87. PARABOLIC CYLINDERS 



Consider the transformation 



X = {^2 - ^i)^ y =- ■^i ^2' '^1 = '■ - ^' -^2 '" ■•" '''' 



r = yjx'^ + y^. 



[87a, b] 



[87c,d,e,f] 



[87g] 



The surfaces A, = constant, or X, = constant, constitute two families of orthogonal confocal 

 parabolas, with the ar-axis as their axis and the focus at the origin; they open toward x-*oa 

 and ar-»-c«, respectively; see Figure 139. The parabola for Aj = is the positive a!-axis, that 

 for X, = the negative. 



The variables X^, X, may be used as parabolic coordinates on the a;y-plane. They are 

 double valued, and changes of sign of X^ or X2 on the same parabola are necessary in order 

 to cover the entire plane. 







2.0 1.9 



1.8 



1.7 



1.6 



1.5 



1.4 ' 



1.3 



1.2 



1 



1.1 



1.0 





0.9 







"^ 





y^ 



-^ 



/ ^ 



^ 











X" 



V 



^ 



>■ 



/\ 





\, 



\. 





0.8 



- 







y- 



y 





V 





/ V 





^M^ 



^ 



y \ 



<^ 





Y 



V 



Y 



' 



^ 



0.7 









;;^ 



l^ 





k 





2^ 





/i'x 



\ 





\ 





-^ 



^ 



^ 



A 





0.6, 









~-y 



/ 









^ 





>!ik 



\ 





\ 



\^ 



V^ 











• J 









Y- 



^ 









V 





■nTI.O"/ 



\ 





^ 



Am 











04 









-jK 



yL 









^ 







XI 





\ 





^J 











3 









/^ 



W 









y 





y XV 



<S 





\\ 





L 











2 









L 



~\_ 









-H 









-X, 



H 



4- 





r 



h-' 





1 









I 



1" 









Tl 







n 





+ 





h- 





1 







n.n 



0.0^ 



.0-1 



.8-1 



h -H 



.6-1 



1 — t- 



.4-1 



.2-1 



0-0 



aU 



n 



uvf'i^^Rj 



.4 



[0 



f\ 



.8 1 







1.2 



1.4 1.6 1.8 2 



.0 



0.1 





J4 



J- 









\\ 



^M'l^^ 





~^ 





4- 







L; 











p- 



^ 









\ 







V 





H 





p 













0.3"^ 







Y' 



V^ 









V 







^ 



ScT^ 



y-jM 



jr 











A/-4' 







-\ 



'^ 









(\ 





vojt' 



>^ 







y 













0.5 







-A 



"\ 









\ 





1 9^ 



y. 







;C 













0.6 







\^ 



\ 





\ 





\^ 





xlfx 



Y 





7 



-Al 











0.7' 



^ 





\ 



^ 





A 





\ y 





vjV 



^ 





<^ 



A 



fSt 









- 



0.8- 







^ 



^ 



\^ 



r 



V 



\ 





I 





y^ 





y 



7 



\,/i / 



^ 







■^ 







O.J 





1 



.0 



1. 





1 



2 



1.3 



1. 



4 1. 



5 





1.6 



1.7 



1.8 



1.9 



2. 











Figure 139 - Diagram for parabolic coordinates. 

 208 



