Gernot Hoffmann Sphere Tessellation by Icosahedron Subdivision. Contents

Transcription

1 Grnot Hoffmann Sphr Tssllation by Icosahdron Subdivision Contnts 1. Vrtx Coordinats. Edg Subdivision 3 3. Triangl Subdivision 4 4. Edg lngths 5 5. Normal Vctors 6 6. Subdividd Icosahdrons 7 7. Txtur Mapping Principls 8 8. Txtur Mapping Exampls 9 9. Résumé Rfrncs 11 Bst Viw Acrobat 5 / Zoom 100% at 7 dpi Smooth lin art / Smooth txt, Do not smooth imags

2 1. Vrtx Coordinats An Icosahdron has 1 vrtics, 30 dgs and 0 triangls. Th vrtics 1 and 1 ar at th pols, th vrtics to 11 ar on two pntagons. Th only unknown angl is th latitud θ. As a rsult of som basic gomtry w find θ = Th vrtics ar asily found in sphr coordinats. z 1 1- s 6 5 c ψ 3 / s 4 θ x c y dps:=*pi/5; sps:=*sqr(sic(dps/)); Quadgli(sps,-1,1-sps,r1,r,i1,i,flag); sth:=r; { smallr root sin(thta) } cth:=sqrt(1-sqr(sth)); { cos(thta) } psi:=0; For k:= to 6 Do With xr[k] Do x:=cth*coc(psi); y:=cth*sic(psi); z:=sth; psi:=psi+dps; sth:=-sth; psi:=0.5*dps; For k:=7 to 11 Do With xr[k] Do x:=cth*coc(psi); y:=cth*sic(psi); z:=sth; psi:=psi+dps; With xr[ 1] Do x:=0; y:=0; z:=+1; With xr[1] Do x:=0; y:=0; z:=-1; r = 1 radius dg lngth ψ = π / 10 c s = = cos θ sinθ / = cos θsinψ = 4 cos θ sin ψ = cos θ + ( 1 sin θ) Quadratic quation for s sin ψ s s + 1 sin ψ = 0 Smallr root dlivrs θ = o

3 . Edg Subdivision An dg is dividd by th normalizd sum of two vrtx vctors, nw point on th sphr with radius 1. z x y Procdur SubDivi(x1,x: XYZ; Var x3: XYZ); Var r: Singl; With x3 Do x:=x1.x+x.x; y:=x1.y+x.y; z:=x1.z+x.z; r:=1/rootsqr(x,y,z); x:=r*x; y:=r*y; z:=r*z; Som important variabls XYZ is a coordinat rcord x,y,z PQE is a pixl and z-buffr rcord p,q, PC is a color rcord Typ Xtr=Rcord t1,t,t3: XYZ; Typ Ptr=Rcord pa1,pb1,pa,pb,pa3,pb3: Singl; Const ntr=0*4*4*4; Var lv,lx,tri : Intgr; Ra,hu : Singl; x1,x,x3,n : XYZ; p1,p,p3 : PQE; c0,c1,c,c3 : PC; Pbody,Psoft,Pgrid: Intgr; xr : Array[1.. 1] Of XYZ; Lt1,Lt : Array[1..ntr] Of ^Xtr; Par1 : Array[1..ntr] Of ^Ptr;

4 3. Triangl Subdivision A triangl 1--3 dlivrs by dg division four triangls 1-4-6, 4--5, and Th vrtx numbrs ar local in this xampl. Two triangl lists ar usd, Lt1 and Lt. St 1--3 is stord in Lt1, thn copid to Lt. Th subdividd st is stord in Lt1. This is much simplr than tru rcursiv programming Procdur IcoRcu; { Rcursiv Subdivision Old 1--3 Nw 1-4-6, 4-5-6, 4--5, } Var x1,x,x3,x4,x5,x6: XYZ; k: Intgr; For i:=1 to tri Do Lt[i]^:=Lt1[i]^; k:=0; For i:=1 to tri Do With Lt[i]^Do x1:=t1; x:=t; x3:=t3; SubDivi(x1,x,x4); SubDivi(x,x3,x5); SubDivi(x3,x1,x6); With Lt1[k]^ Do t1:=x1; t:=x4; t3:=x6; Inc(k); With Lt1[k]^ Do t1:=x4; t:=x5; t3:=x6; Inc(k); With Lt1[k]^ Do t1:=x4; t:=x; t3:=x5; Inc(k); With Lt1[k]^ Do t1:=x6; t:=x5; t3:=x3; tri:=k;

5 4. Edg Lngths An original triangl 1--3 (subdivision lvl 0) dlivrs by subdivision four triangls 1-4-6,4-- 5, and This is subdivision lvl 1. Th vrtx numbrs ar local in this xampl. 3 a α 6 β 3 angls b γ = 60 β 1 a 4 Subdivision lvl 0, 0 triangls: = α = 60 Subdivision lvl 1, 80 triangls: a = b = α = β = γ = a a b d c d c c a a Subdivision lvl, 30 triangls: a = b = c = d = = α = (8 angls) For architctur it would b intrsting to us th sam dglngths with a minor dviation from th tru sphr shap. This is not possibl. Equal dglnghts can b achivd only by a flat subdivision of th original triangls. Any projction towards th sphr surfac crats diffrnt dglngths.

6 5. Normal Vctors Bcaus th Icosahdron is mbddd into a sphr, ach vrtx vctor is also th normal vctor at this position. For Gouraud shading th normal vctor is assignd dirctly to th vrtx. For facttd shading it is ncssary to tak th man valu of thr vrtics for th triangl. Procdur IcoShow; Var i,sl: Intgr; sl:=1; For i:=1 to tri Do With Lt1[i]^ Do x1:=t1; x:=t; x3:=t3; x1.x:=ra*x1.x; x1.y:=ra*x1.y; x1.z:=ra*x1.z; x.x:=ra*x.x; x.y:=ra*x.y; x.z:=ra*x.z; x3.x:=ra*x3.x; x3.y:=ra*x3.y; x3.z:=ra*x3.z; ObjTra3D(x1,x1); { Rotat objct } ObjTra3D(x,x); ObjTra3D(x3,x3); Abbild3R(x1,p1,sl); { Map to Rastr } Abbild3R(x,p,sl); Abbild3R(x3,p3,sl); If Psoft=1 Thn { Gouraud } LicMod(x1,x1,c1,sl); { Luminanc } LicMod(x,x,c,sl); LicMod(x3,x3,c3,sl); End Els { Facttd } n.x:=x1.x+x.x+x3.x; n.y:=x1.y+x.y+x3.y; n.z:=x1.z+x.z+x3.z; LicMod(x1,n,c1,sl); c:=c1; c3:=c1; If Pbody=1 Thn FillTriS(p1,p,p3,c1,c,c3,sl); If Pgrid=1 Thn DrawSTria(p1,p,p3,c0,c0,c0,sl);

7 6. Subdividd Icosahdrons Lft sid facttd shading, right sid Gouraud shading. Grids ar Z-buffrd Lvl 1 80 triangls Lvl 30 triangls Lvl triangls Lvl triangls (p.9)

8 7. Txtur Mapping Principls A part of an imag is mappd as a txtur onto a part of th body. Th rlvant part of th txtur imag is dscribd by a t,u-fram. Th mapping ara on th body is dfind by a paramtr plan a,b-fram. Th Icosahdron dosn t hav a paramtr plan a,b so far. This has to b gnratd additionally. Th paramtr plan is ithr a st of sphr coordinats or a st of cylindr coordinats, which is calculatd for ach vrtx. An altrnativ would b a st of Mrcator cylindr coordinats. Procdur IcoPara; { Assign Paramtrs to vrtics } { a = -pi.. +pi b = -0.5*pi..+0.5*pi or } Var i,flag : Intgr; a1,a,a3,b1,b,b3,r1,r,r3 : Singl; For i:=1 to tri Do With Lt1[i]^Do atangns(t1.y,t1.x,a1,flag); atangns(t.y,t.x,a,flag); atangns(t3.y,t3.x,a3,flag); Us hr on of th sts (right sid) With Par[i]^Do pa1:=a1; pa:=a; pa3:=a3; pb1:=b1; pb:=b; pb3:=b3; Not: atangns is th sam as atan in C A. St of sphr coordinats r1:=sqrt(sqr(t1.x)+sqr(t1.y)); r:=sqrt(sqr(t.x)+sqr(t.y)); r3:=sqrt(sqr(t3.x)+sqr(t3.y)); atangns(t1.z,r1,b1,flag); atangns(t.z,r,b,flag); atangns(t3.z,r3,b3,flag); B. St of cylindr coordinats b1:=t1.z; b:=t.z; b3:=t3.z;

9 8. Txtur Mapping Exampls Th uppr imag shows a lvl 4 subdivision 510 triangls. Th sphr is rotatd by yaw and pitch. Rotation is providd bcaus th txtur is mappd to th body in body fixd coordinats. Th lowr sphr is not rotatd, th unavoidabl singularity at th pols is not visibl.

10 9. Résumé 10 Is it worth to us subdividd Icosahdrons instad of ordinary sphr coordinats? A sphr may b dividd in 10 stps in sphr coordinats. For th latitud from -90 to +90 and th longitud from 0 to 360 w gt a msh with 648 trapzoids or 196 triangls. Th lvl 3 Icosahdron subdivision rsults in 180 triangls. Th originally fiv sgmnts ar subdividd thr tims by two, which rsults in 40 angl stps of 9. Though w hav approximatly th sam numbr of triangls, th rsolution is not considrably bttr. This is a surprising rsult, bcaus th shrinking siz of th trapzoids nar to th pols in sphr coordinats lad to th assumption, that much calculation tim is wastd. Opposd to Icosahdron tssllation, th sphr coordinat tssllation can b don without storing hug coordinat tabls. Furthron, th txtur mapping is much simplr in sphr coordinats. But hr w hav on xcption: if th txtur is mappd pr triangl, a structur lik a golf ball, or qually distributd nois, thn th Icosahdron tssllation is probably bttr. In this xampl w hav mappd on txtur imag to four triangls, which is slightly mor complx than mapping ach imag to on triangl. Th imag itslf has a spcial rotational symmtry.

11 10. Rfrncs 11 Vry ffctiv is Googl sarch, kyword Icosahdron Th author did not us spcial sourcs Computr Graphics by ZEFIR, Imag Procssing by ZEBRA [ 1 ] Dav Ebrly Plnty xcllnt documnts for Gomtry and Computr Graphics. [ ] Hugo Pfortnr A scintific rsarch about optimal sphr subdivisions, optimizd for balancd luminanc in ray-tracr applications. Many rlatd links, a highly intrsting publication. Grnot Hoffmann May 05 / 00 + January 31 / 013 Wbsit Load browsr / Click hr