Criteria for sharksfin and deltoid singularities from the plane into the plane and their applications
aa r X i v : . [ m a t h . G T ] J a n Criteria for sharksfin and deltoid singularities fromthe plane into the plane and their applications
Yutaro Kabata Kentaro SajiJanuary 21, 2021
Abstract
We give criteria for sharksfin and deltoid singularities from the plane intothe plane. We also give geometric meanings for the conditions in the criterionof a sharksfin. As applications, we investigate such singularities appearing onan orthogonal projection of a Whitney umbrella, and a sharksfin appearing onplanar motions with 2-degrees of freedom.
The singularities of maps from the plane into the plane have been one of the fundamentalsubjects in the singularity theory of the smooth maps. Classification of singularitiesof maps from the plane into the plane has been investigated by many researchers, anduseful recognition criteria for main corank one singularities are given (see for example[1,7,10,12,15–17,25]). However, as long as the authors know, useful recognition criteriafor corank two maps have not been given in the literature. In this paper, we giveuseful recognition criteria for “sharksfin” and “deltoid” singularities, which are themost generic singularities of corank two maps from the plane into the plane:
Theorem A.
Let f : ( R , → ( R , be a map-germ satisfying rank df = 0 , and let λ be an identifier of singularities of f . Let λ have a non-degenerate critical point at ,and let η , η be vector fields which satisfy that η (0) and η (0) are linearly independentsolutions of the Hesse quadric of λ at . Then f is a sharksfin ( respectively, deltoid ) at if and only if det Hess( λ )(0) < respectively, det Hess( λ )(0) > , det( η f, η f )(0) = 0 and det( η f, η f )(0) = 0 . (1.1)Here, ηf stands for the directional derivative of f by the vector field η , and η i f = η ( η i − f ). See Section 2.1 for the definitions of identifier of singularities and solutionsof the Hesse quadric of a function at a non-degenerate critical point. We also give a Keywords and Phrases: criteria, sharksfin, deltoid, Whitney umbrella, planar motionsPartly supported by the Japan Society for the Promotion of Science KAKENHI Grants numbered18K03301, 20K14312 and the Japan-Brazil bilateral project JPJSBP1 20190103.Mathematics Subject Classification 2020 Primary 57R45; Secondary 58K05 S ( f ) consists of twotransversal regular curves, say γ i ( i = 1 , f ◦ γ i ( i = 1 ,
2) having 3 / D ± -singularities of fronts, which are similar singularitiesof a sharksfin and a deltoid. A map-germ f : ( R , → ( R ,
0) is called a sharksfin (respectively, deltoid ) if it is A -equivalent to the map-germ( u, v ) ( uv, u + v + u ) (cid:0) respectively, ( uv, − u + v + u ) (cid:1) at the origin 0. Here two map-germs f i : ( R , → ( R ,
0) ( i = 1 ,
2) are said to be A -equivalent if there exist a diffeomorphism-germ ϕ on the source space of f and adiffeomorphism-germ Φ on the target space of f such that Φ ◦ f = f ◦ ϕ holds. We notethat the map-germ ( u, v ) ( uv, u + v + u ) is A -equivalent to ( u, v ) ( u + v , v + u ),and some of the literature uses this form for a sharksfin. A fundamental classificationof map-germs from the plane into the plane is given in [15, 25]. In [16], it is shown thata sharksfin and a deltoid are the only singularities of rank zero and codimension equalto or less than four. See [10, 17, 20, 25] for recognition criteria for fundamental rank onesingularities.Let f : ( R , → ( R ,
0) be a map-germ and let rank df = 0. Let λ be a non-zerofunctional multiple of the Jacobian of f . Since the zeros of λ are the set of singularpoints of f , we call the function λ an identifier of singularities . By the assumptionrank df = 0, the function λ has a critical point at 0.To state the criteria, we define vectors at a non-degenerate critical point of a func-tion. Let h : ( R , → ( R ,
0) be a function which has a non-degenerate critical point at0. The solution of the Hesse quadric of h at 0 is a non-zero vector ( x, y ) which satisfies (cid:18)(cid:0) x y (cid:1) Hess( h )(0) (cid:18) xy (cid:19) = (cid:19) h uu (0) x + 2 h uv (0) xy + h vv (0) y = 0 , where ( · ) u = ∂ ( · ) /∂u , ( · ) v = ∂ ( · ) /∂v and Hess( h )(0) is the Hesse matrix of h at 0.If det Hess( h )(0) >
0, then there exist two C -linearly independent C -valued vectors v , v . If det Hess( h )(0) <
0, then there exist two R -linearly independent R -valuedvectors v , v , and these are the tangent vectors of two branch curves of the zero set of h at 0. For a diffeomorphism ϕ : ( R , → ( R , h ◦ ϕ − t J ϕ (0) − Hess( h )(0) J ϕ (0) − , and (cid:0) x y (cid:1) Hess( h )(0) (cid:18) xy (cid:19) = t (cid:18) J ϕ (0) (cid:18) xy (cid:19)(cid:19) Hess( h ◦ ϕ − )(0) J ϕ (0) (cid:18) xy (cid:19) , the two directions defined by solutions of the Hesse quadric of h at 0 do not depend onthe choice of the coordinate system on the source space. Here, t ( ) stands for the matrixtransposition, and J ϕ stands for the Jacobi matrix of ϕ . We note that if solutions ofthe Hesse quadric η i ( i = 1 ,
2) are complex vectors, then the left-hand sides of (1.1)may be complex numbers.
We give a proof of Theorem A. We first simplify the expression of a rank zero-germ f : ( R , → ( R ,
0) by coordinate changes. Keeping in mind that we will investigategeometric meanings, we restrict to using only particular coordinate changes in the nextlemma.
Lemma 2.1. If rank df = 0 and is a non-degenerate critical point of λ , then thereexist a positive local coordinate system ( u, v ) near , and an orientation preservingisometry M on ( R , , namely, M ∈ SO(2) , such that M ◦ f ( u, v ) = ( uv, O (2)) . Here, O ( i ) stands for the terms whose degrees are greater than or equal to i , namely, O ( i ) is an element of ( f ∈ C ∞ ( R , (cid:12)(cid:12)(cid:12) X a + b ≤ i − ∂ a + b f∂u a ∂v b (0) = 0 ) . A coordinate system is said to be positive if it has the same orientation to the standard R . Proof.
Firstly we show that we may assume 0 is a non-degenerate critical point withindex 1 of f , where f = ( f , f ). If f = O (3) or f = O (3), then the Jacobian of f is O (3). In particular, 0 will never be a non-degenerate critical point of an identifier ofsingularities. Thus, we may assume ( f ) uu (0) = 0 by a suitable rotation on the targetand by choosing a positive coordinate system on the source. By a rotation on the target, f can be written as f = ( f , b uv + b v ) + O (3). If b = 0, then the claim is proved.We assume b = 0. Since f = O (3), it holds that b = 0. Thus f can be writtenas f = ( a u + 2 a uv + a v , b v ) + O (3). If a a − a <
0, then the claim isproved, and if a = 0, then λ degenerates. So, we may assume a a − a > a = 0. By taking the rotation by degree θ on the target, the first component of f canbe written as a cos θu + 2 a cos θuv + ( a cos θ − b sin θ ) v . One can find an angle θ such that cos θ (cos θ ( a a − a ) − sin θa b ) <
0. This shows the claim.Thus we may assume that 0 is a non-degenerate critical point with index 1 of f . Bythe Morse lemma, there exists a positive coordinate system ( u, v ) such that f can bewritten as f = ( ± uv, O (2)). Taking a π -rotation on the target if necessary, this showsthe assertion. 3y Lemma 2.1, we see f is A -equivalent to f ( u, v ) = (cid:16) uv, ε u v X i + j =3 a ij u i v j i ! j ! + O (4) (cid:17) ( ε = ± . (2.1) Lemma 2.2.
Let f be a map-germ of the form (2.1) . When ε = 1 , then f is a sharksfinif and only if ( a − a + 3 a − a )( a + 3 a + 3 a + a ) = 0 . When ε = − , then f is deltoid if and only if a − a = 0 or a − a = 0 .Proof. Following [16, Propositions 2.1.1 and 2.2.1], we will give a proof of this lemma.By the coordinate change u = u + a u / − εa u v / v = v − a u v / εa v / f ( u , v ) = (cid:16) u v , εu v a u a v (cid:17) + ( O (4) , O (4)) , (2.2)where ˜ a = a + 3 εa and ˜ a = a + 3 εa . Let f be written as in (2.2), and ε = 1.By the coordinate change u = u − v + (˜ a − a ) u /
12 + ˜ a u v / − (˜ a + 2˜ a ) v / ,v = u + v + (˜ a − a ) u / − ˜ a u v / − (2˜ a + ˜ a ) v / X, Y ) = ( X + Y, − X + Y ) /
2, we seeΦ ◦ f ( u , v ) = (cid:16) u + (˜ a − ˜ a ) v , v + (˜ a + ˜ a ) u (cid:17) + ( O (4) , O (4)) . (2.3)Since a sharksfin is 3-determined ([16, Proposition 2.1.1]), f in the form (2.1) is asharksfin if and only if (˜ a − ˜ a )(˜ a + ˜ a ) = 0, namely, ( a − a + 3 a − a )( a +3 a + 3 a + a ) = 0.Next, we show the case of a deltoid. Let f be written as in (2.2) and ε = −
1. Let p ( x ) be the polynomial p ( x ) = ˜ a x − a x − a x + 10˜ a x + 5˜ a x − ˜ a , and let x be one of the solutions of p ( x ) = 0. We consider the coordinate changedefined by u = 196 (cid:16) u − x v + u ( − a ( − x ) + ˜ a x (5 + x ))+ u v ( − ˜ a x ( − x ) + 5˜ a x ( − x ))+ v (˜ a ( − x ) + x (˜ a (13 − x ))) (cid:17) ,v = 196 (cid:16) u x + 96 v − u x ( − a ( − x ) + ˜ a x (5 + x )) − u v (˜ a (3 − x ) + ˜ a x (5 + x ))+ v x (˜ a (57 − x ) + ˜ a x ( −
61 + 7 x )) (cid:17) , and we considerΦ( X, Y ) = (cid:16) (1 − x ) X + 2 x Y (1 + x ) , − x − x X + ( x − Y )(1 + x ) ( x − (cid:17) . ◦ f is (cid:18) u v , − u + v + (˜ a + ˜ a )(1 − x + 5 x )3˜ a (1 + x ) v (cid:19) + p ( x )96(1 + x ) (cid:18) − x u + 3( x − u v + 6 x u v + (9 + 7 x ) v , (2.4)2 (cid:0) ( x − u + 6 x u v − x − u v a / ˜ a + 7 x ) v (cid:1)(cid:19) + ( O (4) , O (4)) . The solutions of 5 x − x + 1 = 0 are x = ± (1 ± / √ / , and we see 5 p ( x ) = ± a (10 ± / √ / is not zero if ˜ a = 0. Since a deltoid is 3-determined ([16,Proposition 2.2.1]), f in the form (2.1) is a deltoid if and only if ˜ a = 0 or ˜ a = 0,namely, a − a = 0 or 3 a − a = 0. Lemma 2.3.
Let be a non-degenerate critical point of an identifier of singularities λ of f : ( R , → ( R , , and let ( η , η ) , ( η , η ) ( possibly complex ) be two linearlyindependent solutions of the Hesse quadric of λ at . Let η i be a vector field satisfying η i (0) = η i ∂ u + η i ∂ v ( i = 1 , . Then the condition det( η f, η f )(0) = 0 and det( η f, η f )(0) = 0 does not depend on the choice of η , η .Proof. The condition clearly does not depend on the choice of the coordinate systemon the source space, and the condition does not change by a linear coordinate transfor-mation on the target space. As we remarked just after Lemma 2.1, we may assume f is written as in (2.1). Then the identifier of singularities is − εu + v + O (3). Since thecondition does not change under a constant multiplication of the solutions of the Hessequadric η and η of an identifier of singularities, we may assume η = (1 , , η = (1 , − ε = 1, and η = (1 , i ) , η = (1 , − i ) if ε = −
1, where i = √−
1. By a direct calculation, ξ f = (2 ξ ξ , εξ + ξ ) ,ξ f = (cid:16) ξ ( ξ ) v + ξ ξ (( ξ ) v + ( ξ ) u ) + ξ ( ξ ) u ) ,a ξ + 3 a ξ ξ + 3 a ξ ξ + a ξ + 3 εξ ( ξ ) u + 3 εξ ξ ( ξ ) v + 3 ξ ξ ( ξ ) u + 3 ξ ( ξ ) v (cid:17) , holds at u = v = 0, where ξ = ( ξ , ξ ) is a vector field, and12 det( ξ f, ξ f )(0) = a + 3 a + 3 a + a ( ε = 1 , ξ (0) = η = (1 , − a + 3 a − a + a ( ε = 1 , ξ (0) = η = (1 , − a i − a − ia + a ( ε = − , ξ (0) = η = (1 , i )) − a i − a + 3 ia + a ( ε = − , ξ (0) = η = (1 , − i )) . (2.5)The right-hand side of (2.5) depends only on the value of ξ (0). This shows the assertion.5 roof of Theorem A . The sufficiency follows by the independence of the choice of co-ordinate systems and vector fields. We show the necessity. We assume the assump-tion of Theorem A. Since the condition does not depend on the choice of coordi-nate systems and vector fields, we may assume f is written as in (2.1). Then by(2.5) in the proof of Lemma 2.3, the condition (1.1) in Theorem A is equivalent to( a + 3 a + 3 a + a )( − a + 3 a − a + a ) = 0 when ε = 1, and a − a = 0or 3 a − a = 0 when ε = −
1. By Lemma 2.2, we have the assertion.
We here give a geometric interpretation of condition (1.1). Let f : ( R , → ( R ,
0) bea map-germ with rank df = 0, and let an identifier of singularities λ have an index onecritical point at 0. Then the set λ − (0) consists of images of two transversal regularcurves passing through 0. We set these curves as γ i : ( R , → ( R , c : ( R , → ( R ,
0) at 0 is a 3 / cusp if it is A -equivalent to t ( t , t ). Itis well-known that c : ( R , → ( R ,
0) is a 3 / c ′ (0) = 0, anddet( c ′′ (0) , c ′′′ (0)) = 0. Let c : ( R , → ( R ,
0) be a curve and 0 a 3 / cuspidal direction of c at 0 is the direction defined by c ′′ (0). The cuspidal directionbisects the cusp. Then the following proposition holds. Proposition 3.1.
Under the above setting, f at is a sharksfin if and only if f ◦ γ i ( i = 1 , at are both / -cusps.Proof. Since the condition and the assertion do not depend on the choice of the co-ordinate systems, we may assume that f is written as (2.1) with ε = 1. Then wemay assume that γ = γ + = ( t, a + ( t )) , γ = γ − = ( t, a − ( t )) ( a + (0) = a − (0) = 0).Since λ ( t, a ± ( t )) = 0, we have a ′± (0) = ±
1. We set ˆ γ i ( t ) = f ◦ γ i ( t ). Then we seeˆ γ ′′± (0) = 2( ± , γ ′′′ (0) = (3 a ′′± (0) , a ′′± (0) ± a + 3 a ± a + a ). Thus12 det (cid:0) ˆ γ ′′± (0) , ˆ γ ′′′± (0) (cid:1) = a ± a + 3 a ± a . By (2.5), we have the assertion.
SO(2) -normal form
We give a simplified form of a given rank zero germ by using diffeomorphisms onthe source and isometries on the target space. Since coefficients of such forms aredifferential geometric invariants, this is convenient for studying the differential geometryof singularities. See [2, 4, 5, 8, 9, 11, 13, 14, 19, 22, 24] such studies, for example. Two map-germs f i : ( R , → ( R ,
0) are said to be R + × SO(2) -equivalent if there exist anorientation preserving diffeomorphism-germ ϕ : ( R , → ( R ,
0) and an orientationpreserving isometry M on ( R , M ∈ SO(2), such that M ◦ f = f ◦ ϕ holds. Proposition 3.2.
Let f : ( R , → ( R , be a map-germ with rank df = 0 , andlet an identifier of singularities λ have a non-degenerate critical point at . Then f is + × SO(2) -equivalent to the germ (cid:18) uv, ε a u ε a v a u a v (cid:19) + ( O (4) , O (4)) , (3.1)( a > , ( ε , ε ) ∈ { (1 , , (1 , − , ( − , } ) . We show this proposition by taking R + × SO(2)-equivalent germs step by step. Firstwe show the following lemma.
Lemma 3.3.
Let f : ( R , → ( R , be a map-germ satisfying the condition inProposition . Then f is R + × SO(2) -equivalent to the germ (cid:18) uv, ε a u a uv + ε a v O (3) (cid:19) ( a = 0 , ε , ε ∈ { , − } ) . (3.2) Proof.
By Lemma 2.1, we may assume f is written as f ( u, v ) = ( uv, a u / a uv + a v + O (3)). Since 0 is a non-degenerate critical point of λ , we see a a = 0. Thenby setting u = | a /a | / u , v = | a /a | / v , we see the assertion. Lemma 3.4.
Let f : ( R , → ( R , be a map-germ satisfying the assumption ofProposition . Then f is R + × SO(2) -equivalent to the germ (cid:18) b uv, a u a v (cid:19) + ( O (3) , O (3)) ( b a a = 0) . (3.3) Proof.
By Lemma 3.3, we may assume f is written as (3.2). If a = 0, then no proofis needed. We assume a = 0. We set u = u + v , v = cu + dv ( c, d ∈ R ), and M isthe rotation matrix by degree θ . Then M ◦ f ( u, v ) is (cid:16) (2 c cos θ − ( a + 2 a c + a c ) sin θ ) u ∗ uv (3.4)+ (2 d cos θ − ( a + 2 a d + a d ) sin θ ) v , ∗ u + (( a + a cd + a ( c + d )) cos θ + ( c + d ) sin θ ) uv + ∗ v (cid:17) + ( O (3) , O (3)) , where ∗ stands for a real number which will not be needed in later calculations. Toshow the lemma, we need to solve the equation c cos θ − ( a + 2 a c + a c ) sin θ = 0 , d cos θ − ( a + 2 a d + a d ) sin θ = 0 , ( a + a cd + a ( c + d )) cos θ + ( c + d ) sin θ = 0 (3.5)with respect to c, d, θ . If sin θ = 0, then c = d = 0, and the third equation of (3.5) is a = 0. Thus sin θ = 0 is not a solution. We assume sin θ = 0. Noticing the functioncos θ/ sin θ takes values in R , we set t = cos θ/ sin θ . Then (3.5) is equivalent to2 ct − ( a + 2 a c + a c ) = 0 , (3.6)2 dt − ( a + 2 a d + a d ) = 0 , (3.7)( a + a cd + a ( c + d )) t + c + d = 0 . (3.8)7ote that the equations (3.6) and (3.7) are the same. Thus the solutions c, d are thetwo solutions of the equation − a x + 2( − a + t ) x − a = 0 (3.9)with respect to x . We set c, d are these two solutions satisfying d − c >
0, where we willsee (3.9) has distinct real roots. Hence c + d = 2( − a + t ) /a and cd = 1. Substitutingthis into (3.8), we obtain one of the solutions t = − (1 − a + a ) + p (1 − a + a ) + 4 a a , (3.10)where (1 − a + a ) + 4 a > c and d . Now we see that the equation (3.9) has two distinct real rootsunder (3.10). However this can be easily shown from( − a + t ) − a > (1 − a + a ) + 4 a a . Finally, since 0 is a non-degenerate critical point of λ , neither the coefficients of uv inthe first component nor those of u , v in the second component vanish. Proof of Proposition . By Lemma 3.4, we may assume f is written as in (3.3). By theMorse lemma, there exists a coordinate change u = au + bv + O (2) , v = cu + dv + O (2)such that the first component of f in (3.3) written by the coordinate system ( u , v )is u v . Then we have ac = bd = 0, and we may assume b = c = 0, since the case a = d = 0 is similar. If the coordinate change reverses the orientation, we composewith ( u, v ) ( − u, v ). This implies that we may assume f is written as (cid:18) uv, a u a v a u a u v a uv a v O (3) (cid:19) ( a a = 0) . If the first component is − uv , then we take a π -rotation on the target. By the coordinatechange u = u + a u a − a u v a , v = v − a u v a + a v a ,f is written as (cid:18) u v , a u a v a u a v (cid:19) + ( O (4) , O (4)) . By setting u = | a /a | / u , v = | a /a | / v , and taking ( u, v ) ( v, − u ) and π -rotation on the target if a < a <
0, we see the assertion.We remark that the number of coefficients of the terms whose degrees are less thanor equal to 3 in the form (3.1) in Proposition 3.2 is 3. However, coefficients in the firstcomponent remain. When considering higher degree terms, it should be remarked thatthe form (3.2) in Lemma 3.3 is convenient, since its first component is just uv .8 .3 Geometric meaning of coefficients of the SO(2) -normal form
Let f : ( R , → ( R ,
0) at 0 be a sharksfin. In Proposition 3.1, we showed that theimages of two branch curves of S ( f ) are both (2 , a , a , a in the SO(2)-normal form (3.1) by geometries of these(2 , S ( f )by γ i ( i = 1 ,
2) and set ˆ γ i = f ◦ γ . By the proof of Proposition 3.1, the cuspidaldirections of ˆ γ i ( i = 1 ,
2) at 0 are linearly independent. Thus the angle between twocuspidal directions of ˆ γ i ( i = 1 ,
2) is a geometric invariant of f . On the other hand, let c : ( R , → ( R ,
0) at 0 be a 3 / cuspidal curvature of c at 0 is defined by κ cusp ( c ) = det( c ′′ (0) , c ′′′ (0)) | c ′′ (0) | / . The cuspidal curvature measures the wideness of the 3 / γ i ( i = 1 ,
2) have 3 / f is written as in (3.1). Then in the same notation and by the samearguments as in the proof of Proposition 3.1, we have a ± = ± t + ( a − a ) t / (4 a ) + O (3), and ˆ γ ′′± (0) = ( ± , a ) , ˆ γ ′′′± (0) = (cid:18) ± a − a )2 a , a ∓ a (cid:19) . Thus κ cusp (ˆ γ ± ) = 2( ± a + a )(4 + 4 a ) / , (3.11)and the angle θ γ between the two cuspidal directions of ˆ γ i ( i = 1 ,
2) iscos − (cid:18) | a − | a + 1 (cid:19) . (3.12)By (3.11), (3.12) and a >
0, together with Proposition 3.2, the cuspidal curvaturesof ˆ γ ± and the angle between the two cuspidal directions determines the sharksfin up tothree degrees, namely, κ cusp (ˆ γ ± ) and θ γ determines the R + × SO(2)-class of sharksfinsup to 3-jets. It is known that f ( S ( f )) determines the R -class of a sharksfin, since it is acritical normalization [3,26]. On the other hand, a deltoid is not a critical normalization,and we cannot find such invariants in terms of f ( S ( f )) = { } . Let f : ( R , → ( R ,
0) be a
Whitney umbrella or equivalently, a cross cap , namely, itis A -equivalent to the germ ( u, v ) ( u, v , uv ). A line generated by df ( T R ) is calledthe tangent line of f , and a plane P perpendicular to the tangent line is called the normal plane . Let π : R → P be the orthogonal projection. It is known that if f isa Whitney umbrella, then π ◦ f is a sharksfin or a deltoid, generically. More precisely,if f : ( R , → ( R ,
0) is a Whitney umbrella, then there exist a coordinate system( u, v ) and a rotation Φ on R such that f ( u, v ) = u, uv + c v , X i + j =2 d ij u i v j i ! j ! ! + (0 , O (4) , O (4)) ( c , d > , d ij ∈ R ) . (3.13)9ee [24] or [4]. A Whitney umbrella is said to be elliptic (respectively, hyperbolic ) if d > d < c , d ij ( i + j = 2 ,
3) in (3.13), namely, geometric information of the Whitney umbrella.
Theorem 3.5.
Let f be a Whitney umbrella written in the form (3.13) with d = 0 .Let π : R → R be the orthogonal projection ( X , X , X ) ( X , X ) . Then π ◦ f at is a sharksfin if and only if d > and d ˜ d + 3 δd ˜ d ˜ d + (3 d ˜ d − c ˜ d ) ˜ d + δ ( d − d c ) ˜ d = 0 (3.14) hold for both δ = 1 and δ = − , where d = ˜ d , d = ˜ d . On the other hand, π ◦ f at is a deltoid if and only if d < and d ˜ d − d ˜ d − c ˜ d = 0 or d ˜ d − d ˜ d − d c ˜ d = 0 , (3.15) where d = − ˜ d , d = ˜ d . We remark that the ± -ambiguity of ˜ d , ˜ d is included by the condition (3.14)holding for both δ = ± Proof.
Let λ be an identifier of singularities of π ◦ f . If d > d > λ = − ˜ d u + ˜ d v . In this case, we set η = η + = ( ˜ d , ˜ d ), η = η − =( − ˜ d , ˜ d ). Then we see η ± η ± ( π ◦ f )(0) = (cid:0) d ˜ d , d ˜ d ( d + ˜ d ˜ d ) (cid:1) ,η ± η ± η ± ( π ◦ f )(0) = (cid:0) c ˜ d , d ˜ d + 3 d ˜ d ˜ d + 3 d ˜ d ˜ d + d ˜ d (cid:1) . By a direct calculation, we have the assertion. One can obtain the case of d < η = η + = ( i ˜ d , ˜ d ), η = η − = ( − i ˜ d , ˜ d ),where i = √− π ◦ f is (we allow ( ε , ε ) = ( − , −
1) in (3.1)) uv, w | w | (cid:12)(cid:12)(cid:12)(cid:12) w w (cid:12)(cid:12)(cid:12)(cid:12) / u + w | w | (cid:12)(cid:12)(cid:12)(cid:12) w w (cid:12)(cid:12)(cid:12)(cid:12) / v + w u + w v ! + ( O (4) , O (4)) , (3.16)where w = − d − cot θ ) cos θ + ( − d + d d + d cot θ ) sin θ ) /d ,w = 2( d + cot θ − x )(cot θ cos θ + sin θ ) /d ,w = 2( − d + cot θ + x )(cot θ cos θ + sin θ ) /d , and θ is an angle satisfying thatcot θ = − d − d d + p d + (1 − d + d d ) d . One can obtain the coefficients w and w by following the procedure of the proof ofProposition 3.2. However the formula is quite complicated, and we do not state it here.10 .5 Planar motions As an application of the criteria, we give a concrete condition that a singular pointappearing on a planar motion is a sharksfin.Let S be the 1-dimensional torus S = R / π Z , and let SE (2) be the 3-dimensionalLie group SE (2) = R ⋊ SO(2) = (cid:26) ( a, A ) (cid:12)(cid:12)(cid:12) A = (cid:18) cos θ − sin θ sin θ cos θ (cid:19) , a ∈ R , θ ∈ S (cid:27) . We take a map-germ α : ( R , → SE (2), which is called a planar motion-germ with -degrees of freedom in the context here, see [6] for detail. Take a point ω ∈ R , and seta map-germ ev ω : ( SE (2) , ( a , A )) → R by ev ω ( a, A ) = Aω + a . Then the composition ev ω ◦ α : ( R , → R traces the point w by the action of α ( s ), and is called a trajectoryof ω by α : ev ω ◦ α ( s ) = A ( s ) ω + a ( s ) , where α ( s ) = ( a ( s ) , A ( s )). In [6], a generic classification of singularities of ev ω ◦ α at0 is given when α at 0 is a Whitney umbrella, namely, it is A -equivalent to ( u, v ) ( u, v , uv ). Here, we consider a special case of the above motion. Let α : ( R , → SE (2)and β : ( R , → SE (2) be two curves. Then the composite motion of α and β is definedby ν ( u, v ) = β ( v ) α ( u ) : ( R , × ( R , → SE (2)where the product β ( v ) α ( u ) is that of SE (2). Composite motions are planar motionswith 2-degrees of freedom. In [6], a generic classification of singularities of ev ω ◦ ν at 0is given, where a sharksfin is in the classification. It is also shown that a deltoid neverappears on ev ω ◦ ν . We give a concrete condition for ev ω ◦ ν to be a sharksfin in termsof the geometry of ω and α, β , by using our criterion (Theorem A) when α : ( R , → ( SE (2) , (0 , E )) and β : ( R , → ( SE (2) , (0 , E )) have singular points at 0. We identifySO(2) with S and we set α ( u ) = ((˜ a ( u ) , ˜ a ( u )) , ˜ p ( u )) and β ( u ) = ((˜ b ( v ) , ˜ b ( v )) , ˜ q ( v )),where (˜ a ( u ) , ˜ a ( u )) , (˜ b ( v ) , ˜ b ( v )) ∈ R , ˜ p ( u ) , ˜ q ( v ) ∈ S . By the assumption, we canwrite α ( u ) = (( u a ( u ) , u a ( u )) , u p ( u )) and β ( v ) = (( v b ( v ) , v b ( v )) , v q ( v )). Theorem 3.6.
Under the above notation, is a corank singular point of f = ev ω ◦ ν .Furthermore, ev ω ◦ ν at is a sharksfin if and only if det w − w p (0) a (0) a (0) q (0) b (0) b (0) = 0 , (3.17)det w − w p (0) a (0) a (0) p ′ (0) a ′ (0) a ′ (0) = 0 , det w − w q (0) b (0) b (0) q ′ (0) b ′ (0) b ′ (0) = 0 . (3.18)We remark that the curve α = ( u a ( u ) , u a ( u )) : ( R , → ( R ,
0) is a 3 / (cid:18) a (0) a (0) a ′ (0) a ′ (0) (cid:19) = 0 . roof. By definition, f = (cid:16) v b ( v ) + u a ( u ) cos( v q ( v )) + w cos( u p ( u ) + v q ( v )) − u a ( u ) sin( v q ( v )) − w sin( u p ( u ) + v q ( v )) ,v b ( v ) + u a ( u ) cos( v q ( v )) + w cos( u p ( u ) + v q ( v ))+ u a ( u ) sin( v q ( v )) + w sin( u p ( u ) + v q ( v )) (cid:17) . This shows the first assertion. By a direct calculation, the determinant of the Jacobimatrix of f has the factor uv . Thus the Hesse matrix of det J f at (0 ,
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2, Q. J. Math. (2015), no. 1, 369–391.(Y. Kabata)School of Information and Data Sciences,Nagasaki University,Bunkyocho 1-14, Nagasaki, 852-8131, Japan [email protected] (K. Saji)Department of Mathematics,Graduate School of Science,Kobe University,Rokkodai 1-1, Nada, Kobe, 657-8501, Japan [email protected]@math.kobe-u.ac.jp