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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 • I A GENERALIZATION OF THE FIRST PLUCKER FORMULA BY George William Sparling A Thesis submitted in partial fulfilment of the requirements for the degree of MASTER OF ARTS In the Department of MATHEMATICS THE UNIVERSITY OF BRITISH COLUMBIA April, 1950 Abstract of A GENERALIZATION OF THE FIRST PLUCKER FORMULA by G. W. Spar l ing . The f i r s t Pliioker formula from algebraic geometry gives the c la s s of an algebraic ourve i n terms of the order and the s i n g u l a r i t i e s of the curve. Here a study i s made of r e a l , d i f f e ren t i ab le curves with a view to f ind ing the corres-ponding r e s u l t for such curves . The class of a point P with respect to a r e a l , d i f f erent iable curve C i s defined to be the number of tangents o f C which pass through P. F i r s t i t i s shown how the class of 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 la s s of P depends on the nature, numbers, and r e l a t i v e pos i t ions 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 su 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 olass three. I t i s found that a curve of class three must conta in one of the fol lowing three 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 po int . (2) One cusp and one double tangent. (3) Three cusps. 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 in terms of it s order and i t s singularities. In this thesis an attempt i s made to find the corresponding result for a real, differ-entiable curve. In the f i r s t three chapters a method i s dev-eloped which may be used i n studying the effects of singu-l a r i t i e s on the class of suoh a curve. In the last chapter this method is used to classify differentiable, closed curves of class 3. 1.2 The projective plane. The space considered in this thesis is the projective plane. The chief properties of the projective plane are: (i) Every pair of lines i n the projective plane has : a point in common. (i i ) Lines in the projective plane are closed. ( i i i ) The projective plane i s locally affine. This means that any fi n i t e region of the projective plane has the properties of the affine plane. 1.3 Curves. A ourve i s defined to be a Mngle-valued continuous mapping of the projective line. Such a mapping may be interpreted in two ways. In the f i r s t interpretation the points of the projective line are mapped into the points of the curve. In the other interpretation the points of the s projective line are mapped into the tangents of the curve; Each of these curves is 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 line. 1.5 Differentiability.Tangent. Let a secant a intersect a curve C(s) in points C(sl)^ and C(s2), where s is the ourve parameter. Let s2 approach s i . If as s2 approaches s i , a approaches a limit which is independent of the manner in which s2 approaches s i , then C(s) i s differentiable at C(sl). The.ilimit approached by a i s the tangent to C(s) at C(sl). Throughout this thesis i t is assumed that curves are of fi n i t e order and di f f erentiable at every point. 1.6 The principle of duality. It oan be proved that any proposition concerning lines and points which has been proved for differmentiable curves, holds in the dual. That i s f the proposition holds i f the roles of lines and points are interchanged. 1.7 Dual differentiability. Scherk 1 has proved that i f a curve is d i f f erentiable i t s dual i s di f f erentiable. The dual of the definition given in 1.5 i s : Let A be the point common to two tangents t(sl) 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 letters to which they are affixed. 1 P. Scherk, Czechoslovakian Journal of Mathematics and Physics, Prague, 1936. 3 t(s2) of the ourve whose tangent is t(s). Let s2 approach s i . If as s2 approaches s i , A approaches a limit which i s inde-pendent of the mariner in which s2 approaches s i , then the ourve is differentiable in the dual sense. The limit ap-proached by A i s the point of contact of t ( s l ) . If a ourve C is differentiable (by 1.5) then* by the principle of duality, the dual curve of C i s dif ferentiable in the dual sense. But, by Scherk's \result, the dual curve i s differentiable (by 1.5); so i t s dual, the curve C, i s d i f f e r -entiable in the dual sense. In other words, i f a curve is differentiable i t is differentiable in both senses. 1.8 Elementary Arc. An elemntary aro is an open part of a curve which has at most two points in common with a straight line. 2 It has been shown by Hjelmslev that a curve of finite order is made up of a fini t e number of elementary arcs. 1.9 Lines of support and lines of intersection. A line h which has a point P in common with a curve C is a line 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 line of support. Scherk"L showed that i f one non-tangent of a curve C at a point P i s a line of intersection at P, then a l l non-2, J. Hjelmslev, Om Grundlaget for laeren om simple Kurver, Nyt Tidsk.f. Math., 1907. 4 tangents at P are lines of intersection and i f one non-tangent at P i s a line of support at P, then a l l non-tangents at P are lines of support. 1.10 Characteristic The characteristic 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 lines of intersection or lines of support. (b) The second number i s one or two chosen so that: (i) If the tangent at P is a line of intersection then the sum of the numbers is odd. ( i i ) If the tangent at P i s a line of support then the sum of the numbers is even. 1.11 Classification of curve points. The classification of curve points i s given in the f i r s t two columns of the following table. It can be shown that the dual character-i s t i c s are those which are given in the third column. POINT CHAEACTERISTIC DUAL CHARACTERISTIC DUAL POINT ordinary point (1, 1) (1, 1) ordinary point cusp (2, 1) (1, 2) inflection point inflection point (1, 2) (2, 1) cusp b i l l cusp (2, 2) (2, 2) b i l l cusp 5 It can also be shown that every interior point of an elementary arc i s an ordinary point of the curve of which the arc is a pairt. 1.12 Nodes and double tangents. If in the mapping defined in 1.3, two different points of the projective line are mapped onto the same point of the curve then the point is called a node. The dual of a node is a double tangent.. A double tangent occurs when two different points of the projective line are mapped into the same tangent of the curve. 1.13 Singularities. Cusps, inflection points, b i l l cusps, nodes, and double tangents w i l l be referred to as singular-i t i e s , 1.14 C r i t i c a l tangents. Tangents at inflection 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 is 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 II Given an algebraio curve Ca, the olass with respect to Ca of points in the plane:is oonstant, depending only on the singularities of Ca. In other words the same number of tangents to Ca, real or Imaginary, pass through every point in the plane. For a real, differentiable ourve C,however, the class with respect to C of a point i n the plane depends not only on the singularities of C but also on the location of the point. In this chapter i t w i l l be shown that for 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 in a given region have the same class with respect to C. 2.1 Theorem. On any point on any interior tangent of an elementary aro there exist lines which contain two points of the arc. Proof. Let A be an interior 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 lines through B which contain two points of E. If B^A the line through B and any other point of E is a line 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 is a line of support and there exists a neighborhood N of A such 8 6 A that a l l points of E in N l i e on one side of a. Let a moving point P trace E. E i s continuous so as P approaches A, P intersects a l l lines of the pencil on B which l i e sufficiently 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 lines of the penoil on B» Thus there exist lines 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 interior point of E. Proof. Suppose the tangent a at an end-point A of E contains an interior point P of S. Since E is continuous and differ-entiable there exists a tangent a1 near a which contains a point P on E and in 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 exists a line h on P* which contains two points of the part of E which l i e s in N. Then h contains three points of E. By the definition of an elementary arc 9 this is impossible. Therefore an end-tangent of E cannot contain an interior point of E. According to the definition (1.8) of an elementary-arc, an end-tangent a of an elementary arc E may contain both end-points of E. In the remainder of this 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. If P is any point on an elementary arc E then the class of P with respect to E is one. Proof. E i s differentiable 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. By Theorem 2.2, A is an interior point of E. Let N be: a neighborhood of A which does not contain P. By Theorem 2.1, there exists a line h through P which contains two points of that part of E which l i e s i n N. Then h contains three points of E. By the definition of an elementary arc this i s impossible. Therefore exactly one tangent of E passes through P. Therefore the class of P with respect to E is one. 2 .4 Theorem. On any line through any interior point of an elementary arc there exist points which contain two tangents to the arc. Proof. This is 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 is an interior point. Let N be a neighborhood of S3 which does not contain the end-points of E. As SI and S2 approach S3 along that part of E within N they will be the end-points of an elementary arc E'which i s contained in E. Since s contains only the end points of E', E'will be entirely on one side of s. The remaining part of E in 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 in N l i e s entirely on one side of s. Hence s approaches the tangent at S3. The dual of this lemma i s : Suppose S i s the point common to two tangents s i and s2 of a curve C. Let s3 be the tangent at an ordinary point of C. If 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 let h be a line which intersects C in an ordinary point A. Let a rolling tangent t r o l l along C. As t r o l l s through A the intersection of t with h reverses. Proof. Let E be any elementary arc of C of which A is an interior point. Let P be a point on h which has class 2 with respect to E. Let t(sl) and t(s2), where s is the curve 11 parameter, be the two tangents to E which meet in P. E may be chosen ar 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) for which the intersection of t(s) with h reverses. Let P approach the reversal point. Then si.ap-proaches s3 and s2 approaches s3 and both t(sl) and t(s2) approach t(s3). But from the dual of Lemma 2.5A, as t( s l ) and t(s2) approach a common tangent their intersection approaches a curve point. Therefore the point of reversal of the intersection of t with h is a curve point, namely: A. This proves the theorem. 2.6 Theorem. Let A be an ordinary or an inflection point of a curve C. Let h be any line on A other than the tangent ~at A, and let B. be any point on h other than A. Then, for a sufficiently small neighborhood N of A, any line h» of the pencil on B which is sufficiently close to h intersects the part of C contained in N exactly once. Proof. Let a point P trace C. Since h i s a line of inter-section at A, P w i l l cross lines on both sides of h. If N i s sufficiently small, h has no point other than A in common with the part of C contained in N; hence, P w i l l cross, once, lines of the pencil on B which are sufficiently 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 in a sufficiently small neighborhood of A. Let h be amy line on A other than the tangent at A and let B be amy point on h other than A. Then a line h* of the pencil on B, which is sufficiently 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 is a line of support at A, Na l i e s entirely 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 lines of the pencil on B which are sufficiently close to h and on one side of h. As P recedes from A, P w i l l cross, once more, the same lines of the pencil on B. This proves the theorem. 2.8 Theorem. Let a be tangent to a ourffe C at an ordinary point or a cusp. Let Na be the part of C in a sufficiently 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 let b be any line on H other than a. Then any point H* on b which i s sufficiently close to H has class one with respect to. Na. Proof. This is the dual statement of Theorem 2.6 and i s therefore true by the principle 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 in a sufficiently small neighborhood of the point of oontact of a. Let H be any point on a other 1 3 than i t s point of oontact and let b be any line on H other than a. Then a point H* on b, which is sufficiently close to H, has olass two or zero with respect to Na depending on the side of H on which Hf l i e s . Proof. This i s the dual statement of 2.7 and i s therefore true by the principle of duality. DIAGRAMS ILLUSTRATING THEOREMS 2.8 AND 2.9 14 In the following corollaries a is a tangent to a curve C, H is any point on a other than i t s point of contact, and Na is the part of C in a suitably chosen neighborhood of the point of contact of a. 2.10 Corollary. If a is a non-critical tangent there exists a neighborhood of H in which every point has class one with respect to Na, This follows from 2.8. 2.11 Corollary. If a i s a c r i t i c a l tangent, then in 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 2.9. 2.12 Corollary. If a moving point P crosses a at H then i f a is a non-critical 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 two. This follows from 2.8 and 2.9. 2,15 Theorem. If a fixed line h intersects a curve C in 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 inflection 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 is an interior point of an elementary arc E. Let h be any line through A other than the tangent at A. Let a tangent t r o l l over E and let I be the intersection 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 is a point on E the class of A with respect to E i s one, so I does not return to A. Thus no point on h i s covered by I more than twice. It is also seen that there exists a neighborhood N of A such that points of h which l i e in 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 dis-cussion leads to the two following theorems: 2.14 Theorem. If E is an elementary arc and i f P i s any point in the plane then the class of P with reapect to E Is not greater than two. Proof. If 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. Let h be the line oommon to P and an interior point of 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; If a moving point P crosses an elementary arc E at an interior 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 line h. 16 From the discussion preoeeding 2.14, points of h near E aid on one side of S have class two with respect to E while points of h near E and on the other side of E have class zero with res-pect to E. Therefore as P moves allong h across E i t s class with respect to E ohanges by two. 2.16 Theorem. If 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 or the other end-tangent of E. Let h be any line through H other than b. Let a tangent t start at b and r o l l over E and l e t I be the point of intersection of t with h. As t ro l l s over E, I moves off from H covering points of h near H and on one side of b. I may or may hot return to H. If 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 of h near b and on the other side of b have cl_ass. zero with respect to I, Suppose I does return once to 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 of 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 of b have class one. 17 I pannbt return to 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 one. 2.17 Theorem. Let IS be an elementary aro. Let P be a point, which 4s not contained i n E or in either of the end-tangents of E. If the class of P with respect to § is o (c-0. 1, or 2) then in 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». Let h be any line containing P. Let a tangent t r o l l over E and let I be the intersection of t and h. N contains no points of the end-tangents of E, so the end-points of I are not contained in N. I contains no point of E, so the reversal points of I, i f any exist, l i e outside of N. Therefore every time I covers a point of h in N i t must cover a l l points of h in N. Therefore since P has class; o with respect to E then a l l points of h in K have class c with respect to E. It follows that a l l points in 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 class with respe.ct to C. Proof. Suppose a point P moves about the plane. From 2.18, the only way in which P can change i t s class 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 is composed of a fi n i t e number of elementary arcs the class of P with respect to C is the sum of i t s classes with respect to the oomponent arcs of C. Hence the only ways in which the class of P with respect to 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 in 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 in the neigh-borhood 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 o r i t i a a l tangent. Therefore the ourve C and i t s c r i t i c a l tangents div-ide the plane into regions of equal class with respect to. c. 19 CHAPTER III At the outset of Chapter II i t was stated that for a differentiable curve C, the class of P varies with the loc-ation of P relative to 0. In Chapter II we established regions suoh that in a given region the class of P is 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 for finding the class of points in each region. 3,1 Theorem.; Let E be an elementary arc with an end-point A. Let, a be the tangent at A. Let h be say line through A other than a. Let N be a neighborhood of A which contains no point of the other erid-tangent of E, i f h contains a point B of E other than A choose N small enough to 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 of E on whioh they l i e . 20 Proof. If a point P moves about within N the only way in whioh i t can change i t s class with respeot to E i s by cross-ing either E or a. Suppose P moves within the half of N which li 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 half of N into two parts, one in 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 sufficient to show that h sub-divides the part in 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 in N. The point of intersection I of t with h starts 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 in 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 lie 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 class of P was either zero or two. Then, since the class of P cannot exceed two, the olass of P after crossing a must be one. This proves (a). Given a ourve C, we. wish to obtain a method for finding the class of points in eaoh of the regions defined in Chapter II. Suppose we know the class of some point S. Then we can find the class of any other point A i f we let a moving point P move from S to A and note the changes in 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 find the class of a point S. The method w i l l be to choose an ordinary point S on C, let a moving point P start at S, trace C, and return to S. At any point in the tracing we w i l l let k denote the class of P with respeot to the part of C traced up to that point. If we can keep count of 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. It w i l l be convenient to think of P as oontinually moving out of one elementary arc of C into the adjoining elementary arc. Changes in k w i l l be considered in three parts, 22 (a) the point being traced, (b) the elementary arc E just traced, and (c) that portion of C which preoeeds E. (a) The point being traced. We w i l l think of P as continually "pieking up" a tangent, namely: the tangent at the point being traoed, (b) The elementary arc E just traced. Let A denote the end-point of E. (i) Suppose P traces an ordinary point. Since an ordinary point has characteristic (1,1) then as P traces an ordinary point, P does not cross the tangent at A but P does cross any other line h through A so, by 3.1 (b), P "loses" a tangent to E. But P "picks up" the tangent at the point being traced. The net result i s that k does not change. 23 ( i i ) Suppose P traces a cusp (2,1) or an i n -flection point (1,2). In this case P crosses the tangent at A thus, by 3.1 (a), P "retains" one tangent to E. But P "pioks up", the tangent at the point being traced. The net result is that k increases by one. ( i i i ) Suppose P traces a b i l l cusp (2,2). In this case P does not cross either the tangent at A or any other iine h through A. Therefore, by 3.1 (o), P either "picks up" one tangent to E or "loses" one tangent to E. But P "pioks up" the tangent at the point being traced. The net result is that either k increases by two or k does not change. 24 ( 0 ) That portion of Q which preoeed E. Here Theorems 2.16 and 2.19 apply. 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, k increases or decreases by two. If P crosses the tangent at the starting point S, then the class of P with respect to the elementary arc of which S is an end point increases or decreases by one, depending on the position of cross-ing and the direction of crossing. This follows from 2.16. 3.2 Summary. If a point P starts at an ordinary point S and traces a ourve C and i f , at any point in the tracing, k is the class of P with respect to the part of C traced up to that point, then k changes according to the following: (1) If P traces a cusp or an inflection point, k increases by one. (2) If P traces a b i l l cusp, k either does not change or increases by two. (3) If P orosses the part of C which has been traced, k increases by two or decreases by two. (4) If 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 two. (5) I f P crosses the tangent at the starting 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, differentiable curves of olass three are considered. The methods of Chapters II and III are used to classify such curves. Throughout this chapter C=C(s) (- po < sj= oo_J denotes a closed differentiable curve of class three. t B t ( s ) denotes the tangent to C(s). Ns denotes the part of C in any suitably-chosen; neighborhood of the point C(s). 4.2 Theorem. A c r i t i c a l tangent to C contains no ordinary points of C. Proof. Suppose t(sl) is a c r i t i c a l tangent to G and suppose t(sl) contains an ordinary point C(s2) of C. By:Theorem 2.4, there exists a point P on t(sl) such that the class of P with respect to Ns2 is 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» in M such that the class of P* with respect to Nsl is two. Then the class of Pf with respect to G is at least four. This i s im-possible since C i s a curve of class three. Therefore a c r i t i c a l tangent to c contains no ordinary point of C. 26 4 * 3 Theorem. The class of an ordinary point on C oannot exoeed 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). By 4.2, these are non-c r i t i c a l tangents. By 2 ,10 there exists a neighborhood M of C(sl) in which a l l points have olass one with respect to each of Ns2 and Ns3. But, by Theorem 2 .15 , M contains a point P whose olass with respect to Nsl is two. Then P has class at least four with respect to C, This i s impossible since C i s a ourve of olass 3. Therefore the class of an ordinary point on C cannot exoeed two. 4 . 4 Theorem. Suppose a moving point P traces the ourve C. Let C(0) be the starting point and let k(sl) denote the class of the point C(sl) with respect to 1nat part of C(s) for which O ^ s ^ s l . 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, from 4.3, k cannot exoeed two. Therefore k is always either one or two and k oannot change by more than one. 4.5 Corollary. It follows from 4 i 4 and 3.2 that the only ways in which k oan change are: (i) If P traces an inflection point k increases by one. ( i i ) If P traces a cusp k increases by one. ( i i i ) If P crosses t ( 0 ) either k increases by one or k decreases by one. 27 4.6 Corollary. If C(sl) is a ousp or an inflection 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 Theorem. If C(sl) is an inflection point or a cusp and C(a3) i s an inflection point or a cusp, then for some value s2 where sl<s2<s3, C(s2) l i e s on t(0). Proof. By 4.6, k(s)=2 for s> s i and |s-sll<£. Also by 4.6, k(s)^l for s<s3 and Js-s3l<£. Hence there exists some point C(s2) (sl<s2<s3) 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, is defined to be a si-nglendued, continuous mapping of the projective line onto the projective line 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(sl) where t(sl) contains no singularities of C. Let the tangent t r o l l once over C. Then the intersection of t with t(sl) 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^sl, let I(sl)«C(sl). I(s) defines &. single"-valued mapping of the points of C onto the 28 line t ( s l ) # I(s) is continuous for afsl because of the dif-ferentiability of C and I(s) is also continuous for s^sl since, by the dual diff e r e n t i a b i l i t y of C, 11m I(s)»C(sl.). Further, no point on t(sl) is 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(sl) which satisfies the con-dition for a K2. Therefore I(s) has at most two reversals. This proves the theorem. 4.10 Theorem. The only singularities that can oocur in C are cusps, inflection points and double tangents. Proof. It follows from 4.4 that C can contain no nodes or b i l l cusps since in the tracing of either of these points k would change by two. The only singularities that remain are cusps, inflection points, and double tangents. 4.11 Theorem. C has at least one cusp and, in any case C has an odd number of cusps. Proof. Consider the dual problem. Let t{a) be the tangent to a closed curve C(s) of order three. Let (3(0) be an ordinary point. Let I(s) be the intersection of t with a fixed line h which does not pass through any singularities of C. Sinoe C is closed and con-tinuous, and T is continuous, then I generates on h a contin-uous olosed curve which has an even number of reversals. If a point I(sl) is a reversal point on h then, by 2.13, either ItslJ^Cfsl) or "C(sl) is an inflection point. (By the dual of 4,10 C* oontains no b i l l cusps). 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 to C* and h be V(odd). Let W be the number of inflection points on C. Then from the foregoing W -f- V(odd and at least one) = Uneven),, whence W is odd and at least one. Therefore C has. at least one inflection point and in any case an odd number of infl e c t -ion points. This, dualized, gives the statement which was to be proved. 4.12 Theorem. If A is a cusp or an inflection point of a curve C, then in any neighborhood N of A there exist points which have class three with respect to that part of C which lies in N. Proof. Let E l and E2 be simple arcs i n N which are joined at A. We know from 3.2 that there exists a ourve point P on E l which contains a tangent t of E2. By 2.15, a point PT on t which l i e s sufficiently 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 in N. The dual of this theorem i s : If a is the tangent at an inflection point or a cusp of a ourve C, then there exist lines in a neighborhood,N of a which intersect, three times, that part of C which lies in N. 4.13 Theorem. C cannot have one cusp alone without any 30 other singularity. Proof. Consider the dual problem. Suppose C is a closed curve of order three with one inflection point and no other singularity. Let the inflection point he 0(0) and, with this point as starting point, let a point P trace C. P cannot, cross "t(O) heoause i f P crossed t"(0) in a point C(si) then, by the dual of 4J.2, a line in the neighborhood of T(0) would cut (3 four times, three times in the neighborhood of (3(0) and once in 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-ularities 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 inflection point alone without any other singularities. The dual of the result i s : A closed curve C of class three cannot have one ousp alone without any other singularity. 4.14 Theorem. C oannot have: (a) An inflection point and more than one cusp, or (b) More than one inflection point, or (c) More than three cusps. Proof. Let h be the tangent at an ordinary point of C which contains no singularities of C. Let a tangent t r o l l o v -er C. Suppose C contains a total of n cusps and inflection points. Take the point of contact of h as the starting point 31 and let P trace C. Then, by 4.7, C wl 11 cross h n-1 times. By 2.13 ( i i i ) , each of the n-1 crossing points causes a re-versal of the intersection of t with h. By 2.13 (i),each inflection point causes an additional reversal so, i f m is the number of inflection points on C, the number of reversals may be expressed (n-l)-f-m. By 4.9, this must not exceed two. That i s we must have (n-l)-r-m*s2. 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 tr i p l e tangent since a point of contact of a triple tangent would be a ourve point with class three with respect to 0. This is impossible by 4.3. 4.17 Theorem. If C has a double tangent then Q has at most one cusp. Proof. Suppose 0 has a double tangent and more than one cusp. Taking one of the points of contact of the double tangent as C(0), let P trace C. Then by 4.7, P must cross t(0) at some point say C(sl), Then G(sl) has class three with respect to C. This is impossible by 4.3. Therefore the theorem i s true. 4.18 Theorem. C oannot have a double tangent and an in-flection point. Proof. Suppose G has a double tangent and an inflection point. By Theorem 4.2, the inflection tangent oannot coincide 32 with the double tangent. Let A be the point common to the double tangent and the inflection tangent. Let a point P move along the double tangent. The class of P w i l l be at least two. Therefore as P passes through A, by corollary 2.12, the olass of p changes either from two to four or from four to two. Both changes are impossible since C is a curve of class three. Therefore the theorem i s true. From the foregoing we conclude that any ourve of olass three must contain one of the following three combina-tions of singularities: (a) One cusp and one inflection 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 principle we w i l l confirm the above cla s s i -fication of curves of class three. (a) One ousp and one infledtion point. Consider the third order curve XgXg^ x 3. (1) In the neighborhood of (0,0,1), (1) becomes y 2= x 3. (2) The curve (2) passes through (0,0) where i t has slope zero. 33 Then the tangent to (2) at (0,0) i s y=o. Inspection of (2) shows that (2) l i e s entirely on the right of x^ O and since x^O is not tangent to (2) at (0,0), then non-tangents to (2) at (0,0) are lines of support. It i s also seen that (2) is symmetrical with respeot to the tangent y=0 so the tangent at (0,0) i s a line of inter-section.; Then the characteristic of (0,0) i s (2,1) and (0,0) is a cusp. By 4.11, (1) also has an inflection point, so (1) is a curve of order three with one inflection point and one cusp. Its dual is a ourve of class three with one cusp and one inflection point. (b) One cusp and one double tangent. Consider the third order curve x2?3 = x i + xfx 3. (3) In the neighborhood of (0,0,1) this curve becomes y 2=x 2(x+l). (4) Differentiating, y L~ ±(3x4-2) . (5) 2v rx+T~ Inspection of (4) and (5) shows that (4) passes through the origin and, at the origin, the curve has two distinct taagents. The point (0,0) then, is a node. The curve (3) i s a curve of order three with one inflection point and one node. Its dual is 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 2=x(x s-M). (7) Differentiating, y'»; ±(5x2-h 1) . (8) 2v/x(x2-^ 1) By inspection we see that (7) is 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 line of intersection and the y-axis is a line of support. Equation (8) shows that the y-axis is the tangent to (7) at (0,0), therefore the characteristic 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 singularities (a), (b), and (c) given on Page 32. By duality, a curve of order three must contain one inflection point and one cusp, or one inflection point and one node, or three inflection points. The ourve (6) must have at least two singularities. Therefore (7) must have at least one singularity 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) is not a singularity. Therefore by symmetry any singularities in (7) must occur in pairs. But the only singularity that can occur twice in a curve of order three 35 is an inflection 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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