHAT IS A VENN DIAGRAM? A closed curves divides the plane into two

  1. Simple Symmetric Venn Diagrams with 11 and 13 Curves Khalegh Mamakani and Frank Ruskey Department of Computer Science, University of Victoria, Canada 1 Tuesday, 9 July, 13

  2. WHAT IS A VENN DIAGRAM? A closed curves divides the plane into two open subsets of the points.   2 Tuesday, 9 July, 13

  3. WHAT IS A VENN DIAGRAM? Regions are formed by the intersection of interior and exterior of the curves. 00 Rank of a region : 01 A binary number that indicates the curves containing the region. 11 00 11 Weight of a region : The number of curves containing the region. 10 3 Tuesday, 9 July, 13

  4. 000 100 101 110 111 011 001 010 Venn Diagram : There are exactly 2 n regions where each region is in the interior of a unique subset of the curves. 4 Tuesday, 9 July, 13

  5. SIMPLE VENN DIAGRAMS No more than two curves intersect at any given point. 5 Tuesday, 9 July, 13

  6. MONOTONE VENN DIAGRAMS Every region of weight k is adjacent to at least one region of weight k+1 and is also adjacent to at least one region of weight k-1. 6 Tuesday, 9 July, 13

  7. MONOTONE VENN DIAGRAMS Every region of weight k is adjacent to at least one region of weight k+1 and is also adjacent to at least one region of weight k-1. 6 Tuesday, 9 July, 13

  8. MONOTONE VENN DIAGRAMS Every region of weight k is adjacent to at least one region of weight k+1 and is also adjacent to at least one region of weight k-1. 6 Tuesday, 9 July, 13

  9. MONOTONE VENN DIAGRAMS Every region of weight k is adjacent to at least one region of weight k+1 and is also adjacent to at least one region of weight k-1. 6 Tuesday, 9 July, 13

  10. ROTATIONAL SYMMETRY 7 Tuesday, 9 July, 13

  11. ROTATIONAL SYMMETRY Diagram remains invariant by a rotation of 2 휋 ╱ 푛 radians about a centre point. 7 Tuesday, 9 July, 13

  12. ROTATIONAL SYMMETRY 8 Tuesday, 9 July, 13

  13. ROTATIONAL SYMMETRY For any rotationally symmetric 푛 -Venn diagram, 푛 must be prime (Henderson 1963). 8 Tuesday, 9 July, 13

  14. ROTATIONAL SYMMETRY For any rotationally symmetric 푛 -Venn diagram, 푛 must be prime (Henderson 1963). The existence of non-simple symmetric Venn diagrams has been proved for any prime number of curves (Griggs, Killian and Savage 2004). 8 Tuesday, 9 July, 13

  15. ROTATIONAL SYMMETRY For any rotationally symmetric 푛 -Venn diagram, 푛 must be prime (Henderson 1963). The existence of non-simple symmetric Venn diagrams has been proved for any prime number of curves (Griggs, Killian and Savage 2004). The largest prime number for which a simple symmetric Venn diagram was known : 7 8 Tuesday, 9 July, 13

  16. REPRESENTATION OF SIMPLE MONOTONE VENN DIAGRAMS 9 Tuesday, 9 July, 13

  17. REPRESENTATION OF SIMPLE MONOTONE VENN DIAGRAMS 9 Tuesday, 9 July, 13

  18. REPRESENTATION OF SIMPLE MONOTONE VENN DIAGRAMS 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 A binary matrix of n − 1 rows and 2 n − 2 columns. 9 Tuesday, 9 July, 13

  19. REPRESENTATION OF SIMPLE MONOTONE VENN DIAGRAMS 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 10 Tuesday, 9 July, 13

  20. REPRESENTATION OF SIMPLE MONOTONE VENN DIAGRAMS 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 Crossing Sequence (Wendy Myrvold) : 1, 3, 2, 4, 3, 2, 1, 3, 2, 4, 3, 2, 1, 3, 2, 4, 3, 2, 1, 3, 2, 4, 3, 2, 1, 3, 2, 4, 3, 2 10 Tuesday, 9 July, 13

  21. CROSSCUT SYMMETRY Crosscut : A curve segment that sequentially crosses all other curves once. 11 Tuesday, 9 July, 13

  22. CROSSCUT SYMMETRY M4 (Ruskey) Hamilton (Edwards) 12 Tuesday, 9 July, 13

  23. CROSSCUT SYMMETRY 13 Tuesday, 9 July, 13

  24. CROSSCUT SYMMETRY 13 Tuesday, 9 July, 13

  25. CROSSCUT SYMMETRY Crosscut Symmetry : Reflective symmetry across the crosscut (ignoring the two regions at top and bottom). 13 Tuesday, 9 July, 13

  26. CROSSCUT SYMMETRY Crosscut Symmetry : Reflective symmetry across the crosscut (ignoring the two regions at top and bottom). Curve intersections are palindromic (except for the crosscut). C 5 intersections : [ C 4 , C 6 , C 3 , C 6 , C 4 , C 1 , C 4 , C 6 , C 3 , C 6 , C 4 ] 13 Tuesday, 9 July, 13

  27. CROSSCUT SYMMETRY THEOREM Crossing sequence of the 7-V ossing sequence of the 7-V ossing sequence of the 7-Venn ossing sequence of the 7-Venn ρ α δ α r+ 1, 3, 2, 5, 4, 3, 2, 3, 4, 6, 5, 4, 3, 2, 5, 4, 3, 4 14 Tuesday, 9 July, 13

  28. CROSSCUT SYMMETRY THEOREM Crossing sequence of the 7-V ossing sequence of the 7-Venn ossing sequence of the 7-Venn ossing sequence of the 7-V ρ α δ α r+ 1, 3, 2, 5, 4, 3, 2, 3, 4, 6, 5, 4, 3, 2, 5, 4, 3, 4 A simple monotone rotationally symmetric n -Venn diagram is crosscut-symmetric if and only if it can be represented by a crossing sequence of the form ρ , α , δ , α r + where • ρ is 1 , 3 , 2 , 5 , 4 , . . . , n − 2 , n − 3 and δ is n − 1 , n − 2 , . . . , 3 , 2 . • | α | = | α r + | = (2 n − 1 − ( n − 1) 2 ) /n and α [ i ] ∈ { 2 , . . . , n − 3 } . • α r + is obtained by reversing α and incrementing each element by 1. 14 Tuesday, 9 July, 13

  29. The First Simple Symmetric 11-Venn Diagram 15 Tuesday, 9 July, 13

  30. The First Simple Symmetric 11-Venn Diagram 15 Tuesday, 9 July, 13

  31. THE FIRST SIMPLE SYMMETRIC 11-VENN DIAGRAM α sequence : 3, 2, 3, 4, 3, 4, 5, 4, 3, 2, 3, 4, 3, 4, 5, 4, 3, 4, 5, 4, 5, 6, 5, 4, 5, 6, 5, 6, 7, 6, 5, 4, 3, 2, 5, 4, 3, 4, 6, 5, 4, 5, 6, 7, 6, 7, 8, 7, 6, 5, 6, 5, 4, 3, 4, 5, 7, 6, 5, 4, 6, 5, 8, 7, 6, 5, 4, 5, 7, 6, 5, 6, 8, 7, 6, 5, 4, 6, 5, 7, 6, 5, 6, 7. 16 Tuesday, 9 July, 13

  32. ITERATED CROSSCUT SYMMETRY Simple Symmetric 7-Venn Diagram Hamilton (Edwards) 17 Tuesday, 9 July, 13

  33. ITERATED CROSSCUT SYMMETRY 17 Tuesday, 9 July, 13

  34. A simple symmetric 13-Venn diagram with iterated crosscut symmetry 18 Tuesday, 9 July, 13

  35. 18 Tuesday, 9 July, 13

  36. α sequence of the simple symmetric 13-Venn : 3 2 4 3 5 4 3 2 4 3 5 4 6 5 4 3 5 4 6 5 7 6 5 4 3 2 3 4 3 4 5 4 5 6 5 4 3 6 5 6 5 4 5 4 7 6 5 4 6 5 7 6 8 7 6 5 4 3 7 8 6 7 5 6 7 8 5 6 5 6 7 6 7 4 5 6 7 6 5 6 5 4 5 4 9 8 7 6 5 4 3 2 3 4 3 4 5 4 5 6 5 4 3 5 4 6 5 4 5 6 7 6 5 4 5 6 5 6 7 6 5 6 7 6 7 8 7 6 5 4 3 5 4 6 5 7 6 5 4 6 5 7 6 8 7 8 7 6 5 4 5 6 7 6 5 4 7 6 8 7 6 5 7 6 5 8 7 6 9 8 7 6 5 4 8 7 8 7 6 7 6 5 9 8 7 6 8 7 6 5 9 8 7 6 10 9 8 7 6 5 4 3 7 8 9 10 6 7 8 9 7 8 9 10 6 7 8 7 8 9 8 9 5 6 7 8 9 10 7 8 9 6 7 8 6 7 8 9 7 8 5 6 7 8 7 6 5 6 7 8 9 8 9 7 8 6 7 5 6 7 8 6 7 5 6 4 5 6 7 8 9 8 7 8 7 6 7 8 7 6 7 6 5 6 7 8 7 6 5 6 7 5 6 4 5 6 7 6 5 6 5 4 5 4 19 Tuesday, 9 July, 13

  37. α sequence of the simple symmetric 13-Venn : 3 2 4 3 5 4 3 2 4 3 5 4 6 5 4 3 5 4 6 5 7 6 5 4 3 2 3 4 3 4 5 4 5 6 5 4 3 6 5 6 5 4 5 4 7 6 5 4 6 5 7 6 8 7 6 5 4 3 7 8 6 7 5 6 7 8 5 6 5 6 7 6 7 4 5 6 7 6 5 6 5 4 5 4 9 8 7 6 5 4 3 2 3 4 3 4 5 4 5 6 5 4 3 5 4 6 5 4 5 6 7 6 5 4 5 6 5 6 7 6 5 6 7 6 7 8 7 6 5 4 3 5 4 6 5 7 6 5 4 6 5 7 6 8 7 8 7 6 5 4 5 6 7 6 5 4 7 6 8 7 6 5 7 6 5 8 7 6 9 8 7 6 5 4 8 7 8 7 6 7 6 5 9 8 7 6 8 7 6 5 9 8 7 6 10 9 8 7 6 5 4 3 7 8 9 10 6 7 8 9 7 8 9 10 6 7 8 7 8 9 8 9 5 6 7 8 9 10 7 8 9 6 7 8 6 7 8 9 7 8 5 6 7 8 7 6 5 6 7 8 9 8 9 7 8 6 7 5 6 7 8 6 7 5 6 4 5 6 7 8 9 8 7 8 7 6 7 8 7 6 7 6 5 6 7 8 7 6 5 6 7 5 6 4 5 6 7 6 5 6 5 4 5 4 19 Tuesday, 9 July, 13

  38. α sequence of the simple symmetric 13-Venn : 3 2 4 3 5 4 3 2 4 3 5 4 6 5 4 3 5 4 6 5 7 6 5 4 3 2 3 4 3 4 5 4 5 6 5 4 3 6 5 6 5 4 5 4 7 6 5 4 6 5 7 6 8 7 6 5 4 3 7 8 6 7 5 6 7 8 5 6 5 6 7 6 7 4 5 6 7 6 5 6 5 4 5 4 9 8 7 6 5 4 3 2 3 4 3 4 5 4 5 6 5 4 3 5 4 6 5 4 5 6 7 6 5 4 5 6 5 6 7 6 5 6 7 6 7 8 7 6 5 4 3 5 4 6 5 7 6 5 4 6 5 7 6 8 7 8 7 6 5 4 5 6 7 6 5 4 7 6 8 7 6 5 7 6 5 8 7 6 9 8 7 6 5 4 8 7 8 7 6 7 6 5 9 8 7 6 8 7 6 5 9 8 7 6 10 9 8 7 6 5 4 3 7 8 9 10 6 7 8 9 7 8 9 10 6 7 8 7 8 9 8 9 5 6 7 8 9 10 7 8 9 6 7 8 6 7 8 9 7 8 5 6 7 8 7 6 5 6 7 8 9 8 9 7 8 6 7 5 6 7 8 6 7 5 6 4 5 6 7 8 9 8 7 8 7 6 7 8 7 6 7 6 5 6 7 8 7 6 5 6 7 5 6 4 5 6 7 6 5 6 5 4 5 4 19 Tuesday, 9 July, 13

  39. Thank you! 20 Tuesday, 9 July, 13

Recommend

Boolean Operators and Venn Diagrams Michael Halleen References ESRI Help, Wikipedia, Getting

Boolean Operators and Venn Diagrams Michael Halleen References ESRI Help, Wikipedia, Getting

10/6/2009 Boolean Operators and Venn Diagrams Michael Halleen References ESRI Help, Wikipedia, Getting To Know ArcGIS, Introduction to Geographic Information Systems Boolean Logic It's a 0, or It' s a 1 Yes or No On or Off 1

67 views • 4 slides
Informatics 1 Computation and Logic Sets of States: Venn Diagrams and Truth Tables Michael

Informatics 1 Computation and Logic Sets of States: Venn Diagrams and Truth Tables Michael

Informatics 1 Computation and Logic Sets of States: Venn Diagrams and Truth Tables Michael Fourman @mp4man 1 This course provides a first glimpse of the deep connections between computation and logic. We will focus primarily on the simplest

347 views • 21 slides
Minimum Area Venn Diagrams Bette Bultena, Matthew Klimesh, Frank Ruskey University of Victoria

Minimum Area Venn Diagrams Bette Bultena, Matthew Klimesh, Frank Ruskey University of Victoria

Introduction Minimum Area Expansion Omitting the Empty Set Summary Minimum Area Venn Diagrams Bette Bultena, Matthew Klimesh, Frank Ruskey University of Victoria & California Institute of Technology June 12, 2013 CanaDAM, St. Johns

867 views • 64 slides
MATH 105: Finite Mathematics 6-2: The Number of Elements in a Set Prof. Jonathan Duncan Walla

MATH 105: Finite Mathematics 6-2: The Number of Elements in a Set Prof. Jonathan Duncan Walla

Counting with Venn Diagrams Story Problems Conclusion MATH 105: Finite Mathematics 6-2: The Number of Elements in a Set Prof. Jonathan Duncan Walla Walla College Winter Quarter, 2006 Counting with Venn Diagrams Story Problems Conclusion

1.19k views • 61 slides
Simple Description of Relational Databases Simple Diagrams A schema is represented by a

Simple Description of Relational Databases Simple Diagrams A schema is represented by a

Simple Description of Relational Databases Simple Diagrams A schema is represented by a networked diagram Nodes represent tables Name of the table labels the node Interior of the node are the name of the attributes Underline

501 views • 22 slides
Coxeter groups and Artin groups Day 2: Symmetric Spaces and Simple Groups Jon McCammond (U.C.

Coxeter groups and Artin groups Day 2: Symmetric Spaces and Simple Groups Jon McCammond (U.C.

Coxeter groups and Artin groups Day 2: Symmetric Spaces and Simple Groups Jon McCammond (U.C. Santa Barbara) 1 Symmetric Spaces Def: A Riemannian manifold is called a symmetric space if at each point p there exists an isometry fixing p and

295 views • 25 slides
Plan Lecture 1 - String diagrams and symmetric monoidal categories Lecture 2 -

Plan Lecture 1 - String diagrams and symmetric monoidal categories Lecture 2 -

Plan Lecture 1 - String diagrams and symmetric monoidal categories Lecture 2 - Resource-sensitive algebraic theories Lecture 3 - Interacting Hopf monoids and graphical linear algebra Lecture 4 - Signal Flow Graphs and recurrence

1.2k views • 61 slides
Classification of curves Simple, not closed Simple, closed Closed, not simple Not simple, not

Classification of curves Simple, not closed Simple, closed Closed, not simple Not simple, not

Classification of curves Simple, not closed Simple, closed Closed, not simple Not simple, not closed MA202 Sections 5 & 401 Chapter 11-2 Slides Interior and exterior of a simple closed curve MA202 Sections 5 & 401 Chapter 11-2

490 views • 36 slides
Restricted Voronoi Diagrams for (re)meshing Surfaces and Volumes Curves and Surfaces 2014 Bruno

Restricted Voronoi Diagrams for (re)meshing Surfaces and Volumes Curves and Surfaces 2014 Bruno

Mathmatiques - Informatique Restricted Voronoi Diagrams for (re)meshing Surfaces and Volumes Curves and Surfaces 2014 Bruno Lvy ALICE Gomtrie & Lumire CENTRE INRIA Nancy Grand-Est OVERVIEW Part. 1. Introduction Motivations The

1.64k views • 136 slides
Virtually symmetric representations and marked Gauss diagrams Manpreet Singh Indian Institute of

Virtually symmetric representations and marked Gauss diagrams Manpreet Singh Indian Institute of

Virtually symmetric representations and marked Gauss diagrams Manpreet Singh Indian Institute of Science Education and Research (IISER) Mohali, India Joint work with Valeriy Bardakov and Mikhail Neshchadim Department of Mathematics, IISER Mohali

1.17k views • 86 slides
Sets and Objects Bernd Schr oder logo1 Bernd Schr oder Louisiana Tech University, College

Sets and Objects Bernd Schr oder logo1 Bernd Schr oder Louisiana Tech University, College

Two Initial Axioms Venn Diagrams Models Sets and Objects Bernd Schr oder logo1 Bernd Schr oder Louisiana Tech University, College of Engineering and Science Sets and Objects Two Initial Axioms Venn Diagrams Models Remember

564 views • 39 slides
Topic 6: 3D Curves Intro to curve interpolation & approximation Polynomial

Topic 6: 3D Curves Intro to curve interpolation & approximation Polynomial

Topic 6: 3D Curves Intro to curve interpolation & approximation Polynomial interpolation Bzier curves Cardinal splines Interactive Design of Curves Goal: Expand the capabilities of shapes beyond lines and conics, simple

163 views • 13 slides
Pseudo symmetric monomial curves Mesut S ahin HACETTEPE UNIVERSITY Joint work with Nil S

Pseudo symmetric monomial curves Mesut S ahin HACETTEPE UNIVERSITY Joint work with Nil S

Pseudo symmetric monomial curves Mesut S ahin HACETTEPE UNIVERSITY Joint work with Nil S ahin (Bilkent University) Supported by Tubitak No: 114F094 IMNS 2016, July 4-8, 2016 Mesut S ahin (HACETTEPE UNIVERSITY) Pseudo symmetric

672 views • 32 slides
Dynamic operation 20 A simple model for the propagation delay Symmetric inverter (rise and

Dynamic operation 20 A simple model for the propagation delay Symmetric inverter (rise and

Dynamic operation 20 A simple model for the propagation delay Symmetric inverter (rise and fall delays are identical) Total capacitance is linear t p Minimum length devices = = t 0.69 R C t p0 p W L C L + = 0.69 R C (1

626 views • 26 slides
Visualizing Geometric Proofs 3D surface diagrams Lukas Heidemann Diagrams 0-diagrams 1

Visualizing Geometric Proofs 3D surface diagrams Lukas Heidemann Diagrams 0-diagrams 1

Visualizing Geometric Proofs 3D surface diagrams Lukas Heidemann Diagrams 0-diagrams 1 1-diagrams: bead diagrams 2 1-diagrams: bead diagrams 2 2-diagrams: string diagrams 3 2-diagrams: string diagrams 4 2-diagrams: string diagrams 5

696 views • 40 slides
spaces on simple closed rectifiable curves in the complex plane and P olya singularity results

spaces on simple closed rectifiable curves in the complex plane and P olya singularity results

The closed span of an exponential system in L p spaces on simple closed rectifiable curves in the complex plane and P olya singularity results for Taylor-Dirichlet series Elias Zikkos Cyprus Ministry of Education and Culture New

1.58k views • 135 slides
Outline Notes: Simple waves, rarefaction waves Integral curves in phase plane

Outline Notes: Simple waves, rarefaction waves Integral curves in phase plane

Outline Notes: Simple waves, rarefaction waves Integral curves in phase plane Approximate Riemann solvers Dam break and tsunami modeling Adaptive mesh refinement R.J. LeVeque, University of Washington IPDE 2011, July 7, 2011

540 views • 19 slides
1 Properties of Bzier Curves Important Algorithms for Bzier Curves Affine invariance Convex

1 Properties of Bzier Curves Important Algorithms for Bzier Curves Affine invariance Convex

Topics Parametric curves and surfaces Polynomial curves Advanced Modeling 2 Rational curves Katja Bhler, Andrej Varchola, Eduard Tensor product surfaces Grller Triangular surfaces Institute of Computer Graphics and Algorithms Vienna

78 views • 6 slides
Fast and simple constant-time hashing to the BLS12-381 elliptic curve (and other curves, too!)

Fast and simple constant-time hashing to the BLS12-381 elliptic curve (and other curves, too!)

Fast and simple constant-time hashing to the BLS12-381 elliptic curve (and other curves, too!) Riad S. Wahby, Dan Boneh Stanford December 3 rd , 2019 Motivation Our initial motivation: BLS signatures [BLS01] Motivation Our initial

1.38k views • 126 slides
Curves http://www.ugrad.cs.ubc.ca/~cs314/Vjan2013 Reading FCG Chap 15 Curves Ch 13 2nd

Curves http://www.ugrad.cs.ubc.ca/~cs314/Vjan2013 Reading FCG Chap 15 Curves Ch 13 2nd

University of British Columbia CPSC 314 Computer Graphics Jan-Apr 2013 Tamara Munzner Curves http://www.ugrad.cs.ubc.ca/~cs314/Vjan2013 Reading FCG Chap 15 Curves Ch 13 2nd edition 2 Curves 3 Parametric Curves parametric form

615 views • 32 slides
Neatening sketched strokes using piecewise French Curves James McCrae, Karan Singh French Curves

Neatening sketched strokes using piecewise French Curves James McCrae, Karan Singh French Curves

Neatening sketched strokes using piecewise French Curves James McCrae, Karan Singh French Curves Physical tools, used to model curves French Curves Smoothly connect pre-determined curve points French Curves French Curves Digital French

845 views • 42 slides
Todays Lecture INF 111 / CSE 121: UML Software Tools and Methods Package Diagrams

Todays Lecture INF 111 / CSE 121: UML Software Tools and Methods Package Diagrams

Todays Lecture INF 111 / CSE 121: UML Software Tools and Methods Package Diagrams State Transition Diagrams Activity Diagrams y g Communication Diagrams Lecture Notes for Summer Quarter, 2008 Michele Rousseau Set 8

462 views • 7 slides
parametric spline curves 1 curves used in many contexts fonts (2D) animation paths (3D) shape

parametric spline curves 1 curves used in many contexts fonts (2D) animation paths (3D) shape

parametric spline curves 1 curves used in many contexts fonts (2D) animation paths (3D) shape modeling (3D) different representation implicit curves parametric curves (mostly used) 2D and 3D curves are mostly the same use vector notation

724 views • 35 slides
1 Class Diagrams and Entity Relationship Diagrams (ERD) Class diagrams and ERDs both model the

1 Class Diagrams and Entity Relationship Diagrams (ERD) Class diagrams and ERDs both model the

Tutorial W eek 7 - Class and Entity-Relationship Diagram s 1 Class Diagrams and Entity Relationship Diagrams (ERD) Class diagrams and ERDs both model the structure of a system. Class diagrams represent the dynamic aspects of a system: both

165 views • 14 slides

More recommend