Curves in the space E

Transcription

1 3 Cues i he spae E

2 Smooh piee-wise oiuous ue i spae E 3 is hoogaph of a eo fuio = efie o a ieal i eal umes I R while. all hee sala ooiae fuios ae oiuous o ieal I. he hae o I oiuous eiaies fom whih a leas oe is o-eo. Fuio alues of sala ooiae fuios a a eal ume I efie Caesia ooiaes of poi o ue P = [ ] as e poi of eo = ha is he alue of fuio i he poi. Nume I is alle uiliea ooiae of poi P o ue.

3 Fo I = a we spea aou ue segme wih e pois Pa P while ue segme is oiee equall wih he paameiaio heefoe wih he oieaio of ieal I. Fo Pa = P we spea aou lose ue. Poi o ue is alle m-muliple poi if hee eis suh alues of aiale i fo all i j =... m I i =...m fo whih hols i = i i i = j = j j j Pa = P is a oule poi of he lose ue efie o ieal I = a.

4 Value of eo fuio = efie o ieal I i he poi I eo = is alle age eo o he ue i he poi P. R Rage of eo fuio heefoe efies ieio eo fiel of ages o ue i is pois.

5 Poi P is alle a egula poi of he ue if i hols = I he opposie ase we spea aou sigula poi o ue. A a egula poi P hee eiss age lie o he ue lie passig hough he gie poi wih he ieio eo. Oieaio of eo is equal o he oieaio of he ue. Poi o ue i whih his oieaio is hagig is alle uspial poi o usp. I is a sigula poi o ue i whih =.

6 The followig elaios hol i a egula oule poi o ue = m. m R I he ifle poi o ue i hols = o = m. m R

7 Legh of lie segme efie o I = a is alue of iegal Sala fuio s s a s s a a s.. a a a Defies legh of ue segme as fuio of aiale o ieal I. Popeies of ues ae eie i egula pois o ue segmes i meas eemig eo fuios ae a leas oe iffeeiale o omais ieals I so he hae o I oiuous eiaies up o oe 3. Thee eis a leas fis hee eiaies = = =

8 Elemes of Fee See moig iheo Ui age eo of ue i he poi P is eo Veo i aia poi P I a e epesee also i he fom.. whee ooiaes of eo ae ieio osies of age lie o he ue i his poi I Equaio of age o ue i poi P I eo fom Tu = + u u R implii fom Z Y X

9 Ui iomal eo of ue i he poi P is eo Veo i aia poi P I a e epesee also i he fom.. whee ooiaes of eo ae ieio osies of iomal lie o ue I Equaio of iomal o ue i poi P I eo fom Bu = + u u R implii fom Z Y X

10 Ui piiple omal eo o ue i he poi P is eo Veo i aia poi P I a e also epesee i he fom.. a a while ooiaes of eo ae ieio osies of piiple omal o ue i his poi I ] :[ ] :[ ] [ : : Equaio of omal o ue i poi P I eo fom Nu = + u u R implii fom Z Y X

11 Veos a i he egula poi P o ue ae 3 pepeiula ui eos fomig ohoomal epe of ue alle iisi ohogoal ooiae ssem moig ogehe wih he poi o he ue a hagig is posiio i he spae aoigl u eepig hei muual ohogoali. Lies - age iomal a omal ae 3 muuall pepeiula lies wih oe ommo poi i egula poi P fomig moig iheo of ue alle also Fee-See iheo. ee egula poi P o ue eges halflies o age piiple omal a iomal faes i pepeiula plaes = P = P = P

12 Osulaig plae = P Osulaig plae i a egula poi P o ue is pepeiula o he iomal i he gie poi is ieio eos ae ui age eo a ui piiple omal eo. Veo equaio R -. = ] [ Z Y X- Z Y X- R Plaa ue has he same osulaig plae i all pois i is loae i his uique osulaig plae. Spae ue has a iffee osulaig plae a eah poi P a i is loae i his plae i a small eighouhoo of he espeie poi.

13 Reifiaio plae = P Reifiaio plae i a egula poi P o ue is pepeiula o he piiple omal i he gie poi is ieio eos ae ui age eo a ui iomal eo. Veo equaio R -. = ] [ Z Y X- R Reifiaio plae of a plaa ue is pepeiula o he ue plae. Legh of a spae ue a e eemie as legh of a plaa ue * ha is eelopme of ue o is eifiaio plae.

14 Nomal plae = P Nomal plae i a egula poi P o ue is pepeiula o age i he gie poi is ieio eos ae ui eos a. Veo equaio R -. = ] [ Z Y X- R

15 Deiaio of ue fom is age i a egula poi P is epesee he fis uaue - fleio. Fuio of he fis uaue is sala o-egaie fuio. 3 Osulaig ile hs o ue is loae i he osulaig plae i he espeie egula poi P a i has a ommo age wih he ue. Cee of osulaig ile is a poi o piiple omal i he poi P i ieio of ui omal eo. Raius of osulaig ile = aius of he fis uaue i he gie poi a i equals o iee alue of fuio fo espeie. I

16 Regula poi o ue i whih = is ifleio poi as =. If = i all pois o ue ue is a lie lie segme. = a + + e + f R = a e = e a I e a Cue wih osa o-eo fis uaue is e.g. ile. si os os si si os os si si os 3 R 4 Cile is self-osulaig o iself i all pois.

17 Deiaio of ue fom is osulaig plae i a egula poi P is epesee he seo uaue - osio. Fuio of he seo uaue [ ] is a sala fuio. Sig of alue of fuio i he poi eemies oieaio of eiaio of ue fom he osulaig plae i he poi P. ue is eiaig i he ieio of ui iomal eo oe he osulaig plae i gie poi ue is eiaig i he opposie ieio o he ieio of ui iomal eo elow osulaig plae i gie poi = poi is alle plaa poi ue is o eiaig fom osulaig plae i gie poi i is loae i his plae i he poi eighouhoo If = i all pois o ue ue is i oe plae i.e. i is plaa. I

18 Fo ile hols os si si os os si 4 Cue of osa o-eo seo uaue is e.g. heli. ] [ os si os si si os os si si os R

19 Two ues p I ae alle equiisa paallel ues if hee eiss ume R suh ha I i hols p = +. A oei iles ae equiisa o ellipses wih ommo ee while fo hei semi-aes hols elaio a =.a =. fo small. Cues of osa slope Le e a fie agle eemie ieio - gaie os si. Spae ue whose fuios of he fis a he seo uaue saisf equaio si - os = ae alle ues wih osa slope while slope is he ages of agle a i hols a os Cliial heli is a ue of osa slope a = /.

20 Fee-See fomulas epese elaios ewee ui age omal a iomal eos a hei fis eiaies i a ue egula poi 3 ] [