Advanced Topics in Computer Graphics
Advanced Topics in Computer Graphics CMPS 290
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This 16 page Class Notes was uploaded by Dr. Elyssa Ratke on Monday September 7, 2015. The Class Notes belongs to CMPS 290 at University of California - Santa Cruz taught by Staff in Fall. Since its upload, it has received 23 views. For similar materials see /class/182267/cmps-290-university-of-california-santa-cruz in ComputerScienence at University of California - Santa Cruz.
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Date Created: 09/07/15
Review of Type inference Un ryped rer39ms Ax e In rr39oduce Type variables x 0c Typing r39ules generate cons rr39ain rs 39oc3y ocy3 BinT Solve cons rr39ain rs 39oczin l39 in r BzinT yzin r Conclude Axin r in r e 15 Jan 2004 290G Lecture 4 1 16 Representation Analysis and Polymorphic Types Lec rur39e 4 15 Jan 2004 290G Lecture 4 2 16 Representation Analysis Which values in a program must have the same representation Not all values of a type need be represented identically Shows abstraction boundaries Which values must have the same representation Those that are used quottogetherquot 15 Jan 2004 290G Lecture 4 3 16 The Idea Old Type language T06T gtTin139 New Type language Ever39y Type is a pair old rype x variable T06 I T gt T8in1398 15 Jan 2004 290G Lecture 4 4 16 Old Type Inference Rules Akelnl Akezme AX0X Axaxker lerea Akchx Awlxech n Akelezg Akelnl Akelnl Akezne Akezne AI QJT3 7172am lein r 7273 AHHm Ake1e2im Akifelez nz 15 Jan 2004 290G Lecture 4 5 16 New Type Inference Rules A I e1r1 A I e2 2 72 Ax 05XUBX AX aX8X I e 239 71 72 gt 053 A I X20X8X A I 1Xeax x gt my A I el e2 053 AI e1271 AI e1r1 AI ezm2 AI e2272 AI es22393 2391 2392 infug Z391im398 2392 2393 AI iin r8 AI e1e2in r8 AI ifelezgm2 15 Jan 2004 290G Lecture 4 6 16 Example A lambda Ter39m x xx yk 2x wif x y 2 1 w Equivalence classes x xx yk 2x wif x y z 1 w 15 Jan 2004 290G Lecture 4 7 16 Lackwif Repr39esen ra rion analysis for39 C Very simple efficien r and probably useful Some ugly pieces Eg handling of cas rs 15 Jan 2004 290G Lecture 4 8 16 Applications Reengineeringquot Make some values more abs rrac r Find bugs Every equivalence class wi rh a maloc should have a free Jus r explain wha r pieces of The program inferacf 15 Jan 2004 290G Lecture 4 9 16 Polymorphism 15 Jan 2004 290G Lecture 4 10 16 Polymorphism Wha r is Type of Xxx Is i r in r in r 0c 0c bool bool 01 B 0L all of The above 15 Jan 2004 290G Lecture 4 11 16 Context Sensitivity Polymorphic Types Add a new class of types called fype schemes UVOLC7T Example A polymorphic identity function Van gt 05 Note All quontifiers are at top level 15 Jan 2004 290G Lecture 4 12 16 The Key Idea AI e r a no r free in A AI e Vow This is called general2mm 15 Jan 2004 290G Lecture 4 13 16 Insfam iafion Polymorphic assump rions can be used as usual Bu r we s rill need To Turn a polymorphic rype info a monomor39phic Type for39 The o rher39 rype rules To work AI e Vom AI e 039T39a 15 Jan 2004 290G Lecture 4 14 16 Where is Type Inference Strong Handles da ra s rr39uc rur39es smoo rhly Works in infini re domains Se r of Types is unlimi red No forwards backwards dis rincTion Type polymorphism for con rex r sensi rivi ry 15 Jan 2004 290G Lecture 4 15 16 CMPS 2900 Topics Spring 708 1 Convex Sets 0 A ine sets 0 Convex sets 0 Convex combinations and convex hull o cones and conic combinations 0 hyperplanes and half spaces o Norms and Norm balls 0 polyhedra o Convexity preserving operations intersection image of convex set under af ne transformation 0 Perspective function 0 dual cones generalized conic inequalities minimumminimal elements 0 Separating hyperplane theorem 0 Supporting hyperplane theorem 2 Convex functions 0 De nitions of convexity concavity strict convexity o convex if every restriction to a line is convex 0 First order condition 0 Second order condition 0 Epigraphs and sub level sets Jensen s inequality 0 Closure under non negative weighted sums composition with a ine functions point wise max and sup composition with scalar function minimization over convex set 0 conjugate function 0 quasi convexity 3 Optimization problems 0 Problems in standard form 0 Domain feasible optimal and locally optimal points