Evolute and involute (evolvent)

Transcription

1 Evolue and involue (evolven Ineracive soluions o problem s of differenial geomery S.N Nosulya, V.V Shelomovskii. Topical ses for geomery, 1. D.V. Shelomovskii. GInMA, 1. hp:// Maerials of he ineracive se can be used in Undergraduae educaion for he sudy of he foundaions of differenial geomery, i can be used for individual sudy mehods of differenial geomery. All figures are ineracive if you insall GInMA program from hp:// Concepion of Evolue In he differenial geomery of curves, he evolue of a curve is he locus of all is ceners of curvaure. Equivalenly, i is he envelope of he normals o a curve. Apollonius (c. BC discussed evolues in Book V of his Conics. However, Huygens is someimes credied wih being he firs o sudy hem (1673. Equaion of evolue Le γ be a plane curve conaining poins X T =(, y. The uni normal vecor o he curve is n. The curvaure of γ is K. The cener of curvaure is he cener of he osculaing circle. I lies on he normal 1 line hrough γ a a disance of from γ in he direcion deermined by he sign of K. In symbols, K he cener of curvaure lies a he poin ζ T =(ξ,η. As X varies, he cener of curvaure races ou a plane curve, he evolue of γ and evolue equaion is: ζ = X + n K. (1 Case 1 Le γ is given a general parameerizaion, say he evolue can be epressed in erms of he curvaure ( y', ' curve is n=. Evolue equaion is: ' + y' X T =((, y( Then he parameric equaion of ' ' ' y ' K =. The uni normal vecor o he ( ' + y' 3/ ζ =( y' ( ' + y ' ' y ' ' ' ' y' ' ( ' + y ' ' ' ' y '. ( Case Le γ is given an equaion y= f (. Then he curvaure K =, he uni normal (1+ y ' 3/ ( y',1 vecor o he curve is n= 1+ y '. Evolue equaion is: ζ =( y' (1+ y ', y + 1+ y ' y ' ' y ' '. (3 Case 3 Le γ is given an equaion f (, y=. Then he curvaure K = f f y f y f f yy f y f. ( f + f y 3/ The uni normal vecor o he curve is n= ( f, f y. Evolue equaion is: f + f y f y ( f + f y f f y f y f f yy f y f f f y f y f f yy f y f. ( ζ =( f ( f + f y Case Le γ is given an equaion R= f (s, where s длина дуги, отсчитываемая от некоторой точки. Le evolue equaion is: R= f ( s. Обозначим углы, составляемые касательными с 1

2 d α d s = 1 R, d α d s = 1 R, осью абсцисс α и α. Then d s=d R,α= α + π, d α=d α. Evolue equaion is: R= R d R, s=r+c. (5 d s Тypical figure and i's using In each figure poins and 1 on he -ais specify he Caresian coordinaes. The graph of he original curve is shown by he blue line. I is deermined by he parameers which values are indicaed in he figure. The parameers are se by he acive poins. The evolue is shown in red. The rial poins labeled С, D, E,... are locaed on he curve. Tangen circle and is cener poin C ', D ',..., belonging o evolue are shown in pink. The process of learning and invesigaion one can begin from he verificaion of he basic properiy of he evolue. By moving poin C, make sure ha he cener of curvaure belongs o he evolue a all posisions of poin C,. By acivaing he "Properies" buon find and check he equaions used for he curves consrucion. Eplore several evolue curves. Save he file in a convenien locaion and change he original curve. Perform calculaions and consruc evolue of he seleced curve. Build a angen circle o check your consrucion. Evolues samples The evolue of he parabola The curve given by he equaion y= f (=k. Then he curvaure of he curve is K = (1+ y' = k y',1 ( k,1, he normal is n=( = 3 / (1+( k 3 / 1+ y' 1+(k. The equaion of he evolue is: ζ =( y' (1+( y ' 1+( y ' y ' ' ( = (1+ k, k + 1+ k k = ( k, 3 k + 1 k. The evolue of he ellipse The curve given by he equaion f (, y= a + y 1=. Then he equaion of he evolue is: b =( ζ f ( f + f y f y ( f + f y f f y f y f f yy f y f f f y f y f f yy f y f ( = 3 y3, a b (a b. Fig.1. Evolue of he parabola Fig.. Evolue of he ellipse

3 The evolue of a hyperbola The curve given by he equaion f (, y= a y 1=. We perform he parameerizaion b =a cosh, y=bsinh. Then he equaion of he evolue is: ζ =( y' ( ' + y ' ' y ' ' ' ' y' ' ( ' + y ' ' ' ' y ' ( = cosh3, sinh 3 a b (a +b =( 3 y3, a b (a +b. The evolue of an asroid The curve given by he equaion {, y }={ acos 3,a sin 3 }. The curvaure of he curve K = 3 a sin. Then he equaion of he evolue is: ζ =a(cos(3 cos,sin (1+cos. is Рис.3. Evolue of a hyperbola Рис.. Evolue of an asroid Involue The involue of he plane curve is a curve wih respec o which he given curve is he evolue. The involue is a curve, for which he normal a each poin is angen o he iniial curve. Tha is, he involue conains he angens o he given curve. Imagine ha he lile pebble lies on he curve a an arbirary poin P. Le he hread run from he pebble along he curve. Then he involue deermines he rajecory of pebble, moving away from he curve and associaed wih i by he sreched hread. They say ha he seleced poin moves along he involue when unwinding he hread, which lies on he curve. The family of he involues generaed by differen poins of he surface eiss for each curve. Furher we consider only one of hem, saring a he origin. Case 1. Le his curve be given paramerically by he epression (1, y ' is τ =. The lengh of he hread unwound from he beginning is 1+( y' s= equaion of he involue ζ T =(ξ,η has he form of: ζ = X sτ =( 1+( y' d, y 1+( y' y= f (. Then he uni angen vecor 1+( y ' d. The y' 1+( y' d. (6 1+( y' 3

4 Case. Le he curve be given paramerically by he epression X T =((, y(. Then he uni angen ( ' (, y' ( vecor is τ =. The lengh of he hread unwound from he beginning is ( ' ( +( y ' ( s= ( d( d +( dy( d. d The equaion of he involue ζ T =(ξ,η has he form of: ζ = X sτ =( ' ' + y' d, y ' + y' y' ' + y' d. (7 ' + y' Case 3. Le he curve be given by he epression involue: R= f ( s Using (5, we find he equaion of he R= s, s= s d s R. (8 Тypical figure and i's using In each figure poins and 1 on he -ais specify he Caresian coordinaes. The graph of he original curve is shown by he blue line. I is deermined by he parameers which values are indicaed in he figure. The parameers are se by he acive poins. The involue is shown in red. The rial poins labeled С, D, E,... are locaed on he curve. Tangen circle and is cener poin C ', D ',..., belonging o involue are shown in pink. The process of learning and invesigaion one can begin from he verificaion of he basic properiy of he involue. By moving poin C, make sure ha he segmen CC ' (DD ',... is angen o he he given original curve a all posisions of poin C, and poin С (D, E,... is he cener of curvaure of he involue. By acivaing he "Properies" buon find and check he equaions used for he curves consrucion. Eplore several involue curves. Save he file in a convenien locaion and change he original curve. Perform calculaions and consruc involue of he seleced curve. Build a angen circle o check your consrucion. Involues samples The involue of he parabola The curve given by he equaion y=k (1,k. Then he uni angen vecor is τ =. The lengh 1+ k of he hread unwound from he beginning is: s= 1+k d = 1+ k + 1 k ln(k + 1+k. The equaion of he involue has he form of: ζ =( s 1+ k, y k s 1+ k ( = arcsinh( k arcsinh(k, k 1+ k 1+k.

5 The involue of he circle The curve given by he equaion R=a. We use (8 and obain: s d s R= s, s= = sd s = s R a a. The equaion of he involue has he form of: R =a s. The curve in paramericallyform: ζ =a(cos+ sin, sin cos. Involue of he parabola Involue of he circle The involue of he asroid The curve given by he equaion X T =(a cos 3,asin 3. Then he uni angen vecor is τ =( cos,sin. The lengh of he hread unwound from he beginning is: s=1,5 a sin. The equaion of he involue has he form of: ζ = a (cos (3 cos, sin. Involue of an asroid Involue of Archimedean spiral Lieraure 1. И.Н. Бронштейн, К.А. Семендяев. Справочник по математике. М.: «НАУКА», с.. J.W.Bruce and P.J.Giblin, Curves and Singulariies, Cambridge Universiy Press,198, pp. 3. A.A. Савелов. Плоские кривые. М.: «ГИФМЛ», с. 5