Tema 3. Superficies No Circulares

 

 

AWUQJ JKFHKG CG GQS@EHFHC@C CG S@FWCGQ AJK QFHCG R?.4 SGN@ 1

 

Aursj jkfhkg cg Qfhcg v?.4 / Sgn` 1

Sgn` 1. Qupgrbh Qupgrbhahgs ahgs kj ahrauf`rg ahrauf`rgs. s. 

Hktrjcuaahñk

Gk gstg tgn` tr`t`rgnjs ajk supgrbhahgs ajk uk` ogjngtrî` chbgrgktg. L`st` `ljr` tjcjs fjs a`sjs ajktgnpf`cjs hkafuî`k uk` supgrbhahg cg cgsfhz`nhgktj ahrauf`r, f`s au`fgs pjsggk cjs a`r`atgrîstha`s9 sjk nuy l`ehtu`fgs y sjk bàahfgs cg njcgfhz`r. Tgrj jevh`ngktg kj shgnprg sg prjcuag uk` rjtur` ajk gst` ogjngtrî`. R`njs ` uthfhz`r gf nhsnj t`fuc qug p`r` uk a`sj prgvhj, gk gf qug ajktgnpfàe`njs uk t`fuc ajk khvgf cg `ou`, uk` a`p` cg n`tgrh`f ajk pja` rgshstgkah` cgktrj cg jtr` n`yjrht`rh` ajk nàs rgshstgkah`, y f` `chahñk cg uk` a`ro` gxtgrk` ukhbjrngngktg chstrheuhc` gk f` p`rtg supgrhjr cgf t`fuc. Kjs shrvg pgrbgat`ngktg p`r` kugstrj a`sj. Njcgfj

Tjr t`ktj, uk` vgz qug tgkgnjs gf Qfhcg `ehgrtj, cgegnjs `erhr gf `ralhvj qug tgkgnjs ou`rc`cj gk f` a`rpgt` cgf tgn` 5 `ktgrhjr, y qug l`eî`njs ff`n`cj ‖N`tgrh`fgs y `pfha`ahñk cg a`ro`s„. T`r` gffj v`njs ` Bhfg, Jpgk y `erhnjs gf `ralhvj ajrrgspjkchgktg. Ugajrcgnjs qug tuvhnjs qug ou`rc`rfj `f ngkjs `ktgs cg rg`fhz`r gf aàfaufj. Cg tjc`s bjrn`s, vjfvgnjs ` rgpgthr qug gs eugk` hcg` tgkgr f` ajstunerg cg l`agr brgaugktgs s`fv`cjs pjr fj qug pugc` p`s`r, sjerg tjcj sh gst`njs njcgf`kcj uk prjygatj ajnpfgij gk gf qug uk prjefgn` hkgspgr`cj pugcg c`r `f tr`stg ajk uk` or`k a`kthc`c cg tr`e`ij. @ ajkthku`ahñk, v`njs ` ou`rc`r gf `ralhvj, ajk Q`vg @s, gk f` a`rpgt` cgf tgn` 1 ajrrgspjkchgktg, ajk gf kjnerg9 Qupgrbhahgs kj ahrauf`rgs. Eùsqugc` cg supgrbhahg

Ajk gf njcgfj `ehgrtj, fj prhngrj cg tjcj gs cgahrfg `f prjor`n` qug gk gst` ja`shñk kj qugrgnjs eusa`r uk` supgrbhahg ahrauf`r. 0

 

Aursj jkfhkg cg Qfhcg v?.4 / Sgn` 1

Qgfgaahjk`njs f` pgst`ð` Qurb`ags y v`njs ` Qurb`ag Jpthjks, gk gf ngkù Qurb`ags j gk f` e`rr` cg lgrr`nhgkt`s= gk thpj cg supgrbhahg a`neh`njs ` kj ahrauf`r. @utjnàtha`ngktg, gf rgstj cg gst` vgkt`k` a`neh`. Gk gf rgau`crj cg `f f`cj sg a`neh`k f`s jpahjkgs. T`r` supgrbhahgs kj ahrauf`rgs f`s jpahjkgs cg eùsqugc` sgràk Efjam Qg`ral, T`tl Qg`ral, Qhnuf`tgc @kkg`fhko, @utj Ugbhkg y Auamjj Qg`ral. Sr`e`i`rgnjs ajk f` prhngr`, f` eùsqugc` pjr efjqug.

Eùsqugc` cg supgrbhahg kj ahrauf`r pjr efjqug.

Fjs v`fjrgs cg f`s chbgrgktgs a`shff`s qug prgsgkt` pjr cgbgatj gst` jpahñk kjs shrvgk, `sî qug kj njchbha`njs k`c`. Agrr`njs f` vgkt`k` ajk jm. Rgnjs qug l` cgs`p`rgahcj f` au`crîauf` cg fjs agktrjs cg cgsfhz`nhgktj. Gs fñohaj, y` qug gs` au`crîauf` sg rgf`ahjk`e` ajk gf thpj cg eùsqugc` `ktgrhjr p`r` supgrbhahgs ahrauf`rgs. 5

 

Aursj jkfhkg cg Qfhcg v?.4 / Sgn` 1

T`r` agktr`r gf njcgfj sgfgaahjk`njs Yjjn @ff cg f` e`rr` cg lgrr`nhgkt`s, cgf ngkù Rhgw, j cgf ngkù gngrogktg qug s`fg `f phkal`r ajk ejtñk cgrgalj cgf r`tñk sjerg f` p`kt`ff`. Eùsqugc` pjr efjqug (Efjam Qg`ral)

ºPuè gkthgkcg Qfhcg pjr uk` eùsqugc` pjr efjqug, y` qug gst`njs gk gstg a`sj: Tugs hcgkthbha` uk` n`s` cg sugfj qug cgsfhz` qug gstà ajnpugst` pjr uk` p`rtg supgrhjr `athv`, uk` p`rtg hktgrngch` agktr`f –qug jaup` f` n`yjrî` cgf n`tgrh`f hkvjfuar`cj–, y uk` p`rtg hkbgrhjr p`shv`. F` ogjngtrî` gs hrrgouf`r, fj qug c` chsthktjs pf`kjs cg cgsfhz`nhgktj, `f ajktr`rhj qug gk uk` supgrbhahg shnpfg ahrauf`r.

T`rtgs qug ajnpjkgk uk` supgrbhahg cg cgsfhz`nhgktj kj ahrauf`r.

Gf prjor`n` cgeg `c`pt`rsg ` gst`s ahraukst`kah`s. Fj l`ag pgrnhthgkcj uthfhz`r fjs jeigtjs cg eùsqugc` cg efjqug, qug pugcgk ajkshsthr gk uk` fîkg`, uk` vgkt`k`, uk puktj j uk` pjfhfîkg`. Gstjs jeigtjs sg us`k p`r` ogkgr`r cg n`kgr` `fg`tjrh` fjs vèrthags cg f` supgrbhahg cg cgsfhz`nhgktj. C`c`s f`s a`r`atgrîstha`s cg kugstrj t`fuc, ajk f` prgsgkah` cg uk` a`p` pjaj rgshstgktg y gstrgal`, cjkcg ` prhjrh y` supjkcrî`njs qug gs uk` zjk` susagpthefg cg hkbfugkah`r uk` supgrbhahg cg cgsfhz`nhgktj, f` ngijr jpahñk qug sg kjs prgsgkt` p`r` eusa`r gs` supgrbhahg gs f` cg f` pjfhfîkg`. 1

 

Aursj jkfhkg cg Qfhcg v?.4 / Sgn` 1

Bukahjk` cgf shouhgktg njcj. Qjerg f` pjfhfîkg` sg cgtgrnhk`k cjs puktjs. F` supgrbhahg cg cgsfhz`nhgktj sg hkafuhrà gktrg gsjs cjs puktjs, cgktrj cg f` pjfhfîkg`. X fjs àkoufjs cg prjygaahñk shrvgk p`r` shnpfgngktg prjygat`r gs` supgrbhahg qug `ljr` gstà gk prjbukchc`c l`ah` `rrhe`, l`st` qug hktgrsgatg f` supgrbhahg cgf t`fuc.

Tjfhfîkg` y àkoufjs cg prjygaahñk p`r` uk` eùsqugc` pjr efjqug.

T`r` a`c` efjqug qug sg l`o` sg l`rà gf nhsnj prjagsj. Kjs chrhohnjs `ljr` ` Efjam Qg`ral, gk f` pgst`ð` Qurb`ags –y` qug v`njs ` l`agr uk` eùsqugc` pjr efjqug–, y cgktrj cg gsg ngkù sgfgaahjk`njs @cc Tjfyfhkg. Tjcgnjs c`r f` nhsn` jrcgk cgscg gf hajkj ajrrgspjkchgktg cg f` e`rr` cg lgrr`nhgkt`s, pugs èst` l` a`neh`cj au`kcj sgfgaahjk`njs uk` eùsqugc` cg supgrbhahg kj ahrauf`r. Kjs `p`rgag f` vgkt`k` Efjam Qg`ral Tjfyfhkg.

Rgkt`k` Efjam Qg`ral Tjfyfhkg.

Gk gst` vgkt`k`, tgkgnjs qug cgahchr añnj sg v`k ` eusa`r fjs puktjs gk f` pjfhfîkg`. Qfhcg kjs jbrgag trgs `ftgrk`thv`s. 3

 

Aursj jkfhkg cg Qfhcg v?.4 / Sgn` 1

Tjcgnjs cgi`r f` qug vhgkg pjr cgbgatj, gs cgahr, f` qug eusa`rà gsjs puktjs cg n`kgr` `fg`tjrh` gk au`fquhgr sgongktj cg f` pjfhfîkg`. Gst` jpahñk, ajnj gs fñohaj, ajkt`rà ajk f` n`yjr ajegrtur` pjshefg cgktrj cg fjs fînhtgs cg f` pjfhfîkg`, pugs ajktgnpf` tjc` su gxtgkshñk. F`s jtr`s cjs `ftgrk`thv`s sjk nàs fhnht`ktgs gk gstg `spgatj. Cgi`njs sgfgaahjk`c` gst` `ftgrk`thv` gktjkags, p`r` `nejs puktjs, y s`fhnjs cg f` vgkt`k` afha`kcj gk gf ejtñk jm. Sgkgnjs qug cgbhkhr f` pjfhfîkg`. T`r` gffj pjcgnjs l`agrfj, ajnj gs l`ehtu`f qug kjs cgig gf prjor`n`, cg cjs bjrn`s. Wk` gs hktrjcuahgkcj ajjrcgk`c`s gk f` fîkg` cg ajn`kcjs –f` au`f sg l` `athv`cj–, y f` jtr` gs cg n`kgr` n`ku`f ajk gf aursjr. Gfgohnjs f` prhngr`, qug kjs pgrnhthrà uk` n`yjr prgahshñk. F`s ajjrcgk`c`s sjk9 (18, 51) y (