UBC Theses and Dissertations

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UBC Theses and Dissertations

A generalization of the first Plücker formula Sparling, George William 1950

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(fro f\t Cop  A GENERALIZATION OF THE FIRST PLUCKER FORMULA BY  George William Sparling  A Thesis submitted i n p a r t i a l f u l f i l m e n t of the requirements for the degree of MASTER OF ARTS In the Department of MATHEMATICS  THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1950  • I  Abstract of A GENERALIZATION OF THE FIRST PLUCKER FORMULA by G. W. S p a r l i n g .  The f i r s t Pliioker formula from a l g e b r a i c geometry gives the c l a s s of an a l g e b r a i c ourve i n terms o f the o r d e r and the s i n g u l a r i t i e s o f the c u r v e . Here a study i s made of r e a l , d i f f e r e n t i a b l e curves with a view t o f i n d i n g the c o r r e s ponding r e s u l t f o r such c u r v e s . The c l a s s of a p o i n t P with r e s p e c t to a r e a l , d i f f e r e n t i a b l e curve C i s d e f i n e d t o be the number of tangents o f C which pass through P. First i t is shown how the c l a s s o f P depends on i t s p o s i t i o n r e l a t i v e to C, then i t i s shown how the c l a s s o f P depends on the n a t u r e , numbers, and r e l a t i v e p o s i t i o n s of the s i n g u l a r i t i e s of C. In the l a s t Chapter the r e s u l t s are apjplied to. c l a s s i f y r e a l , d i f f e r e n t i a b l e curves of o l a s s t h r e e . It i s found that a curve o f c l a s s three must c o n t a i n one of the f o l l o w i n g t h r e e combinations of s i n g u l a r i t i e s : (1)  One cusp and one i n f l e c t i o n p o i n t .  (2)  One cusp and one double  (3)  Three c u s p s .  tangent.  A GENERALIZATION OF THE FIRST PLUCKER FORMULA CHAPTER I 1.1  Introduction. The f i r s t Plucker formula from algebraic  geometry expresses the class of an algebraic curve i n terms of i t s order and i t s s i n g u l a r i t i e s .  In this thesis an attempt  i s made t o f i n d the corresponding r e s u l t for a r e a l , entiable curve.  differ-  In the f i r s t three chapters a method i s dev-  eloped which may be used i n studying the e f f e c t s of singul a r i t i e s on the class of suoh a curve.  In the l a s t chapter  this method i s used to c l a s s i f y d i f f e r e n t i a b l e , closed curves of class 3. 1.2  The projective plane.  The space considered i n t h i s  thesis i s the projective plane.  The chief properties of the  projective plane are: (i)  Every pair of l i n e s i n the projective plane  has :  a point i n common. (ii)  Lines i n the projective plane are closed.  (iii)  The projective plane i s l o c a l l y a f f i n e .  This  means that any f i n i t e region of the projective plane has the properties of the a f f i n e plane. 1.3  Curves.  A ourve i s defined to be a  continuous mapping of the projective l i n e . may be interpreted i n two ways.  Mngle-valued Such a mapping  In the f i r s t  interpretation  the points o f the projective l i n e are mapped into the points of the curve.  In the other interpretation the points of the  s projective l i n e are mapped into the tangents of the curve; Each of these curves i s called the dual of the other. 1.4  Order. The order of a curve i s the greatest number  of points oommon to the curve and any straight l i n e . 1.5  Differentiability.Tangent. Let a secant a intersect  a curve C(s) i n points C ( s l ) ^ and C(s2), where s i s the ourve parameter. approaches  Let s2 approach s i .  I f as s2 approaches  si, a  a l i m i t which i s independent of the manner i n  which s2 approaches The.ilimit approached  s i , then C(s) i s d i f f e r e n t i a b l e at C ( s l ) . by a i s the tangent to C(s) at C ( s l ) .  Throughout this thesis i t i s assumed that curves are of f i n i t e order and d i f f erentiable at every point. 1.6  The p r i n c i p l e of duality.  I t oan be proved that any  proposition concerning l i n e s and points which has been proved for differmentiable curves, holds i n the dual.  That i s f  the proposition holds i f the roles of l i n e s and points are interchanged. 1.7  Dual d i f f e r e n t i a b i l i t y .  Scherk  1  has proved that i f a  curve i s d i f f erentiable i t s dual i s d i f f erentiable. The dual o f the d e f i n i t i o n given i n 1.5 i s : Let A be the point common to two tangents t ( s l ) and #  Due to typing d i f f i c u l t i e s , subscripts are placed on the same lines as the l e t t e r s t o which they are affixed.  1  P. Scherk, Czechoslovakian Journal of Mathematics and Physics, Prague, 1936.  3  t(s2) of the ourve whose tangent i s t ( s ) .  Let s2 approach s i .  If as s2 approaches s i , A approaches a l i m i t which i s independent of the mariner i n which s2 approaches s i , then the ourve i s d i f f e r e n t i a b l e i n the dual sense.  The l i m i t ap-  proached by A i s the point of contact o f t ( s l ) . If a ourve C i s differentiable  (by 1.5) then* by  the p r i n c i p l e of duality, the dual curve of C i s d i f ferentiable i n the dual sense. differentiable entiable  (by 1.5) so i t s dual, the curve C, i s d i f f e r ;  i n the dual sense.  differentiable 1.8  But, by Scherk's \result, the dual curve i s  In other words, i f a curve i s  i t i s differentiable i n both senses.  Elementary Arc.  An elemntary aro i s an open part of  a curve which has at most two points i n common with a straight line. 2 It has been shown by Hjelmslev  that a curve of  f i n i t e order i s made up of a f i n i t e number of elementary arcs. 1.9  Lines of support and l i n e s of intersection.  A line h  which has a point P i n common with a curve C i s a l i n e of intersection of C at P i f in. any neighborhood of P, however small, there exist points of C on both sides of h.  Otherwise  h i s a l i n e of support. Scherk" showed that i f one non-tangent of a curve C L  at a point P i s a l i n e of intersection 2,  at P, then a l l non-  J. Hjelmslev, Om Grundlaget for laeren om simple Kurver, Nyt Tidsk.f.  Math., 1907.  4 tangents at P are l i n e s of intersection and i f one non-tangent at P i s a l i n e of support at P, then a l l non-tangents at P are lines of support. 1.10  Characteristic  The c h a r a c t e r i s t i c of a point P i s  given by a pair of numbers determined as follows: (a)  The f i r s t number i s one or two according as  non-tangents at P are l i n e s of intersection or l i n e s of support. (b)  The second number i s one or two chosen so that: (i)  I f the tangent at P i s a l i n e of intersection  then the sum of the numbers i s odd. ( i i ) I f the tangent at P i s a l i n e of support then the sum of the numbers i s even. 1.11  C l a s s i f i c a t i o n of curve points.  The c l a s s i f i c a t i o n  of curve points i s given i n the f i r s t two columns of the following table.  I t can be shown that the dual character-  i s t i c s are those which are given i n the t h i r d column.  CHAEACTERISTIC  DUAL CHARACTERISTIC  DUAL POINT  ordinary point  (1, 1)  (1, 1)  ordinary point  cusp  (2,  1)  (1,  2)  i n f l e c t i o n point  i n f l e c t i o n point  (1, 2)  (2,  1)  cusp  b i l l cusp  (2,  (2, 2)  POINT  2)  b i l l cusp  5  It  can also be shown that every i n t e r i o r point of  an elementary arc i s an ordinary point of the curve of which the arc i s a pairt. 1.12  Nodes and double tangents.  I f i n the mapping defined  in 1.3, two different points of the projective l i n e are mapped onto the same point of the curve then the point i s c a l l e d a node.  The dual of a node i s a double tangent..  A double  tangent occurs when two d i f f e r e n t points of the projective line are mapped into the same tangent of the curve. 1.13  Singularities.  Cusps, i n f l e c t i o n points, b i l l cusps,  nodes, and double tangents w i l l be referred t o as singularities, 1.14  C r i t i c a l tangents.  Tangents at i n f l e c t i o n points  and b i l l cusps w i l l be called c r i t i c a l tangents. I.'l5  The class of a point.  The class of a point P with  respect to a curve C i s the number of tangents to C which pass through P. The tangent to a curve C at a point P on C w i l l be counted as one tangent through P.  i  6  Node  Double tangent  7  CHAPTER I I Given an algebraio curve Ca, the olass with respect to Ca of points i n the plane:is oonstant, depending only on the s i n g u l a r i t i e s of Ca.  In other words the same number of  tangents to Ca, r e a l or Imaginary, pass through every point i n the plane.  For a r e a l , d i f f e r e n t i a b l e ourve C,however,  the class with respect to C of a point i n the plane depends not only on the s i n g u l a r i t i e s of C but also on the l o c a t i o n of the point.  In this chapter i t w i l l be shown that f o r a  given ourve C, the curve C and i t s c r i t i c a l tangents divide the plane into regions such that a l l points i n a given region have the same class with respect to C. 2.1  Theorem.  On any point on any i n t e r i o r tangent of an  elementary aro there exist l i n e s which contain two points of the arc. Proof.  Let A be an i n t e r i o r point on an elementary arc E.  Let a be the tangent at A.  Let B be any point on a. The  problem i s to show that there exist l i n e s through B which contain two points of E. I f B^A the l i n e through B and any other point of E i s a l i n e through B which contains two points of E.  Therefore  the theorem i s true i f B*=A. Suppose BfA.  Since A i s an ordinary point, a i s a  line of support and there exists a neighborhood N of A such  8  A  6  that a l l points of E i n N l i e on one side of a. point P trace E.  Let a moving  E i s continuous so as P approaches A, P  intersects a l l lines of the p e n c i l on B which l i e s u f f i c i e n t l y near a and on the same side of a as E. As P recedes from A, P remains on the same side of a and so intersects" the same l i n e s of the penoil on B»  Thus there exist l i n e s through B  which oontain two points of E.  This proves the theorem  completely. 2.2  Theorem.  An end-tangent to an elementary arc E cannot  contain an i n t e r i o r point of E. Proof.  Suppose the tangent a at an end-point A of E contains  an i n t e r i o r point P of S.  Since E i s continuous and d i f f e r -  entiable there exists a tangent a near a which contains a 1  point P on E and i n a neighborhood of P.  Let N be a neighbor-  hood of the point of contact of a'which does not contain P . By Theorem 2.1  there e x i s t s a l i n e h on P* which contains two  points of the part of E which l i e s i n N. three points of E.  Then h contains  By the d e f i n i t i o n of an elementary arc  9 t h i s i s impossible.  Therefore an end-tangent of E cannot  contain an i n t e r i o r point of E. According to the d e f i n i t i o n (1.8) of an elementaryarc,  an end-tangent a of an elementary arc E may  both end-points of E.  contain  In the remainder of t h i s thesis i t  w i l l be convenient to ass.ume that an end-tangent to an elementary arc E does not contain both end-points of E. 2.3 Theorem.  I f P i s any point on an elementary arc E  then the class of P with respect to E i s one. Proof.  E i s d i f f e r e n t i a b l e at P; therefore there exists  at least one tangent of E which passes through P, namely: the  tangent at P i t s e l f . Suppose a second tangent a of.E passes through P.  Let A be the point of contact of a. i n t e r i o r point of E. not  contain P.  By Theorem 2.2, A i s an  Let N be: a neighborhood of A which does  By Theorem 2.1, there exists a l i n e h through  P which contains two points o f that part of E which l i e s i n N. Then h contains three points of E.  By the d e f i n i t i o n of an  elementary arc t h i s i s impossible.  Therefore exactly one  tangent of E passes through P.  Therefore the class of P  with respect to E i s one. 2.4 Theorem.  On any l i n e through any i n t e r i o r point of an  elementary arc there exist points which contain two tangents to the arc. Proof.  This i s the dual of Theorem 2.1 and i s therefore  true by the prlnoiple of duality.  10 2.5A -Lemma..  Suppose a seoant s contains two points SI  and S2 of a curve C.  Let S3 be an ordinary point of C.  If  SI and S2 both approach S3, along C then s approaches the tangent at S3. Proof.  Let E be a f i n i t e elementary arc of which S3 i s an  i n t e r i o r point.  Let N be a neighborhood  contain the end-points of E.  of S3 which does not  As SI and S2 approach S3 along  that part of E within N they w i l l be the end-points of an elementary arc E'which i s contained i n E.  Since s contains  only the end points of E', E ' w i l l be e n t i r e l y on one side of s. The remaining part of E i n N w i l l l i e on the side of s opposite to E \  As SI and S2 approach S3, E ' vanishes and the part of  E i n N l i e s e n t i r e l y on one side of s.  Hence s approaches  the tangent at S3. The dual of t h i s  lemma i s :  Suppose S i s the point common to two tangents s i and s2 of a curve C. point of C.  Let s3 be the tangent at an ordinary  I f s i and s2 both approach s3 along G then S  approaches the point of contact of s3. 2.5 Theorem.  Let C be a curve and l e t h be a line which  intersects C i n an ordinary point A. r o l l along C.  Let a r o l l i n g  tangent t  As t r o l l s through A the i n t e r s e c t i o n of t  with h reverses. Proof.  Let E be any elementary arc of C of which A i s an  i n t e r i o r point. respect to E.  Let P be a point on h which has class 2 with Let t ( s l ) and t(s2), where s i s the curve  11  parameter, be the two tangents t o E which meet i n P. E may be chosen a r b i t r a r i l y small so i t may be assumed, that; some points on h have class zero with respect to E. Then as t r o l l s along E, t does not intersect a l l points of h yet t intersects P twice.  Therefore there exists a value  s3 (sl<s3<s2) f o r which the i n t e r s e c t i o n of t(s) with h reverses. Let P approach the reversal point.  Then si.ap-  proaches s3 and s2 approaches s3 and both t ( s l ) and t(s2) approach t ( s 3 ) .  But from the dual of Lemma 2.5A, as t ( s l )  and t(s2) approach a common tangent their i n t e r s e c t i o n approaches a curve point.  Therefore the point of r e v e r s a l  of the i n t e r s e c t i o n of t with h i s a curve point, namely: A. This proves the theorem. 2.6  Theorem.  of a curve C.  Let A be an ordinary or an i n f l e c t i o n point  Let h be any l i n e on A other than the tangent  ~at A, and l e t B. be any point on h other than A.  Then, for a  s u f f i c i e n t l y small neighborhood N of A, any line h» of the pencil on B which i s s u f f i c i e n t l y close to h i n t e r s e c t s the part of C contained Proof.  i n N exactly once.  Let a point P trace C.  Since h i s a line of i n t e r -  section at A, P w i l l cross l i n e s on both sides of h.  If N i s  s u f f i c i e n t l y small, h has no point other than A i n common with the part of C contained  i n N; hence, P w i l l cross, once,  l i n e s of the p e n c i l on B which are s u f f i c i e n t l y close to h.  12 2.7  Theorem.  Let A be a cusp or a b i l l cusp on a curve C.  Let Na be the part of G i n a s u f f i c i e n t l y small neighborhood of A.  Let h be amy  l i n e on A other than the tangent at A and  l e t B be amy point on h other than A.  Then a l i n e h* of the  p e n c i l on B, which i s s u f f i c i e n t l y close to h, intersects Na twice or not at a l l depending on the side of h on which h* lies_ Proof.  Let a point P trace Na.  Since h i s a l i n e of  support at A, Na l i e s e n t i r e l y on one side of h so as P traces Na, P w i l l not cross h.  As P approaches A, since C  is continuous, P w i l l cross, once, a l l l i n e s of the p e n c i l on B which are s u f f i c i e n t l y close to h and on one side of h.  As  P recedes from A, P w i l l cross, once more, the same l i n e s of the p e n c i l on B. 2.8  Theorem.  point or a cusp.  This proves the theorem. Let a be tangent to a ourffe C at an  ordinary  Let Na be the part of C i n a s u f f i c i e n t l y  small neighborhood of the point of contact of a.  Let H be  any point on a other than i t s point of oontact and l e t b be any line on H other than a.  Then any point H* on b which i s  s u f f i c i e n t l y close to H has class one with respect to. Na. Proof.  This i s the dual statement of Theorem 2.6  and i s  therefore true by the p r i n c i p l e of duality. 2.9  Theorem.  Let a be a c r i t i c a l tangent to. a curve C.  Let Na be the part of G i n a s u f f i c i e n t l y small neighborhood of the point of oontact of a.  Let H be any point on a other  13  than i t s point of oontact and l e t b be any line on H other than a.  Then a point H* on b, which i s s u f f i c i e n t l y close to  H, has olass two or zero with respect to Na depending on the side of H on which H Proof.  f  lies.  This i s the dual statement of 2.7 and i s therefore  true by the p r i n c i p l e of d u a l i t y . DIAGRAMS ILLUSTRATING THEOREMS 2.8 AND 2.9  14 In the following c o r o l l a r i e s a i s a tangent to a curve C, H i s any point on a other than i t s point of contact, and Na i s the part of C i n a suitably chosen neighborhood of the point of contact of a. 2.10  Corollary.  I f a i s a n o n - c r i t i c a l tangent there  exists a neighborhood of H i n which every point has class one with respect to Na, This follows from 2.11  Corollary.  2.8.  I f a i s a c r i t i c a l tangent, then i n any  neighborhood of H, however small, there exist points with class zero with respect to Na and points with class two  with  respect to Na. This follows from 2.12  Corollary.  2.9.  I f a moving point P crosses a at H then  i f a i s a n o n - c r i t i c a l tangent th© class of P with respect to Na does not change but i f a i s a c r i t i c a l tangent the class of P with respect to Na changes by This follows from 2.8 2,15  Theorem.  and  two. 2.9.  I f a fixed l i n e h intersects a curve C i n  ordinary points only, then as a tangent t r o l l s along C, i t s intersection with h reverses i f and only i f : (i)  t r o l l s through an i n f l e c t i o n point,  or ( i i ) t r o l l s through a b i l l ousp. or ( i l l ) t r o l l s through a point common to C and h. Proof.  This theorem follows immediately from 2.5 and  2.12.  15 Suppose A i s an i n t e r i o r point of an elementary arc E.  Let h be any l i n e through A other than the tangent at A.  Let  a tangent t r o l l over E and l e t I be the i n t e r s e c t i o n of t  with h.  As t r o l l s over E, I moves along h to A, reverses at  A, and moves back along h.  I does not reverse again, by  theorem 2;. 13, and since A i s a point on E the class of A with respect to E i s one, so I does not return to A. on h i s covered by I more than twice.  Thus no point  I t i s also seen that  there exists a neighborhood N of A such that points of h which l i e i n N and on one side of E are covered twice whilepoints of h on the other side of E are not covered by I. This discussion leads to the two following theorems: 2.14  Theorem.  I f E i s an elementary  arc and i f P i s any  point i n the plane then the class of P with reapect to E Is not greater than two. Proof.  I f P l i e s on E then, by Theorem 2.3, the class of P  with respect to E i s one. Suppose P does not l i e on E. oommon to P and an i n t e r i o r point of E.  Let h be the l i n e From the foregoing  discussion, no point on h has class greater than two with respect to E.  Therefore the class of P with respect to E i s  not greater than two. 2.15  Theorem;  I f a moving point P crosses an elementary  arc E at an i n t e r i o r point then the class of P with respect to E changes by two. Proof.  Assume that as P crosses 1 i t moves along a l i n e h.  16 From the discussion preoeeding  2.14, points of h near E aid on  one side of S have c l a s s two with respect to E while points of h near E and on the other side of E have class zero with respect t o E.  Therefore as P moves allong h across E i t s class  with respect to E ohanges by two. 2.16  Theorem.  I f a moving point P crosses an end-tangent b  of an elementary arc E at any point other than the point of contact of b or the point common to b and the other end-tangent of E, than the class of P with respect to E changes by one. Proof.  Let H be any point on b which i s not on E o r the  other end-tangent of E. than b.  Let h be any l i n e through H other  Let a tangent t start at b and r o l l over E and l e t  I be the point of i n t e r s e c t i o n of t with h.  As t r o l l s over  E, I moves o f f from H covering points of h near H and on one side of b.  I may or may hot return t o H.  I f I does not return to H, then points of h near b and on one side of b have class one with respect to E, while points o f h near b and on the other side of b have cl_ass. zero with respect to I, Suppose I does return once t o H.  I w i l l not stop at  H since H i s not a point on the other end-tangent of E and, by Theorem 2.13, I w i l l not reverse at H.  Therefore I w i l l move  through H covering once a l l points of h near b and on both sides o f b.  So i f I returns once to H, points of h near b and  on one side of b have olass two with respect to E, while points of h near b and on the other side o f b have class one.  17 I pannbt return t o H a second time since the olass of H with respect to E cannot exceed  two.  It i s seen that points of h near b and on one side of b have class one greater than points of h on the other side of b.  Therefore i f a point P moves across b i t s class with  respect to E changes by 2.17  Theorem.  one.  Let IS be an elementary aro.  Let P be a point,  which 4s not contained i n E or i n either of the end-tangents of E.  I f the c l a s s of P with respect to § i s o (c-0. 1, or 2)  then i n any neighborhood of P, which contains no points of E or of the end-tangents of E, a l l points have class c with respect to E. Proof.  Let N be a neighborhood of P which contains no points  of E or of the end-tangents of i». P.  Let h be any l i n e containing  Let a tangent t r o l l over E and l e t I be the i n t e r s e c t i o n  of t and h.  N contains no points of the end-tangents of E, so  the end-points of I are not contained i n N.  I contains no  point of E, so the r e v e r s a l points of I, i f any exist, l i e outside of N.  Therefore every time I covers a point of h i n  N i t must cover a l l points of h i n N.  Therefore since P has  class; o with respect to E then a l l points of h i n K have class c with respect to E.  I t follows that a l l points i n N have  class c with respect to E , 2.18  Corollary.  A moving point P ohanges i t s class with  respect to an elementary arc E i f and only i f i t crosses E or an end-tangent of E.  18 2_19  Theorem..  A ourve C'and the c r i t i c a l tangents to. C  divide the plane into regions of uniform c l a s s with respe.ct to C. Proof.  Suppose a point P moves about the plane.  From 2.18,  the only way i n which P can change i t s c l a s s with respect to an elementary arc i s by crossing either the aro i t s e l f or one of i t s end-tangents. Since C i s composed o f a f i n i t e number of elementary arcs the c l a s s of P with respect to C i s the sum of i t s classes with respect to the oomponent arcs o f C.  Hence the only ways  i n which the class o f P with respect t o C can change are: (a)  P may cross C i t s e l f , thereby changing i t s class  with respect to the elementary arc i n the neighborhood of the crossing point. (b)  P may cross a tangent to c, thus ohanging i t s class  with respect to the two elementary arcs i n the neighborhood of the point of oontact.of the tangent.  It  follows from 2.12 that the class of P with respect to C changes only i f the tangent crossed i s a  oritiaal  tangent. Therefore the ourve C and i t s c r i t i c a l tangents divide  the plane into regions o f equal class with respect to. c.  19  CHAPTER I I I At the outset of Chapter I I i t was stated that f o r a d i f f e r e n t i a b l e curve C, the class o f P varies with the location of P r e l a t i v e to 0.  In Chapter I I we established regions  suoh that i n a given region the class of P i s constant and such that as P crosses from one region into an adjoining region i t s class increases or decreases by two.  In this part  we wish to obtain a method f o r finding the class of points i n each region. 3,1 A.  Theorem.;  Let E be an elementary arc with an end-point  Let, a be the tangent at A.  other than a.  Let h be say l i n e through A  Let N be a neighborhood of A which contains no  point o f the other erid-tangent of E,  i f h contains a point B  of E other than A choose N small enough t o exclude B.  Then a  and h divide N into four parts as follows: (a)  Two parts, those on the opposite side of a from E,  in whioh a l l points have olass one with respect to E. (b)  One part, on the same side of a as E but on the  opposite side of h,in which a l l points have class zero, with respect to E. (o)  One part, on the same side of a as E and on the  same side of h as E,in which points not on E have class zero or two with respect to E depending on the side o f E on whioh they l i e .  20  Proof.  I f a point P moves about within N the only way i n  whioh i t can change i t s class with respeot t o E i s by crossing  either E or a. Suppose P moves within the h a l f of N which l i e s on  the same side of a as E.  As P moves across E i t s class with  respect to E changes by two.  Therefore, except when P i s on  E i t s e l f , the olass of P with respect to JB can never be one. Therefore E divides this h a l f of N i n t o two parts, one i n which a l l points have olass two with respeot to E and one i n which a l l points have olass zero with respect to E.  To prove  (b) and (c) i t w i l l be s u f f i c i e n t to show that h sub-divides the part i n which a l l points have olass zero with respeot to E, Let  a tangent t start at a and r o l l over the part of  E which i s contained i n N.  The point of intersection I of t  with h s t a r t s at A and moves along h.  I w i l l not reverse, and  i t w i l l not return to A sinoe A has class one with respect to E.  Therefore no point on h i s covered twioe by I.  Therefore  21  h oontains no points of class two with respect to E and so the points of h which l i e i n N and on the same side of a as E have olass zero with respect to E.  This proves (b) and (c).  Suppose P moves across a into the half of N which l i e s on the side of a opposite to E.  By 2.16, the class of P  w i l l change by one.  But before crossing a the c l a s s of P was  either zero or two.  Then, since the class of P cannot exceed  two,  the olass of P a f t e r crossing a must be one. This proves  (a). Given a ourve C, we. wish to obtain a method f o r finding the class of points i n eaoh of the regions defined i n Chapter I I .  Suppose we know the class of some point S.  Then  we can f i n d the class of any other point A i f we l e t a moving point P move from S to A and note the changes i n the class of P as i t crosses C and the o r i t i c a l tangents of C.  The problem  is to f i n d the c l a s s of a point S. The method w i l l be to choose an ordinary point S on C, l e t a moving point P start at S, trace C, and return to S. At any point i n the tracing we w i l l l e t k denote the class of P with respeot  to the part of C traced up to that point.  If we can keep count o f k throughout the tracing then as P arrives back at S, k w i l l give us the class of S with respect to the complete ourve C. I t w i l l be convenient to think of P as oontinually moving out o f one elementary arc of C into the adjoining elementary arc.  Changes i n k w i l l be considered  i n three parts,  22 (a) the point being traced, (b) the elementary arc E just traced, and (a)  (c) that portion of C which preoeeds E. The point being traced. We w i l l think of P as  continually "pieking up" a tangent, namely: tangent at the point being (b)  the  traoed,  The elementary arc E just traced. Let A denote  the end-point of E. (i)  Suppose P traces an ordinary point.  an ordinary point has characteristic  Since  (1,1) then  as P traces an ordinary point, P does not cross the tangent at A but P does cross any other h through A so, by 3.1 to E.  line  (b), P "loses" a tangent  But P "picks up" the tangent at the point  being traced. change.  The net result i s that k does not  23 (ii)  Suppose P traces a cusp (2,1) or an i n -  f l e c t i o n point (1,2).  In t h i s case P crosses the  tangent at A thus, by 3.1 tangent to E.  But P "pioks up", the tangent at  the point being traced. k increases by  (a), P " r e t a i n s " one  The net result i s that  one.  ( i i i ) Suppose P traces a b i l l cusp (2,2).  In  t h i s case P does not cross either the tangent at A or any other i i n e h through A. 3.1  Therefore,  by  (o), P either "picks up" one tangent to E or  "loses" one tangent to E.  But P "pioks up"  tangent at the point being traced.  the  The net r e s u l t  i s that e i t h e r k increases by two or k does not change.  24 (0)  That portion of Q which preoeed E.  2.16  and 2.19  apply.  Here Theorems  That i s , i f P crosses a part of  the ourve which has been traced, or a c r i t i c a l tangent to that part of the curve, then by 2.19, or decreases by two.  k increases  I f P crosses the tangent at the  starting point S, then the class of P with respect to the elementary arc of which S i s an end point  increases  or decreases by one, depending on the p o s i t i o n of crossing  and the d i r e c t i o n of crossing.  This follows from  2.16. 3.2  Summary.  I f a point P starts at an ordinary point S  and traces a ourve C and i f , at any point i n the tracing, k is the class o f P with respect to the part of C traced up to that point, then k changes according to the following: (1)  I f P traces a cusp or an i n f l e c t i o n point, k  increases by (2)  one.  I f P traces a b i l l cusp, k either does not change  or increases by (3)  two.  I f P orosses the part of C which has been traced,  k increases by two or decreases by (4)  two.  I f P orosses a o r i t i c a l tangent of that part of C  which has been traced, k increases or decreases by (5)  two.  I f P crosses the tangent at the s t a r t i n g point,  k inoreases or decreases by  one.  25  CHAPTER IV 4.1  The olass of a curve.  The olass of a curve C i s given  by the greatest number of tangents to C which contain a common point. In this chapter closed, d i f f e r e n t i a b l e curves of olass three are considered.  The methods of Chapters I I and I I I  are used to c l a s s i f y such curves. Throughout t h i s chapter C=C(s) (- po < s j = oo_J denotes a closed d i f f e r e n t i a b l e curve of class three. denotes the tangent to C(s). suitably-chosen; 4.2  Theorem.  t t(s) B  Ns denotes the part of C i n any  neighborhood of the point C(s). A c r i t i c a l tangent to C contains no ordinary  points o f C. Proof.  Suppose t ( s l ) i s a c r i t i c a l tangent to G and suppose  t ( s l ) contains an ordinary point C(s2) of C.  By:Theorem 2.4,  there exists a point P on t ( s l ) such that the class of P with respect to Ns2 i s two.  By Theorem 2,17, there exists a  neighbrohood M of P wherein a l l points have class two with respeot to Ns2.  By 2.11, there exists a point P» i n M such  that the c l a s s of P* with respect to N s l i s two. class of P  f  with respect to G i s at least four.  possible since C i s a curve of class three.  Then the This i s im-  Therefore a  c r i t i c a l tangent to c contains no ordinary point of C.  26 4  *  3  Theorem.  exoeed  The class of an ordinary point on C oannot  two.  Proof.  Suppose an ordinary point C(sl) has class at Deast  three with respect to 0 ,  Let the three tangents containing  C(-sl) be t ( s l ) , t(s2), and t(s3). c r i t i c a l tangents. C(sl)  By 2,10  By 4.2, these are  non-  there exists a neighborhood M of  i n which a l l points have olass one with respect to each  of Ns2 and Ns3.  But, by Theorem 2 . 1 5 ,  whose olass with respect to Nsl i s two.  M contains a point P Then P has class at  least four with respect to C,  This i s impossible since C i s  a ourve of olass 3.  the class of an ordinary point  on C cannot exoeed 4.4 Let C(0)  Theorem.  Therefore two.  Suppose a moving point P traces the ourve C.  be the starting point and l e t k ( s l ) denote the class  of the point C(sl) with respect to 1nat part of C(s) f o r which O^s^sl .  As P traces C, k is aLways at least one since there  is a tangent at the point P i t s e l f and, exoeed two.  from 4.3, k cannot  Therefore k i s always e i t h e r one or two and k  oannot change by more than one. 4.5  Corollary.  I t follows from 4 i 4 and 3.2  that the only  ways i n which k oan change are: (i)  I f P traces an i n f l e c t i o n point k increases by  ( i i ) I f P traces a cusp k increases by ( i i i ) I f P crosses t ( 0 ) k decreases by  one.  one.  either k increases by one or  one.  27 4.6  Corollary.  I f C(sl) i s a ousp or an i n f l e c t i o n point  then  for s ^ s l and |s-sl|<e, kCsJ^l and for s > s l and |s-sl|<£, k(s)-2 since as P traces 0(81), k increases by one., 4.7 C(a3)  Theorem.  I f C(sl) i s an i n f l e c t i o n point or a cusp and  i s an i n f l e c t i o n point or a cusp, then f o r some value s2  where s l < s 2 < s 3 , C(s2) l i e s on t ( 0 ) . Proof.  By 4.6, k(s)=2 for s> s i and |s-sll<£.  k ( s ) ^ l for s < s 3 and Js-s3l<£. C(s2)  Also by 4.6,  Hence there exists some point  ( s l < s 2 < s 3 ) where k deoreases by one. By 4,5 the point  C(s2) must l i e on t ( 0 ) . 4.8  A K2.  A ourve of order two on a projective one-spaoe,  which i s denoted by the symbol K2, i s defined to be a si-nglendued, continuous mapping of the projective l i n e onto the projective l i n e where no point i s covered more than twice, A well-known property of auK2 i s that i t has at most two reversals. 4.9  Theorem.  Let C(sl) be an ordinary point on C with  tangent t ( s l ) where t ( s l ) contains no s i n g u l a r i t i e s of C. Let the tangent t r o l l once over C.  Then the i n t e r s e c t i o n of  t with t ( s l ) has at most two reversals. Proof.  Consider the following mapping:  let I(s) be the intersection of t(s) with t ( s l ) for s ^ s l , l e t I(sl)«C(sl). I(s) defines &. single"-valued mapping o f the points of C onto the  28 line t ( s l )  #  I(s) i s continuous for afsl because of the d i f -  f e r e n t i a b i l i t y of C and I(s) i s also continuous for s ^ s l since, by the dual d i f f e r e n t i a b i l i t y of C, 11m I(s)»C(sl.). no point on t ( s l )  Further,  i s covered more than twice since three coin-  cident values of I , say I(s2), I(s3) and I(s4),,would imply four concurrent tangents namely:  t ( s l ) , t(s2), t(s3), and t(s4).  Thus I(s) generates a ourve on t ( s l ) which s a t i s f i e s the cond i t i o n for a K2.  Therefore I(s) has at most two reversals.  This proves the theorem. 4.10 are  Theorem.  The only s i n g u l a r i t i e s that can oocur i n C  cusps, i n f l e c t i o n points and double tangents.  Proof.  I t follows from 4.4 that C can contain no nodes or  b i l l cusps since i n the tracing of either of these points k would change by two.  The only s i n g u l a r i t i e s that remain are  cusps, i n f l e c t i o n points, and double tangents. 4.11  Theorem.  C has at least one cusp and, i n any case C  has an odd number of cusps. Proof.  Consider the dual problem. Let  order three.  t{a)  be the tangent to a closed curve C(s) of  Let (3(0) be an ordinary point.  Let I(s) be the  intersection of t with a fixed l i n e h which does not pass through any s i n g u l a r i t i e s of C.  Sinoe C i s closed and con-  tinuous, and T i s continuous, then I generates on h a continuous olosed curve which has an even number of reversals. a point I ( s l )  i s a reversal point on h then, by 2.13,  I t s l J ^ C f s l ) or "C(sl) i s an i n f l e c t i o n point. 4,10 C* oontains no b i l l cusps).  If  either  (By the dual of  29  Let the number of reversals of I be U(even).  Sinoe C~ i s closed  and of odd order, the number of points common to "c and h i s odd and at least one. Let the number of points common t o C* and h be V(odd).  Let W be the number of i n f l e c t i o n points on C.  Then from the foregoing W -f- V(odd and at least one) = Uneven),, whence W i s odd and at least one.  Therefore C has. at least  one i n f l e c t i o n point and i n any case an odd number of i n f l e c t ion points.  This, dualized, gives the statement which was to  be proved. 4.12  Theorem.  I f A i s a cusp or an i n f l e c t i o n point of a  curve C, then i n any neighborhood N of A there exist points which have class three with respect to that part of C which l i e s i n N. Proof. at A.  Let E l and E2 be simple arcs i n N which are joined  We know from 3.2 that there e x i s t s a ourve point P on E l  which contains a tangent t of E2. By 2.15, a point P  T  on t  which l i e s s u f f i c i e n t l y near P w i l l have class two with respect to E l .  Then P» has class three with respect to that part of C  which l i e s i n N. The dual of this theorem i s : I f a i s the tangent at an i n f l e c t i o n point or a cusp of a ourve C, then there exist l i n e s i n a neighborhood,N of a which intersect, three times, 4.13  Theorem.  that part of C which l i e s i n N.  C cannot have one cusp alone without any  30 other s i n g u l a r i t y . Proof.  Consider the dual problem. Suppose C i s a closed curve of order three with one  i n f l e c t i o n point and no other s i n g u l a r i t y .  Let the i n f l e c t i o n  point he 0(0) and, with t h i s point as starting point, l e t a point P trace C.  P cannot, cross "t(O) heoause i f P crossed t"(0)  i n a point C(si) then, by the dual of 4J.2, a l i n e i n the neighborhood of T(0) would cut (3 four times, three times i n the neighborhood of (3(0) and once i n the neighborhood of (3(sl). Consider k (the class of P with respect to the part traced) as P approaches (3(0).  Since P has traced no sing-  u l a r i t i e s and has not orossed t"(0), then k"=l as P approaches C(O').  But from 3.2,  k must be at least two.  Therefore C  oannot have one i n f l e c t i o n point alone without any other singularities. The dual of the r e s u l t i s : A closed curve C of class three cannot have one ousp alone without any other 4.14  Theorem. (a)  singularity.  C oannot have:  An i n f l e c t i o n point and more than one  or (b)  More than one i n f l e c t i o n point,  or (c)  More than three cusps.  Proof.  Let h be the tangent at an ordinary point of C  which contains no s i n g u l a r i t i e s of C. er  C.  points.  cusp,  Let a tangent t r o l l o v -  Suppose C contains a t o t a l of n cusps and i n f l e c t i o n Take the point of contact of h as the s t a r t i n g point  31 and l e t P trace C. By 2.13  (iii),  Then, by 4.7, C wl 11 cross h n-1 times.  each of the n-1 crossing points causes a re-  versal of the i n t e r s e c t i o n of t with h.  By 2.13  (i),each  i n f l e c t i o n point causes an additional reversal so, i f m i s the number of i n f l e c t i o n points on C, the number of reversals may  be expressed (n-l)-f-m.  By 4.9, t h i s must not exceed  That i s we must have (n-l)-r-m*s2.  two.  By checking each of (a), (b),  and (c) with t h i s formula i t i s seen that the theorem i s true. 4.15  Theorem.  C has at most one double tangent because  the  point of intersection of two double tangents wauld have olass four with respect to C. 4.16  Theorem.  G cannot have a t r i p l e tangent since a point  of contact of a t r i p l e tangent would be a ourve point with class three with respect to 0. 4.17  Theorem.  This i s impossible by  4.3.  I f C has a double tangent then Q has at most  one cusp. Proof. cusp.  Suppose 0 has a double tangent and more than one  Taking one of the points of contact of the double tangent  as C(0), l e t P trace C. some point say C ( s l ) , to C.  Then by 4.7, P must cross t(0) at Then G(sl) has class three with respect  This i s impossible by 4.3.  Therefore the theorem i s  true. 4.18  Theorem.  C oannot have a double tangent and an i n -  f l e c t i o n point. Proof. point.  Suppose G has a double tangent and an i n f l e c t i o n By Theorem 4.2,  the i n f l e c t i o n tangent oannot coincide  32 with the double tangent.  Let A be the point common to the  double tangent and the i n f l e c t i o n tangent. move along the double tangent. least two. the  Let a point P  The class of P w i l l be at  Therefore as P passes through A, by c o r o l l a r y  2.12,  olass of p changes e i t h e r from two to four or from four to  two. three.  Both changes are impossible since C i s a curve of class Therefore the theorem i s true. From the foregoing we conclude that any ourve of  olass three must contain one of the following three combinations of s i n g u l a r i t i e s : (a)  One cusp and one i n f l e c t i o n point.  (b)  One cusp and one double tangent.  (c)  Three ousps.  It remains to show the existence of curves of class three for each of the types (a),'(b) and (c). This w i l l be done by examples. The method w i l l be to consider curves which, from algebraic geometry, are know to be of order three *  Then by  using-the duality p r i n c i p l e we w i l l confirm the above c l a s s i f i c a t i o n of curves o f class three. (a)  One ousp and one i n f l e d t i o n point.  Consider the t h i r d  order curve XgXg^ x . 3  (1)  In the neighborhood of (0,0,1), (1) becomes y= x . 2  3  The curve (2) passes through (0,0) where i t has slope zero.  (2)  33 Then the tangent to (2) at (0,0) i s  y=o. Inspection of (2) shows that (2) l i e s e n t i r e l y on the right of x^O and since x^O i s not tangent to (2) at (0,0), then non-tangents to (2) at (0,0) are l i n e s of support. It i s also seen that (2) i s symmetrical with respeot to the tangent y=0 so the tangent at (0,0) i s a l i n e of i n t e r section.;  Then the c h a r a c t e r i s t i c  is a cusp.  By 4.11,  of (0,0) i s (2,1) and  (0,0)  (1) also has an i n f l e c t i o n point, so (1)  is a curve of order three with one i n f l e c t i o n point and one cusp.  I t s dual i s a ourve of class three with one cusp and  one i n f l e c t i o n point. (b)  One cusp and one double tangent.  Consider the t h i r d  order curve x  2?3  xi + xfx .  =  3  (3)  In the neighborhood of (0,0,1) this curve becomes y =x (x+l).  (4)  y ~ ±(3x4-2) .  (5)  2  2  Differentiating, L  2v x+T~ r  Inspection of (4) and (5) shows that (4) passes through the o r i g i n and, at the origin, the curve has two d i s t i n c t taagents. The point (0,0) then, i s a node. The curve (3) i s a curve of order three with one i n f l e c t i o n point and one node.  I t s dual i s a curve of class  three with one cusp and one double tangent.  34 (o)  Three cusps.  Consider the third order curve XgXg-XiUf+Xg).  (6)  In the neighborhood of (0,0,1) the curve becomes y =x(x -M).  (7)  y'»; ±(5x -h 1) .  (8)  2  s  Differentiating, 2  2v/x(x2-^ 1) By inspection we see that (7) i s symmetrical with respect to the x-axis,- that (7) has no points on the l e f t of the y-axis, and that (7) cuts the x-axis at the point (0,0) only.  Prom this i t follows that, at the point (0,0), the x-axis  i s a l i n e of i n t e r s e c t i o n and the y-axis i s a line of support. Equation  (8) shows that the y-axis i s the tangent to (7) at  (0,0), therefore the c h a r a c t e r i s t i c of the point (0,0) i s (1,1) so (0,0) i s an ordinary  pOint.  We have seen that a curve of class three must contain one of the three combinations of s i n g u l a r i t i e s (a), (b), and (c) given on Page 32.  By duality, a curve of order three must  contain one i n f l e c t i o n point and one cusp, or one i n f l e c t i o n point and one node, or three i n f l e c t i o n points. The ourve (6) must have at l e a s t two s i n g u l a r i t i e s . Therefore  (7) must have at least one s i n g u l a r i t y since it  contains a l l the points of (6) except the point at i n f i n i t y . The point (0,0) i s not a s i n g u l a r i t y . Therefore by symmetry any s i n g u l a r i t i e s i n (7) must occur i n pairs.  But the only  s i n g u l a r i t y that can occur twice i n a curve of order  three  35 i s an i n f l e c t i o n point.  Therefore (7) has three  inflection  points and i t s dual curve w i l l be a ourve of class three with three cuspsi  

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