"Inviluppo di una iperbole"

x^2/a^2-y^2/b^2=1

"Prendiamo una generica iperbole avente come asse focale l'asse delle x"

"per le formule di sdoppiamento la tangente ad una iperbole in suo punto ha e~
quazione"

x*x*0/a^2-y*y*0/b^2=1

"quindi poichŠ le equazioni parametriche dell'iperbole si possono scrivere x=~
-a/cos t, y=b*tan t"

"l'equazione della tangente all'iperbole in un suo punto diventa"

-b*x-a*SIN(t)*y=a*b*COS(t)

"supponendo a=2, b=3 si ottiene"

-3*x-2*y*SIN(t)=6*COS(t)

x^2/4-y^2/9=1

VECTOR(-3*x-2*y*SIN(t)=6*COS(t),t,0,6.28,0.1)

;Expd(#12)
[-3*x=6,-2*y*SIN(1/10)-3*x=6*COS(1/10),-2*y*SIN(1/5)-3*x=6*COS(1/5),-2*y*SIN(~
3/10)-3*x=6*COS(3/10),-2*y*SIN(2/5)-3*x=6*COS(2/5),-2*y*SIN(1/2)-3*x=6*COS(1/~
2),-2*y*SIN(3/5)-3*x=6*COS(3/5),-2*y*SIN(7/10)-3*x=6*COS(7/10),-2*y*SIN(4/5)-~
3*x=6*COS(4/5),-2*y*SIN(9/10)-3*x=6*COS(9/10),-2*y*SIN(1)-3*x=6*COS(1),-2*y*S~
IN(11/10)-3*x=6*COS(11/10),-2*y*SIN(6/5)-3*x=6*COS(6/5),-2*y*SIN(13/10)-3*x=6~
*COS(13/10),-2*y*SIN(7/5)-3*x=6*COS(7/5),-2*y*SIN(3/2)-3*x=6*COS(3/2),-2*y*SI~
N(8/5)-3*x=6*COS(8/5),-2*y*SIN(17/10)-3*x=6*COS(17/10),-2*y*SIN(9/5)-3*x=6*CO~
S(9/5),-2*y*SIN(19/10)-3*x=6*COS(19/10),-2*y*SIN(2)-3*x=6*COS(2),-2*y*SIN(21/~
10)-3*x=6*COS(21/10),-2*y*SIN(11/5)-3*x=6*COS(11/5),-2*y*SIN(23/10)-3*x=6*COS~
(23/10),-2*y*SIN(12/5)-3*x=6*COS(12/5),-2*y*SIN(5/2)-3*x=6*COS(5/2),-2*y*SIN(~
13/5)-3*x=6*COS(13/5),-2*y*SIN(27/10)-3*x=6*COS(27/10),-2*y*SIN(14/5)-3*x=6*C~
OS(14/5),-2*y*SIN(29/10)-3*x=6*COS(29/10),-2*y*SIN(3)-3*x=6*COS(3),-2*y*SIN(3~
1/10)-3*x=6*COS(31/10),-2*y*SIN(16/5)-3*x=6*COS(16/5),-2*y*SIN(33/10)-3*x=6*C~
OS(33/10),-2*y*SIN(17/5)-3*x=6*COS(17/5),-2*y*SIN(7/2)-3*x=6*COS(7/2),-2*y*SI~
N(18/5)-3*x=6*COS(18/5),-2*y*SIN(37/10)-3*x=6*COS(37/10),-2*y*SIN(19/5)-3*x=6~
*COS(19/5),-2*y*SIN(39/10)-3*x=6*COS(39/10),-2*y*SIN(4)-3*x=6*COS(4),-2*y*SIN~
(41/10)-3*x=6*COS(41/10),-2*y*SIN(21/5)-3*x=6*COS(21/5),-2*y*SIN(43/10)-3*x=6~
*COS(43/10),-2*y*SIN(22/5)-3*x=6*COS(22/5),-2*y*SIN(9/2)-3*x=6*COS(9/2),-2*y*~
SIN(23/5)-3*x=6*COS(23/5),-2*y*SIN(47/10)-3*x=6*COS(47/10),-2*y*SIN(24/5)-3*x~
=6*COS(24/5),-2*y*SIN(49/10)-3*x=6*COS(49/10),-2*y*SIN(5)-3*x=6*COS(5),-2*y*S~
IN(51/10)-3*x=6*COS(51/10),-2*y*SIN(26/5)-3*x=6*COS(26/5),-2*y*SIN(53/10)-3*x~
=6*COS(53/10),-2*y*SIN(27/5)-3*x=6*COS(27/5),-2*y*SIN(11/2)-3*x=6*COS(11/2),-~
2*y*SIN(28/5)-3*x=6*COS(28/5),-2*y*SIN(57/10)-3*x=6*COS(57/10),-2*y*SIN(29/5)~
-3*x=6*COS(29/5),-2*y*SIN(59/10)-3*x=6*COS(59/10),-2*y*SIN(6)-3*x=6*COS(6),-2~
*y*SIN(61/10)-3*x=6*COS(61/10),-2*y*SIN(31/5)-3*x=6*COS(31/5)]

"Š possibile ottenere delle formule generali"

TANGENTE_IPERBOLE(a,b):=-b*x-a*SIN(t)*y=a*b*COS(t)

IPERBOLE_INVILUPPO(a,b):=VECTOR(TANGENTE_IPERBOLE(a,b),t,0,6.28,0.1)

"___________________________________________________________"

"               Esempio di utilizzo                         "

"___________________________________________________________"

IPERBOLE_INVILUPPO(3,2)

"expand poi plot"

;Simp(User)
[-3*x=6,-2*y*SIN(1/10)-3*x=6*COS(1/10),-2*y*SIN(1/5)-3*x=6*COS(1/5),-2*y*SIN(~
3/10)-3*x=6*COS(3/10),-2*y*SIN(2/5)-3*x=6*COS(2/5),-2*y*SIN(1/2)-3*x=6*COS(1/~
2),-2*y*SIN(3/5)-3*x=6*COS(3/5),-2*y*SIN(7/10)-3*x=6*COS(7/10),-2*y*SIN(4/5)-~
3*x=6*COS(4/5),-2*y*SIN(9/10)-3*x=6*COS(9/10),-2*y*SIN(1)-3*x=6*COS(1),-2*y*S~
IN(11/10)-3*x=6*COS(11/10),-2*y*SIN(6/5)-3*x=6*COS(6/5),-2*y*SIN(13/10)-3*x=6~
*COS(13/10),-2*y*SIN(7/5)-3*x=6*COS(7/5),-2*y*SIN(3/2)-3*x=6*COS(3/2),-2*y*SI~
N(8/5)-3*x=6*COS(8/5),-2*y*SIN(17/10)-3*x=6*COS(17/10),-2*y*SIN(9/5)-3*x=6*CO~
S(9/5),-2*y*SIN(19/10)-3*x=6*COS(19/10),-2*y*SIN(2)-3*x=6*COS(2),-2*y*SIN(21/~
10)-3*x=6*COS(21/10),-2*y*SIN(11/5)-3*x=6*COS(11/5),-2*y*SIN(23/10)-3*x=6*COS~
(23/10),-2*y*SIN(12/5)-3*x=6*COS(12/5),-2*y*SIN(5/2)-3*x=6*COS(5/2),-2*y*SIN(~
13/5)-3*x=6*COS(13/5),-2*y*SIN(27/10)-3*x=6*COS(27/10),-2*y*SIN(14/5)-3*x=6*C~
OS(14/5),-2*y*SIN(29/10)-3*x=6*COS(29/10),-2*y*SIN(3)-3*x=6*COS(3),-2*y*SIN(3~
1/10)-3*x=6*COS(31/10),-2*y*SIN(16/5)-3*x=6*COS(16/5),-2*y*SIN(33/10)-3*x=6*C~
OS(33/10),-2*y*SIN(17/5)-3*x=6*COS(17/5),-2*y*SIN(7/2)-3*x=6*COS(7/2),-2*y*SI~
N(18/5)-3*x=6*COS(18/5),-2*y*SIN(37/10)-3*x=6*COS(37/10),-2*y*SIN(19/5)-3*x=6~
*COS(19/5),-2*y*SIN(39/10)-3*x=6*COS(39/10),-2*y*SIN(4)-3*x=6*COS(4),-2*y*SIN~
(41/10)-3*x=6*COS(41/10),-2*y*SIN(21/5)-3*x=6*COS(21/5),-2*y*SIN(43/10)-3*x=6~
*COS(43/10),-2*y*SIN(22/5)-3*x=6*COS(22/5),-2*y*SIN(9/2)-3*x=6*COS(9/2),-2*y*~
SIN(23/5)-3*x=6*COS(23/5),-2*y*SIN(47/10)-3*x=6*COS(47/10),-2*y*SIN(24/5)-3*x~
=6*COS(24/5),-2*y*SIN(49/10)-3*x=6*COS(49/10),-2*y*SIN(5)-3*x=6*COS(5),-2*y*S~
IN(51/10)-3*x=6*COS(51/10),-2*y*SIN(26/5)-3*x=6*COS(26/5),-2*y*SIN(53/10)-3*x~
=6*COS(53/10),-2*y*SIN(27/5)-3*x=6*COS(27/5),-2*y*SIN(11/2)-3*x=6*COS(11/2),-~
2*y*SIN(28/5)-3*x=6*COS(28/5),-2*y*SIN(57/10)-3*x=6*COS(57/10),-2*y*SIN(29/5)~
-3*x=6*COS(29/5),-2*y*SIN(59/10)-3*x=6*COS(59/10),-2*y*SIN(6)-3*x=6*COS(6),-2~
*y*SIN(61/10)-3*x=6*COS(61/10),-2*y*SIN(31/5)-3*x=6*COS(31/5)]