You are an expert at summarizing long articles. Proceed to summarize the following text: 
let @xmath1 be a @xmath0 immersion such that the set @xmath3 is not the empty set , where @xmath4 is identified with @xmath5 . the perspective projection of the given plane curve @xmath6 from any point of ( 1.1 ) does not give the silhouette of @xmath6 because it is non - singular . by this reason , the set ( 1.1 ) is called the _ no - silhouette _ of @xmath7 and is denoted by @xmath8 ( see figure 1 ) . .,width=151 ]    the notion of no - silhouette was first defined and studied from the viewpoint of perspective projection in @xcite . in @xcite it has been shown that the topological closure of no - silhouette is a wulff shape , which is the well - known geometric model of crystal at equilibrium introduced by g.  wulff in @xcite .    in this paper , we show that by rotating all tangent lines about their tangent points simultaneously with the same angle , we always obtain a geometric model of crystal growth ( proposition [ proposition 4 ] ) , our model never formulates a polygon while it is growing ( theorem [ theorem 1 ] ) , our model always dissolves to a point ( theorems [ theorem 2 ] ) , and our model is growing in a relatively simple way when the given @xmath7 has no inflection points ( theorem 3 ) . for any @xmath0 immersion @xmath1 and any real number @xmath9 , define the new set @xmath10 where @xmath11 is the rotation defined by @xmath12 ( see figure 2 ) . when the given @xmath7 has its no - silhouette @xmath8 , by definition , it follows that @xmath13 . [ lemma 1.1 ] for any @xmath0 immersion @xmath1 , @xmath14 is the empty set .    for any point @xmath15 , let @xmath16 be the function defined by @xmath17 where the dot in the center stands for the scalar product of two vectors . since @xmath18 is a @xmath0 function and @xmath19 is compact , there exist the maximum and the minimum of the set of images @xmath20 . let @xmath21 ( resp . , @xmath22 ) be a point of @xmath19 at which @xmath18 attains its maximum ( resp . , minimum ) . then , both @xmath21 and @xmath22 are critical points of @xmath18 . thus , differentiating ( 1.2 ) with respect to @xmath23 yields that the vector @xmath24 is perpendicular to the tangent line to @xmath7 at @xmath25 . it follows that @xmath26 . @xmath27    in section [ section 2 ] , it turns out that with respect to the pompeiu - hausdorff metric the topological closure of @xmath28 varies continuously depending on @xmath9 while @xmath28 is not empty ( proposition [ proposition 5 ] ) . therefore , by lemma 1.1 , the following notion of aperture angle @xmath29 @xmath30 is well - defined . [ definition 1 ] let @xmath1 be a @xmath0 immersion with its no - silhouette @xmath8 . then , @xmath29 @xmath31 is defined as the largest angle which satisfies @xmath32 for any @xmath9 ( @xmath33 ) . the angle @xmath29 is called the _ aperture angle _ of the given @xmath7 .    in section 2 , it turns out also that @xmath34 is a wulff shape for any @xmath9 @xmath35 , where @xmath34 stands for the topological closure of @xmath28 ( proposition [ proposition 4 ] ) . we are interested in how the wulff shape @xmath34 dissolves as @xmath9 goes to @xmath29 from @xmath36 . [ theorem 1 ] let @xmath1 be a @xmath0 immersion with its no - silhouette @xmath8 . then , for any @xmath9 @xmath37 , @xmath38 is never a polygon even if the given @xmath39 is a polygon .    by theorem [ theorem 1 ] , none of @xmath40 , @xmath41 and @xmath42 in figure [ figure 2 ] is a polygon although @xmath43 is a polygon constructed by four tangent lines to @xmath7 at four inflection points . [ theorem 2 ]    here , @xmath44 is the pompeiu - hausdorff metric ( for the pompeiu - hausdorff metric , see section 2 ) . theorem [ theorem 2 ] justifies the following definition . [ definition 2 ] let @xmath1 be a @xmath0 immersion with its no - silhouette @xmath8 . then , the set @xmath45 is called the _ aperture _ of @xmath7 and the unique point @xmath46 is called the _ aperture point _ of * r*. here , @xmath29 @xmath31 is the aperture angle of @xmath7 . the simplest example is a circle . the aperture of a circle is the topological closure of its inside region and the aperture point of it is its center . in this case , the aperture angle is @xmath47 . in general , in the case of curves with no inflection points , the crystal growth is relatively simpler than curves with inflections as follows .    [ theorem 3 ] let @xmath1 be a @xmath0 immersion with its no - silhouette @xmath8 . suppose that @xmath7 has no inflection points . then , for any two @xmath48 satisfying @xmath49 , the following inclusion holds : @xmath50    figure [ figure 2 ] shows that in general it is impossible to expect the same property for a curve with inflection points .    in section 2 , preliminaries are given . theorems [ theorem 1 ] , [ theorem 2 ] and [ theorem 3 ] are proved in sections [ section 3 ] , [ section 4 ] and [ section 5 ] respectively . let @xmath51 be a @xmath0 immersion . let @xmath52 be the mapping defined by @xmath53 where @xmath54 stands for differentiating @xmath55 with respect to @xmath56 . let @xmath57 be the mapping defined by @xmath58 the mapping @xmath57 is called the _ spherical dual of _ @xmath59 . the singularities of @xmath60 belong to the class of legendrian singularities which are relatively well - investigated ( for instance , see @xcite ) . let @xmath61 be an open arc of @xmath19 . suppose that @xmath62 for any @xmath63 . then , for the orthogonal moving frame @xmath64 , @xmath65 , the following serre - frenet type formula has been known . [ lemma 2.1 ] @xmath66    here , @xmath67 is defined by @xmath68    let @xmath69 be the north pole @xmath70 of the unit sphere @xmath71 and let @xmath72 be the northern hemisphere @xmath73 , where @xmath74 stands for the scalar product of two vectors @xmath75 . then , define the mapping @xmath76 , which is called the _ central projection _ , as follows : @xmath77 where @xmath78 . let @xmath1 be a @xmath0 immersion . then , from @xmath7 we can naturally obtain a spherical curve @xmath51 as follows : @xmath79 where @xmath80 is the mapping defined by @xmath81 . for any @xmath56 , let @xmath82 be the intersection @xmath83 . the following clearly holds :    [ lemma 2.2 ] by the central projection @xmath76 , @xmath84 is mapped to the line @xmath85 . one of the merit of considering inside the sphere @xmath86 is the following :    [ lemma 2.3 ] let @xmath87 be a legendrian mapping . then , the following two are equivalent conditions .    1 . the set @xmath88 is not empty and @xmath69 is inside this open set . the connected subset @xmath89 is inside @xmath72 , where @xmath60 is the dual of @xmath59 . let @xmath90 be the mapping defined by @xmath91 the mapping @xmath92 is very useful for studying spherical pedals , pedal unfoldings of spherical pedals , hedgehogs , and wulff shapes ( see @xcite ) . there is also a hyperbolic version of @xmath92 ( @xcite ) . the fundamental properties of @xmath92 is as follows :    1 .   for any @xmath93 , the equality @xmath94 holds , 2 .   for any @xmath93 , the property @xmath95 holds , 3 .   for any @xmath93 , the property @xmath96 holds , 4 . the restriction @xmath97 is a @xmath0 diffeomorphism .    by these properties , we have the following :    the mapping @xmath98 is the inversion of @xmath99 .      for any point @xmath100 of @xmath86 , we let @xmath101 be the following set : @xmath102 here , the dot in the center stands for the scalar product of @xmath103 .    [ spherical polar set ] let @xmath104 be a subset of @xmath86 . then , the set @xmath105 is called the _ spherical polar set _ of @xmath104 and is denoted by @xmath106 . figure [ figure 3 ] illustrates definition [ spherical polar set ] . it is clear that @xmath107 is closed for any @xmath108 . .,width=151 ]    [ definition 2.1 ] a subset @xmath108 is said to be _ hemispherical _ if there exists a point @xmath109 such that @xmath110 . figure [ figure 4 ] illustrates definition [ definition 2.1 ] . .,width=151 ]    [ definition 2.2 ] a hemispherical subset @xmath108 is said to be _ spherical convex _ if @xmath111 for any @xmath112 .    here , @xmath113 stands for the following arc : @xmath114 note that @xmath115 for any @xmath112 and any @xmath116 $ ] if @xmath108 is hemispherical . note further that @xmath106 is spherical convex if @xmath104 is hemispherical and it has an interior point .    [ definition 2.2 ] let @xmath104 be a hemispherical subset of @xmath86 . then , the _ spherical convex hull of _ @xmath104 ( denoted by @xmath117 ) is the following set . @xmath118    [ maehara s lemma ] for any hemispherical finite subset @xmath119 , the following holds : @xmath120    lemma [ maehara s lemma ] is called _ maehara s lemma _ ( see @xcite ) . [ pompeiu - hausdorff ]    let @xmath121 be a complete metric space .    1 . let @xmath122 be a point of @xmath123 and let @xmath124 a non - empty compact subset of @xmath123 . define @xmath125 then , @xmath126 is called the _ distance from the point @xmath122 to the set @xmath124_. 2 . let @xmath127 be two non - empty compact subsets of @xmath123 . define @xmath128 then , @xmath129 is called the _ distance from the set @xmath130 to the set @xmath124_. 3 . let @xmath127 be two non - empty compact subsets of @xmath123 . define @xmath131 then , @xmath132 is called the _ pompeiu - hausdorff distance between @xmath130 and @xmath124_.    let @xmath121 be a complete metric space . the set consisting of non - empty compact subsets of @xmath123 is denoted by @xmath133 , which is the metric space with respect to the pompeiu - hausdorff metric @xmath134 , where @xmath135 is the metric naturally induced by the pompeiu - hausdorff distance . it is well - known also that the metric space @xmath136 is complete . for more details on the complete metric space @xmath136 , see for instance @xcite . [ spherical polar transform ] let @xmath137 be the mapping defined by @xmath138    the mapping @xmath137 is called the _ spherical polar transform_.    [ continuity ] the spherical polar transform is continuous with respect to the pompeiu - hausdorff metric . let @xmath139 be a convergent sequence . set @xmath140 . in order to prove proposition [ continuity ] , it is sufficient to show that @xmath141 . suppose that there exists a real number @xmath142 such that for any @xmath143 there exists an @xmath144 @xmath145 such that @xmath146 . then , by definition [ pompeiu - hausdorff ] , it follows that for any @xmath143 , at least one of @xmath147 and @xmath148 holds . by taking a subsequence if necessary , from the first we may assume that one of the following holds :    1 . @xmath147 for any @xmath143 . 2 .   @xmath148 for any @xmath143 ,    we first show that ( 1 ) implies a contradiction . by definition [ pompeiu - hausdorff ] , it follows that for any @xmath143 there exists a point @xmath149 such that @xmath150 . again by definition [ pompeiu - hausdorff ] , it follows that for any @xmath143 there exists a point @xmath149 such that the inequality @xmath151 holds for any @xmath152 . it is known that @xmath130 can be characterized as follows ( @xcite ) . @xmath153 let @xmath100 be a point of @xmath130 . by ( 2.1 ) , for any @xmath143 we may choose a point @xmath154 such that @xmath155 . then , since @xmath156 , it follows that @xmath157 . since @xmath86 is compact , there exists a convergent subsequence @xmath158 of the sequence @xmath159 . set @xmath160 . then , the inequality @xmath151 implies the inequality @xmath161 for any @xmath152 . on the other hand , the inequality @xmath157 implies the inequality @xmath162 for any @xmath163 . therefore , the point @xmath122 is an element of @xmath164 such that the inequality @xmath161 holds for any @xmath152 . this is a contradiction . we next show that ( 2 ) implies a contradiction . by the same argument as in ( 1 ) , we have that for any @xmath143 there exists a point @xmath165 such that the inequality @xmath166 for any @xmath167 . this implies that there exists an @xmath168 such that for any @xmath143 there exists @xmath169 such that @xmath170 . since @xmath86 is compact , there exists a subsequence @xmath171 of @xmath172 such that both @xmath158 , @xmath173 are convergent sequences . set @xmath160 and @xmath174 . then , the inequality @xmath170 implies the inequality @xmath175 . on the other hand , since @xmath164 is compact , @xmath122 belongs to @xmath164 . moreover , by ( 2.1 ) , @xmath100 belongs to @xmath130 . hence , by definition [ spherical polar set ] , the scalar product @xmath176 must be non - negative . therefore , we have a contradiction . @xmath27      let @xmath177 be the set @xmath178 and let @xmath179 be a continuous function . for any @xmath180 , set @xmath181 where the dot in the center stands for the scalar product of two vectors @xmath182 . the following set is called the _ wulff shape associated with the support function @xmath183 _ ( see figure [ figure 5 ] ) : @xmath184    .,width=226 ]    for any crystal at equilibrium the shape of it can be constructed as the wulff shape @xmath185 by an appropriate support function @xmath183 ( @xcite ) . it is clear that any wulff shape @xmath186 is a convex body ( namely , it is compact , convex and the origin of @xmath187 is contained in @xmath186 as an interior point ) . it has been known that its converse , too , holds as follows . [ taylor ] let @xmath104 be a subset of @xmath188 . then , there exists a parallel translation @xmath189 such that @xmath190 is the wulff shape associated with an appropriate support function if and only if @xmath104 is a convex body . [ proposition 1 ]    [ proposition 2 ] let @xmath191 be a cauchy sequence of wulff shapes in @xmath192 with respect to the pompeiu - hausdorff metric @xmath135 . suppose that @xmath193 does not have an interior point . then , it must be a point or a segment . [ dual wulff shape ] let @xmath194 be a continuous function . then , for the wulff shape @xmath186 , the set @xmath195 is the wulff shape associated with an appropriate support function . the wulff shape @xmath195 is called the _ dual wulff shape of _ @xmath186 .    [ proposition c1 ] let @xmath194 be a function of class @xmath196 . then , the wulff shape @xmath186 is never a polygon . [ proposition 3 ]    [ proposition 4 ] let @xmath1 be a @xmath0 immersion with its no - silhouette @xmath8 . then , for any @xmath197 , there exists a parallel translation @xmath198 such that @xmath199 is a wulff shape @xmath200 by an appropriate support function @xmath201 . we first show that @xmath28 is an open set for any @xmath197 . let @xmath100 be a point of @xmath28 . suppose that for any positive integer @xmath202 , there exists a point @xmath203 , where @xmath204 is the disc @xmath205 . then , since @xmath19 is compact , by taking a subsequence if necessary , we may assume that there exists a convergent sequence @xmath206 @xmath207 such that @xmath208 belongs to @xmath209 . then , we have that @xmath210 where @xmath211 , which implies @xmath212 . hence , @xmath28 is an open set .    since @xmath213 , it follows that @xmath32 . let @xmath100 be a point of @xmath28 . let @xmath214 be the point such that the vector @xmath215 is perpendicular to the line @xmath216 . then , by obtaining the concrete expression of @xmath217 , it follows that the mapping @xmath218 defined by @xmath219 is of class @xmath0 . let @xmath220 be the plane inversion defined by @xmath221 . since @xmath222 , the composed mapping @xmath223 is well - defined . let the boundary of convex hull of @xmath224 be denoted by @xmath225 . then , by the construction , @xmath225 intersect the half line @xmath226 exactly at one point for any @xmath56 . thus , the composed image @xmath227 intersect the half line @xmath226 exactly at one point for any @xmath56 . moreover , the intersecting points depend on @xmath23 continuously . hence , by corresponding @xmath56 to the distance between @xmath100 and the unique intersecting point @xmath228 , we obtain the well - defined continuous function @xmath201 . since @xmath229 is of class @xmath0 , it is easily seen that the obtained function @xmath230 satisfies the assumption of theorem 6.3 in @xcite . let @xmath198 be the parallel translation given by @xmath231 . then , by theorem 6.3 of @xcite , it follows that @xmath232 @xmath27    [ proposition 5 ] let @xmath1 be a @xmath0 immersion with its no - silhouette @xmath8 . then , the map @xmath233 defined by @xmath234 is continuous ,    let @xmath235 be the set consisting of continuous function @xmath236 . the set @xmath235 is a ( non - complete ) metric space with respect to the metric @xmath237 . let @xmath238 ( resp . @xmath239 ) be the mapping defined by @xmath240 ( resp . @xmath241 ) , where @xmath230 is the continuous function defined in the proof of proposition [ proposition 4 ] . then , in order to show that @xmath242 is continuous , it is sufficient to show that both @xmath243 are continuous . we first show that @xmath244 is continuous . let @xmath245 be the function defined by @xmath246 where the set @xmath247 , which appeared in the proof of proposition [ proposition 4 ] , is a one point set and it is regarded as a point . by obtaining the concrete expression of @xmath229 given in the proof of proposition , it is easily seen that @xmath229 is smoothly depending on @xmath248 . thus , @xmath249 is continuously depending on @xmath248 . since @xmath250 is a @xmath0 diffeomorphism of @xmath251 , it follows that @xmath230 is continuously depending on @xmath248 . hence , @xmath244 is a continuous mapping . we next show that @xmath252 is continuous . let @xmath253 be a convergent sequence to an element of @xmath235 . set @xmath254 . we also set @xmath255 then , it is easily seen that @xmath256 is an open set . thus , @xmath104 is a closed set . we show @xmath257 . let @xmath100 be an interior point of @xmath186 . then , since @xmath254 , @xmath100 must be an interior point of @xmath258 for any sufficiently large @xmath259 . thus , @xmath100 is contained in @xmath104 . since both @xmath186 and @xmath104 are closed , it follows that @xmath260 . next , let @xmath261 be a point of @xmath104 . suppose that @xmath261 is not contained in @xmath186 . then , there exists @xmath262 such that @xmath263 , where @xmath264 stands for the scalar product of two vectors @xmath265 . set @xmath266 . since @xmath254 , it follows that @xmath267 for any sufficiently large @xmath259 . this contradicts to the assumption that @xmath268 . hence , we have that @xmath269 , and it follows that @xmath257 .    the remaining part of the proof that @xmath252 is continuous is to show the following : @xmath270 in order to show ( 2.2 ) , by the construction of @xmath104 , it is sufficient to show that @xmath271 is a cauchy sequence of @xmath272 . since @xmath273 is a cauchy sequence of @xmath235 , it is clear that @xmath271 is a cauchy sequence . therefore , we have that @xmath274 and it follows that @xmath252 is continuous . set @xmath275 and @xmath276 for @xmath56 . we investigate the singularities of @xmath277 . let @xmath61 be an open arc of @xmath19 . by using the arc - length parameter of @xmath278 , without loss of generality , from the first we may assume that @xmath62 for @xmath279 . then , by lemma [ lemma 2.1 ] , we have the following : @xmath280 since the angle @xmath9 satisfies @xmath281 in theorem [ theorem 1 ] , it follows that @xmath282 . therefore , @xmath277 is non - singular even at the point @xmath56 such that @xmath283 . next , we show that @xmath284 for any @xmath56 . let the dual of @xmath277 be denoted by @xmath285 . then , it follows that @xmath285 is a legendrian mapping and the following equality holds . @xmath286 since @xmath213 , by lemma [ lemma 2.3 ] , we have that @xmath284 for any @xmath56 . thus , the spherical convex hull of @xmath287 is well - defined . since @xmath277 is non - singular , the boundary of @xmath288 is a submanifold of class @xmath196 ( for instance see @xcite ) . by the property ( 4 ) of @xmath92 , the boundary of @xmath289 is a submanifold of class @xmath196 . it follows that the boundary of @xmath290 is a submanifold of class @xmath196 .    on the other hand , @xmath291 is a wulff shape @xmath186 with the support function @xmath183 whose graph with respect to the polar coordinate expression is the boundary of @xmath290 . therefore , the support function @xmath183 for the wulff shape @xmath291 is of class @xmath196 and it follows that @xmath186 is never a polygon by proposition [ proposition 3 ] . @xmath27      by proposition [ proposition 4 ] , for any @xmath292 there exists a parallel translation @xmath293 such that @xmath294 is a wulff shape @xmath258 by an appropriate support function @xmath295 . by proposition [ dual wulff shape ] , for any @xmath292 the set @xmath296 is a wulff shape too . thus , by proposition [ taylor ] , it follows that both @xmath297 and @xmath298 belong to @xmath299 for any @xmath292 . moreover , proposition [ proposition 5 ] , we may assume that @xmath300 is a cauchy sequence . thus , both @xmath301 and @xmath302 are cauchy sequences .    by proposition [ proposition 2 ] , @xmath303 is a point or segment . suppose that it is a segment . let @xmath304 be two boundary points of this segment . then , by proposition [ continuity ] and lemma [ maehara s lemma ] , we have the following : @xmath305    let @xmath306 be the @xmath0 mapping defined by @xmath307 for any @xmath56 , where @xmath60 and @xmath308 are the same @xmath0 mapping as in the proof of theorem [ theorem 1 ] . then , notice that @xmath309 . for any @xmath310 @xmath311 , we let the set @xmath312 be denoted by @xmath313 . then , the intersection @xmath314 consists of two antipodal points @xmath315 . by lemma [ lemma 2.3 ] and proposition [ proposition 1 ] , there exists @xmath316 @xmath317 such that @xmath318 , @xmath319 .    on the other hand , since @xmath320 , similarly as in the proof of theorem [ theorem 1 ] , it follows that @xmath321 is non - singular . thus , we have a contradiction . @xmath27      for any @xmath9 @xmath322 and any @xmath56 , set @xmath323 let @xmath324 be the affine function which define @xmath325 . set @xmath326 then , since @xmath327 is a convex body for any @xmath9 @xmath35 , it follows that one of @xmath328 or @xmath329 . by proposition [ proposition 4 ] , we may assume that the following holds for any @xmath9 @xmath35 . @xmath330 since @xmath7 does not have inflection points , it follows that @xmath331 contains @xmath28 for any @xmath9 @xmath35 . thus , for any @xmath9 @xmath35 , we have the following : @xmath332 since @xmath7 does not have inflection points , we have that @xmath333 contains @xmath334 for any two @xmath335 satisfying @xmath49 . it follows that @xmath336 if @xmath337 . t.  nishimura is partially supported by jsps kakenhi grant number 26610035 . 99 v.  i.  arnold , _ the geometry of spherical curves and the algebra of quaternions _ , russian math . surveys , 50(1995 ) , 168 . v.  i.  arnold , v.  v.  goryunov , o.  v.  lyashko and v.  a.  vasilev , _ dynamical systems viii . _ encyclopaedia of mathematical sciences , * 39 * , springer - verlag , berlin heidelberg new york , 1989 . v.  i.  arnold , s.  m.  gusein - zade and a.  n.  varchenko , _ singularities of differentiable maps . i. _ monographs in mathematics , * 82 * , birkhuser , boston inc . , boston , ma , 1985 . m.  barnsley , _ fractals everywhere 2nd edition _ , morgan kaufmann pub . , san fransisco , 1993 . k.  falconer , _ fractal geometry mathematical foundations and applications 2nd edition _ , john wiley & sons ltd . , chichester , west sussex , 2003 . s.  izumiya and f.  tari , _ projections of hypersurfaces in the hyperbolic space to hyperhorospheres and hyperplanes _ , rev .  mat . iberoam . * 24*(2008 ) , 895920 . t.  nishimura , _ normal forms for singularities of pedal curves produced by non - singular dual curve germs in @xmath338 _ , geom dedicata , * 43 * ( 2008 ) 5966 . t.  nishimura , _ singularities of pedal curves produced by singular dual curve germs in @xmath338 _ , demonstratio math . , * 63 * ( 2010 ) , 447459 . t.  nishimura , _ singularities of one - parameter pedal unfoldings of spherical pedal curves _ , j.  singul . , * 2*(2010 ) , 160169 . t.  nishimura and y.  sakemi , _ view from inside _ , hokkaido math . j. , * 40 * ( 2011 ) 361373 . t.  nishimura and y.  sakemi , _ topological aspect of wulff shapes _ , j. math . japan , * 66 * ( 2014 ) , 89109 . . s.  a.  robertson and m.  c.  romero - fuster , _ the convex hull of a hypersurface _ , proc . london math . soc . , * 50*(1985 ) , 370384 . j.  e.  taylor , _ crystalline variational problems _ , bull . amer . math . * 84 * ( 1978 ) , 568588 . g.  wulff , _ zur frage der geschwindindigkeit des wachstrums und der auflsung der krystallflachen _ , z. kristallographine und mineralogie , * 34 * ( 1901 ) , 449530 . v.  m.  zakalyukin , _ singularities of convex hulls of smooth maniifolds _ , functional anal . , * 11*(1977 ) , 225227(1978 ) .
for any given @xmath0 immersion @xmath1 such that the set @xmath2 is not empty , a simple geometric model of crystal growth is constructed . 
 it is shown that our geometric model of crystal growth never formulates a polygon while it is growing . 
 moreover , it is shown also that our model always dissolves to a point .