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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 17 "Virtual Seashells" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 19 "" 0 "" {TEXT -1 45 "by Alain
 Goriely, goriely@math.arizona.edu,\n " }{TEXT 260 1 "(" }{URLLINK 17 
"http://www.math.arizona.edu/~goriely" 4 "http://www.math.arizona.edu/
~goriely" "" }{TEXT 261 2 ")\n" }}{PARA 257 "" 0 "" {TEXT 256 9 "Abstr
act:" }{TEXT -1 49 " This worksheet show how to represent seashells.\n
" }{TEXT 259 28 "Application Areas/Subjects: " }{TEXT 257 0 "" }{TEXT 
-1 33 " Education, Biology, 3D graphics\n" }{TEXT 258 9 "Keywords:" }
{TEXT -1 21 " Seashells, 3D plots\n" }{TEXT 263 14 "Prerequisites:" }
{TEXT -1 7 "  plots" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }{TEXT 262 5 "N
ote:" }{TEXT -1 68 "  Send me an e-mail (comments-criticisms) if you u
se this worksheet." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 20 "restart:with(plots):" }}}{SECT 0 {PARA 3 "
" 0 "" {TEXT -1 12 "Introduction" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
159 "Seashells can be represented by  surfaces in 3D. Their self simil
ar shape can be generated by a simple formula with a  certain number o
f free parameters. Let (" }{XPPEDIT 18 0 "x,y,z" "6%%\"xG%\"yG%\"zG" }
{TEXT -1 81 ") be a point on the shell. Then, this point can be writte
n in the following form:" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 1 "\n" 
}{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 1 "=" }{XPPEDIT 18 0 "D*(A*sin(
beta)*cos(Theta)+R*cos(s+phi)*cos(Theta+Omega)-R*sin(mu)*sin(s+phi)*si
n(Theta+Omega))*exp(Theta*cot(alpha));" "6#*(%\"DG\"\"\",(*(%\"AGF%-%$
sinG6#%%betaGF%-%$cosG6#%&ThetaGF%F%*(%\"RGF%-F.6#,&%\"sGF%%$phiGF%F%-
F.6#,&F0F%%&OmegaGF%F%F%**F2F%-F*6#%#muGF%-F*6#,&F6F%F7F%F%-F*6#,&F0F%
F;F%F%!\"\"F%-%$expG6#*&F0F%-%$cotG6#%&alphaGF%F%" }}{PARA 258 "" 0 "
" {XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT -1 1 "=" }{XPPEDIT 18 0 "(-A*cos(
beta)+R*sin(s+phi)*cos(mu))*exp(Theta*cot(alpha));" "6#*&,&*&%\"AG\"\"
\"-%$cosG6#%%betaGF'!\"\"*(%\"RGF'-%$sinG6#,&%\"sGF'%$phiGF'F'-F)6#%#m
uGF'F'F'-%$expG6#*&%&ThetaGF'-%$cotG6#%&alphaGF'F'" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 174 "This formula was obtaine
d by A. Cortie and follows a series of basic rotations around the diff
erent axes (helical axis, body axis, etc..). In this formula the param
eters are:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" 
{XPPEDIT 18 0 "Theta" "6#%&ThetaG" }{TEXT -1 19 ": polar coordinates" 
}}{PARA 15 "" 0 "" {XPPEDIT 18 0 "phi" "6#%$phiG" }{TEXT -1 50 ": tilt
 of ellipse major axis from horizontal plane" }}{PARA 15 "" 0 "" 
{XPPEDIT 18 0 "s" "6#%\"sG" }{TEXT -1 30 ":  angle on each cross secti
on" }}{PARA 15 "" 0 "" {XPPEDIT 18 0 "mu" "6#%#muG" }{TEXT -1 68 ":  a
ngle between the plane of the cross section and the helical axis" }}
{PARA 15 "" 0 "" {XPPEDIT 18 0 "Omega" "6#%&OmegaG" }{TEXT -1 80 ":  r
otation of the elliptical cross section with respect to the horizonatl
 plane" }}{PARA 15 "" 0 "" {XPPEDIT 18 0 "beta" "6#%%betaG" }{TEXT -1 
35 ":  angle between the helical axis (" }{XPPEDIT 18 0 "z" "6#%\"zG" 
}{TEXT -1 25 ") and local aperture line" }}{PARA 15 "" 0 "" {TEXT -1 
121 "R:  length of aperture generating curve (can itself be a function
 of parameters and s to have a nonuniform cross sections" }}{PARA 15 "
" 0 "" {TEXT -1 50 "a:  major semiaxis of the elliptical cross section
" }}{PARA 15 "" 0 "" {TEXT -1 17 "b: minor semiaxis" }}{PARA 15 "" 0 "
" {TEXT -1 19 "D: sense of coiling" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 179 "
The following worksheet is a straightforward application of this formu
la for different typical shapes of seashells. It is not a profound exe
rcice (it is actually straightforwad). " }}{PARA 0 "" 0 "" {TEXT -1 
46 "Just try to have fun with virtual seashells! I" }}{PARA 0 "" 0 "" 
{TEXT -1 221 "f anybody has the courage to do it, it would be nice to \+
transform these plots in vrml (easy) and apply realistic textures file
s (harder). It would be nice to have a real 3D virtual seashells museu
m in vrml (let me know!)." }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 9 "The
 model" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "rad:=proc(aa) aa/3
60*2*Pi end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 303 "x:=D*(A*si
n(beta)*cos(Theta)+R*cos(s+phi)*cos(Theta+Omega)-R*sin(mu)*sin(s+phi)*
sin(Theta+Omega))*exp(Theta*cot(alpha)):\ny:=(-A*sin(beta)*sin(Theta)-
R*cos(s+phi)*sin(Theta+Omega)-R*sin(mu)*sin(s+phi)*cos(Theta+Omega))*e
xp(Theta*cot(alpha)):\nz:=(-A*cos(beta)+R*sin(s+phi)*cos(mu))*exp(Thet
a*cot(alpha)):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "SR:=R=
RE+k;\nSRE:=RE=1/sqrt(cos(s)^2/a^2+sin(s)^2/b^2);\nSk:=k=L*exp(-(2*(s-
P)/W[1])^2)*exp(-(2*g/W[2])^2);\nSg:=g=2*Pi/N*(Theta*N/2/Pi-round(Thet
a*N/2/Pi));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#SRG/%\"RG,&%#REG\"\"
\"%\"kGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$SREG/%#REG*&\"\"\"F(*$
,&*&-%$cosG6#%\"sG\"\"#%\"aG!\"#F(*&-%$sinGF.F0%\"bGF2F(#F(F0!\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#SkG/%\"kG*(%\"LG\"\"\"-%$expG6#,$*(
\"\"%F),&%\"sGF)%\"PG!\"\"\"\"#&%\"WG6#F)!\"#F3F)-F+6#,$*(F/F)%\"gGF4&
F66#F4F8F3F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#SgG/%\"gG,$**\"\"#
\"\"\"%#PiGF*%\"NG!\"\",&**F)F-%&ThetaGF*F,F*F+F-F*-%&roundG6#,$**F)F-
F0F*F,F*F+F-F*F-F*F*" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 8 "Natalina
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "S:=\{D=1,alpha=rad(80),
beta=rad(40),phi=rad(55),mu=rad(10), Omega=rad(30),smM=rad(-270)..rad(
80),A=25,a=12,b=16,P=0,L=0\};\nR1:=subs(S,subs(SR,SRE,Sk,Sg,[x,y,z])):
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG<./%\"AG\"#D/%\"DG\"\"\"/%\"
LG\"\"!/%\"PGF./%\"aG\"#7/%\"bG\"#;/%#muG,$*&\"#=!\"\"%#PiGF+F+/%$phiG
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F<F=F+F+/%&OmegaG,$*&\"\"'F<F=F+F+/%&alphaGFK/%%betaG,$*(FJF+FNF<F=F+F
+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "plot3d(R1,s=subs(S,sm
M),Theta=-2*Pi..4*Pi,grid=[100,100],style=patchnogrid,scaling=constrai
ned,orientation=[-68,80],lightmodel=light2,shading=xy);" }}{PARA 13 "
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{SECT 0 {PARA 3 "" 0 "" {TEXT -1 8 "Nautilus" }}{EXCHG {PARA 0 "> " 0 
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\nR:=simplify(subs(S,subs(SR,SRE,Sk,Sg,[x,y,z])));" }{TEXT -1 0 "" }}
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{PARA 3 "" 0 "" {TEXT -1 8 "Ammonite" }}{EXCHG {PARA 0 "> " 0 "" 
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#>%\"SG<1/%\"AG\"$]%/%\"DG\"\"\"/%\"LG\"\"!/%\"NG\"\")/%\"PGF./%\"aG\"
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"" {TEXT -1 7 "Codakia" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 196 "S
:=\{D=1,alpha=rad(37),beta=rad(90),phi=rad(0),mu=rad(1), Omega=rad(1),
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0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG<1/%\"AG\"&+0\"/%\"DG\"\"
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "plot3d(R2,s=subs(S,smM),The
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1 1 1.000000 82.000000 -58.000000 1 0 "Curve 1" }}}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "Refe
rences" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "A. Cortie : Digital Seas
hells" }{TEXT 265 19 " Comput & Graphics " }{TEXT -1 4 "Vol " }{TEXT 
266 2 "17" }{TEXT -1 18 ", pp. 79-84 (1993)" }}}{EXCHG {PARA 259 "" 0 
"" {TEXT 264 11 "Disclaimer:" }{TEXT -1 240 " While every effort has b
een made to validate the solutions in this worksheet, Waterloo Maple I
nc. and the contributors are not responsible for any errors contained \+
and are not liable for any damages resulting from the use of this mate
rial." }}}}}{MARK "0 0 0" 1 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
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