Page 1. News. Compositing, Clipping, Curves. Week 3, Thu May 26. Schedule Change. Homework 1 Common Mistakes. Midterm Logistics.

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1 Universiy of Briish Columbia CPSC 4 Compuer Graphics May-June 5 Tamara Munzner Composiing, Clipping, Curves Week, Thu May 6 hp:// News era lab coverage: Mon -, Wed -4 P demo slo signup shee handing back H oday we ll ry o ge H back omorrow we will pu hem in bin in lab, ne o era handous soluions will be posed you don have o ell us you re using grace days only if you re urning i in lae and you do *no* wan o use up grace days grace days are ineger quaniies Homework Common Misakes Q4, Q5: oo vague don jus say roae 9, say around which ais, and in which direcion (CCW vs CW) be clear on wheher acions are in old coordinae frame or new coordinae frame Q8: confusion on push/pop and comple operaions wrong: objec drawn in wrong spo! correc: objec drawn in righ spo boh: nice modular funcion ha doesn change modelview mari glpushmari(); gltranslae(..a..); glroae(..); draw hings glpop(); glpushmari(); gltranslae(..a..); glroae(..); gltranslae(..-a..); draw hings glpop(); Schedule Change HW ou Thu 6/, due Wed 6/8 4pm Poll which do you prefer? P4 due Fri, final Sa final Thu in-class, P4 due Sa Tuesday -:5 Miderm Logisics si spread ou: every oher row, a leas hree seas beween you and ne person you can have one 8.5 handwrien onesided shee of paper keep i, can wrie on oher side oo for final calculaors ok Page

2 Miderm Topics H, P, H, P firs hree lecures opics Reading: Today FCG Chaper pp 9-4 only: clipping Inro, Mah Review, OpenGL Transformaions I/II/III Viewing, Projecions I/II FCG Chap RB Chap Blending, Anialiasing,... only Secion Blending Reading: Ne Time FCG Chaper 7 Erraa p 4 f(p) > is ouside he plane p 4 For quadraic Bezier curves, N= w_i^n() = (N-)! / (i! (N-i-)!)... Review: Illuminaion ranspor of energy from ligh sources o surfaces & poins includes direc and indirec illuminaion Review: Ligh Sources direcional/parallel lighs poin a infiniy: (,y,z,)t poin lighs finie posiion: (,y,z,)t spolighs posiion, direcion, angle ambien lighs Images by Henrik Wann Jensen Page

3 Review: Ligh Source Placemen geomery: posiions and direcions sandard: world coordinae sysem effec: lighs fied wr world geomery alernaive: camera coordinae sysem effec: lighs aached o camera (car headlighs) Review: Reflecance specular: perfec mirror wih no scaering gloss: mied, parial speculariy diffuse: all direcions wih equal energy = specular glossy diffuse = reflecance disribuion Review: Reflecion Equaions Review: Reflecion Equaions I diffuse = k d I ligh (n l) l θ n Blinn improvemen I specular = k s I ligh (h n) n shiny l h n v I specular = k s I ligh (v r) n shiny h = (l v)/ full Phong lighing model combine ambien, diffuse, specular componens ( N (N L)) L = R # lighs I oal = k s I ambien I i ( k d (n l i ) k s (v r i ) nshiny ) i= don forge o normalize all vecors: n,l,r,v,h Review: Lighing lighing models ambien normals don maer Lamber/diffuse angle beween surface normal and ligh Phong/specular surface normal, ligh, and viewpoin fla shading Review: Shading Models compue Phong lighing once for enire polygon Gouraud shading compue Phong lighing a he verices and inerpolae lighing values across polygon Phong shading compue averaged vere normals inerpolae normals across polygon and perform Phong lighing across polygon Page

4 Correcion/Review: Compuing Normals per-vere normals by inerpolaing per-face normals OpenGL suppors boh compuing normal for a polygon hree poins form wo vecors pick a poin vecors from (a-b) (c-b) A: poin o previous B: poin o ne c-b AB: normal of plane direcion normalize: make uni lengh which side of plane is up? c a-b counerclockwise a poin order convenion b Review: Non-Phoorealisic Shading n l cool-o-warm shading k w =,c = k w c w ( k w )c c draw silhouees: if (e n )(e n ), e=edge-eye vecor draw creases: if (n n ) hreshold hp:// End of Class Las Time use version conrol for your projecs! CVS, RCS parially work hrough problem wih lighing Composiing Composiing how migh you combine muliple elemens? foreground color A, background color B Premuliplying Colors specify opaciy wih alpha channel: (r,g,b,α) α=: opaque, α=.5: ranslucen, α=: ransparen A over B C = αa (-α)b bu wha if B is also parially ransparen? C = αa (-α) βb = βb αa βb - α βb γ = β (-β)α = β α αβ muliplies, differen equaions for alpha vs. RGB premuliplying by alpha C = γ C, B = βb, A = αa C = B A - αb γ = β α αβ muliply o find C, same equaions for alpha and RGB Page 4 4

5 Rendering Pipeline Geomery Daabase Model/View Transform. Lighing Perspecive Transform. Clipping Clipping Scan Conversion Teuring Deph Tes Blending Frame- buffer Ne Topic: Clipping we ve been assuming ha all primiives (lines, riangles, polygons) lie enirely wihin he viewpor in general, his assumpion will no hold: Clipping analyically calculaing he porions of primiives wihin he viewpor Why Clip? bad idea o raserize ouside of framebuffer bounds also, don wase ime scan convering piels ouside window could be billions of piels for very close objecs! D Line Clipping deermine porion of line inside an ais-aligned recangle (screen or window) D deermine porion of line inside ais-aligned parallelpiped (viewing frusum in NDC) simple eension o D algorihms Page 5 5

6 Clipping naïve approach o clipping lines: for each line segmen for each edge of viewpor find inersecion poin pick neares poin if anyhing is lef, draw i wha do we mean by neares? how can we opimize his? A C D B Trivial Acceps big opimizaion: rivial accep/rejecs Q: how can we quickly deermine wheher a line segmen is enirely inside he viewpor? A: es boh endpoins Trivial Rejecs Q: how can we know a line is ouside viewpor? A: if boh endpoins on wrong side of same edge, can rivially rejec line Clipping Lines To Viewpor combining rivial acceps/rejecs rivially accep lines wih boh endpoins inside all edges of he viewpor rivially rejec lines wih boh endpoins ouside he same edge of he viewpor oherwise, reduce o rivial cases by spliing ino wo segmens Cohen-Suherland Line Clipping Cohen-Suherland Line Clipping oucodes 4 flags encoding posiion of a poin relaive o op, boom, lef, and righ boundary OC(p)= OC(p)= OC(p)= p p = min = p = ma y=y ma y=y min assign oucode o each vere of line o es line segmen: (p,p) rivial cases OC(p)== && OC(p)== boh poins inside window, hus line segmen compleely visible (rivial accep) (OC(p) & OC(p))!= here is (a leas) one boundary for which boh poins are ouside (same flag se in boh oucodes) hus line segmen compleely ouside window (rivial rejec) Page 6 6

7 Cohen-Suherland Line Clipping if line canno be rivially acceped or rejeced, subdivide so ha one or boh segmens can be discarded pick an edge ha he line crosses (how?) inersec line wih edge (how?) discard porion on wrong side of edge and assign oucode o new vere apply rivial accep/rejec ess; repea if necessary Cohen-Suherland Line Clipping if line canno be rivially acceped or rejeced, subdivide so ha one or boh segmens can be discarded pick an edge ha he line crosses check agains edges in same order each ime for eample: op, boom, righ, lef D C A B E Cohen-Suherland Line Clipping inersec line wih edge (how?) Cohen-Suherland Line Clipping discard porion on wrong side of edge and assign oucode o new vere C D E C D B B A A apply rivial accep/rejec ess and repea if necessary Viewpor Inersecion Code (, y ), (, y ) inersec verical edge a righ y inersec = y m( righ ) m=(y -y )/( - ) (, y ), (, y ) inersec horiz edge a y boom inersec = (y boom y )/m m=(y -y )/( - ) (, y ) righ (, y ) (, y ) Cohen-Suherland Discussion use opcodes o quickly eliminae/include lines bes algorihm when rivial acceps/rejecs are common mus compue viewpor clipping of remaining lines non-rivial clipping cos redundan clipping of some lines more efficien algorihms eis (, y ) y boom Page 7 7

8 approach Line Clipping in D clip agains parallelpiped in NDC afer perspecive ransform means ha clipping volume always he same min=ymin= -, ma=yma= in OpenGL Polygon Clipping objecive D: clip polygon agains recangular window or general conve polygons eensions for non-conve or general polygons D: clip polygon agains parallelpiped boundary lines become boundary planes bu oucodes sill work he same way addiional fron and back clipping plane zmin = -, zma = in OpenGL Polygon Clipping no jus clipping all boundary lines may have o inroduce new line segmens Why Is Clipping Hard? wha happens o a riangle during clipping? possible oucomes: riangle riangle riangle quad riangle 5-gon how many sides can a clipped riangle have? How Many Sides? Why Is Clipping Hard? seven a really ough case: Page 8 8

9 Why Is Clipping Hard? a really ough case: classes of polygons riangles conve concave Polygon Clipping holes and self-inersecion concave polygon muliple polygons Suherland-Hodgeman Clipping basic idea: consider each edge of he viewpor individually clip he polygon agains he edge equaion afer doing all edges, he polygon is fully clipped Suherland-Hodgeman Clipping basic idea: consider each edge of he viewpor individually clip he polygon agains he edge equaion afer doing all edges, he polygon is fully clipped Suherland-Hodgeman Clipping basic idea: consider each edge of he viewpor individually clip he polygon agains he edge equaion afer doing all edges, he polygon is fully clipped Suherland-Hodgeman Clipping basic idea: consider each edge of he viewpor individually clip he polygon agains he edge equaion afer doing all edges, he polygon is fully clipped Page 9 9

10 Suherland-Hodgeman Clipping basic idea: consider each edge of he viewpor individually clip he polygon agains he edge equaion afer doing all edges, he polygon is fully clipped Suherland-Hodgeman Clipping basic idea: consider each edge of he viewpor individually clip he polygon agains he edge equaion afer doing all edges, he polygon is fully clipped Suherland-Hodgeman Clipping basic idea: consider each edge of he viewpor individually clip he polygon agains he edge equaion afer doing all edges, he polygon is fully clipped Suherland-Hodgeman Clipping basic idea: consider each edge of he viewpor individually clip he polygon agains he edge equaion afer doing all edges, he polygon is fully clipped Suherland-Hodgeman Clipping Suherland-Hodgeman Algorihm basic idea: consider each edge of he viewpor individually clip he polygon agains he edge equaion afer doing all edges, he polygon is fully clipped inpu/oupu for algorihm inpu: lis of polygon verices in order oupu: lis of clipped poygon verices consising of old verices (maybe) and new verices (maybe) basic rouine go around polygon one vere a a ime decide wha o do based on 4 possibiliies is vere inside or ouside? is previous vere inside or ouside? Page

11 Clipping Agains One Edge p[i] inside: cases Clipping Agains One Edge p[i] ouside: cases inside ouside inside ouside inside ouside inside ouside p[i-] p[i-] p[i] p p[i-] p p[i] p[i] p[i] p[i-] oupu: p[i] oupu: p, p[i] oupu: p oupu: nohing Clipping Agains One Edge Suherland-Hodgeman Eample clippolygontoedge( p[n], edge ) { for( i= ; i< n ; i ) { if( p[i] inside edge ) { } if( p[i-] inside edge ) oupu p[i]; else { // p[-]= p[n-] p= inersec( p[i-], p[i], edge ); oupu p, p[i]; } } else { // p[i] is ouside edge if( p[i-] inside edge ) { p= inersec(p[i-], p[i], edge ); oupu p; } p p7 inside p6 p ouside p4 p p5 p } Suherland-Hodgeman Discussion similar o Cohen/Suherland line clipping inside/ouside ess: oucodes inersecion of line segmen wih edge: window-edge coordinaes clipping agains individual edges independen grea for hardware (pipelining) all verices required in memory a same ime no so good, bu unavoidable anoher reason for using riangles only in hardware rendering Suherland/Hodgeman Discussion for rendering pipeline: re-riangulae resuling polygon (can be done for every individual clipping edge) Page

12 Parameric Curves parameric form for a line: Curves, y and z are each given by an equaion ha involves: parameer = ( ) y = y ( ) y z = z ( ) z some user specified conrol poins, and his is an eample of a parameric curve Splines Splines - Hisory a spline is a parameric curve defined by conrol poins erm spline daes from engineering drawing, where a spline was a piece of fleible wood used o draw smooh curves conrol poins are adjused by he user o conrol shape of curve drafsman used ducks and srips of wood (splines) o draw curves wood splines have secondorder coninuiy, pass hrough he conrol poins a duck (weigh) ducks race ou curve Hermie Spline hermie spline is curve for which user provides: endpoins of curve parameric derivaives of curve a endpoins parameric derivaives are d/d, dy/d, dz/d Hermie Cubic Splines eample of kno and coninuiy consrains more derivaives would be required for higher order curves Page

13 Page Hermie Spline () say user provides cubic spline has degree, is of he form: for some consans a, b, c and d derived from he conrol poins, bu how? we have consrains: curve mus pass hrough when = derivaive mus be when = curve mus pass hrough when = derivaive mus be when = d c b a =,,, Hermie Spline () solving for he unknowns gives rearranging gives d c b a = = = = ) ( ) ( ) ( ) ( = [ ] = or Basis Funcions a poin on a Hermie curve is obained by muliplying each conrol poin by some funcion and summing funcions are called basis funcions ' ' Sample Hermie Curves Splines in D and D so far, defined only D splines: =f(:,,, ) for higher dimensions, define conrol poins in higher dimensions (ha is, as vecors) = z z z z y y y y z y Bézier Curves similar o Hermie, bu more inuiive definiion of endpoin derivaives four conrol poins, wo of which are knos

14 Bézier Curves derivaive values of Bezier curve a knos dependen on adjacen poins Bézier vs. Hermie can wrie Bezier in erms of Hermie noe: jus mari form of previous equaions Bézier vs. Hermie Bézier Basis, Geomery Marices Now subsiue his in for previous Hermie bu why is M Bezier a good basis mari? Bézier Blending Funcions Bézier Blending Funcions look a blending funcions family of polynomials called order- Bernsein polynomials C(, k) k (-) -k ; <= k <= all posiive in inerval [,] sum is equal o every poin on curve is linear combinaion of conrol poins weighs of combinaion are all posiive sum of weighs is herefore, curve is a conve combinaion of he conrol poins Page 4 4

15 Bézier Curves curve will always remain wihin conve hull (bounding region) defined by conrol poins Bézier Curves inerpolae beween firs, las conrol poins s poin s angen along line joining s, nd ps 4 h poin s angen along line joining rd, 4 h ps Comparing Hermie and Bézier Hermie Bézier. Comparing Hermie and Bezier demo: ' ' B B B B ' ' B B B B.. Rendering Bezier Curves: Simple evaluae curve a fied se of parameer values, join poins wih sraigh lines advanage: very simple disadvanages: epensive o evaluae he curve a many poins no easy way of knowing how fine o sample poins, and maybe sampling rae mus be differen along curve no easy way o adap: hard o measure deviaion of line segmen from eac curve Rendering Beziers: Subdivision a cubic Bezier curve can be broken ino wo shorer cubic Bezier curves ha eacly cover original curve suggess a rendering algorihm: keep breaking curve ino sub-curves sop when conrol poins of each sub-curve are nearly collinear draw he conrol polygon: polygon formed by conrol poins Page 5 5

16 Sub-Dividing Bezier Curves sep : find he midpoins of he lines joining he original conrol verices. call hem M, M, M Sub-Dividing Bezier Curves sep : find he midpoins of he lines joining M, M and M, M. call hem M, M M P P P P M M M M M M M P P P P Sub-Dividing Bezier Curves sep : find he midpoin of he line joining M, M. call i M P P M M M M Sub-Dividing Bezier Curves curve P, M, M, M eacly follows original from = o =.5 curve M, M, M, P eacly follows original from =.5 o = P P M M M M M M M M P P P P Sub-Dividing Bezier Curves coninue process o creae smooh curve P P de Caseljau s Algorihm can find he poin on a Bezier curve for any parameer value wih similar algorihm for =.5, insead of aking midpoins ake poins.5 of he way M P P =.5 M M P P P P demo: Page 6 6

17 Longer Curves a single cubic Bezier or Hermie curve can only capure a small class of curves a mos inflecion poins one soluion is o raise he degree allows more conrol, a he epense of more conrol poins and higher degree polynomials conrol is no local, one conrol poin influences enire curve beer soluion is o join pieces of cubic curve ogeher ino piecewise cubic curves oal curve can be broken ino pieces, each of which is cubic local conrol: each conrol poin only influences a limied par of he curve ineracion and design is much easier Piecewise Bezier: Coninuiy Problems demo: Coninuiy when wo curves joined, ypically wan some degree of coninuiy across kno boundary C, C-zero, poin-wise coninuous, curves share same poin where hey join C, C-one, coninuous derivaives C, C-wo, coninuous second derivaives Geomeric Coninuiy derivaive coninuiy is imporan for animaion if objec moves along curve wih consan parameric speed, should be no sudden jump a knos for oher applicaions, angen coninuiy suffices requires ha he angens poin in he same direcion referred o as G geomeric coninuiy curves could be made C wih a re-parameerizaion geomeric version of C is G, based on curves having he same radius of curvaure across he kno Hermie curves Achieving Coninuiy user specifies derivaives, so C by sharing poins and derivaives across kno Bezier curves hey inerpolae endpoins, so C by sharing conrol ps inroduce addiional consrains o ge C parameric derivaive is a consan muliple of vecor joining firs/las conrol poins so C achieved by seing P, =P, =J, and making P, and J and P, collinear, wih J-P, =P, -J C comes from furher consrains on P, and P, B-Spline Curve sar wih a sequence of conrol poins selec four from middle of sequence (p i-, p i-, p i, p i ) Bezier and Hermie goes beween p i- and p i B-Spline doesn inerpolae (ouch) any of hem bu approimaes he going hrough p i- and p i P P P 6 leads o... P P P 4 P 5 Page 7 7

18 B-Spline by far he mos popular spline used C, C, and C coninuous localiy of poins B-Spline demo: bumpy plane Projec vere heigh varies randomly by % of face widh world coordinae ligh, camera coord ligh regenerae errain oggle colors si riangles around a vere [demo] Projec : Normals calculae once (per errain) per-face normals hen inerpolae for per-vere use when drawing specify inerleaved wih verices eplicily drawing normals brisles a verices visual debugging Projec : Daa Srucures suggesion: 4 array for vere coords colors? normals? per-face, per-vere Projec 4 creae your own graphics game or uorial required funcionaliy D, ineracive, lighing/shading euring, picking, HUD advanced funcionaliy pieces wo for -person eam four for -person eam si for -person eam Page 8 8

19 P4: Advanced Funcionaliy (new) navigaion procedural modelling/eures paricle sysems collision deecion simulaed dynamics level of deail conrol advanced rendering effecs whaever else you wan o do proposal is a check wih me due Wed Jun 4pm P4 Proposal eiher elecronic handin, or bo handin for hardcopy shor (< page) descripion how game works how i will fulfill required funcionaliy advanced funcionaliy mus include a leas one annoaed screensho mockup skech hand-drawn scanned or using compuer ools P4 Wrieup wha: a high level descripion of wha you've creaed, including an eplici lis of he advanced funcionaliy iems how: mid-level descripion of he algorihms and daa srucures ha you've used howo: deailed insrucions of he low-level mechanics of how o acually play (keyboard conrols, ec) sources: sources of inspiraion and ideas (especially any source code you looked a for inspiraion on he Inerne) include screen shos wih handin for HOF eligibiliy P4 Grading final projec due :59pm Fri Jun 7 face o face demos again I will be grading grading 5% base: required funcions, gameplay, ec 5% advanced funcionaliy buckes, enaive mapping zero = minus = 4 check-minus = 6 check = 8 check-plus = plus 5 Page 9 9