ASIMPTOTIC LINES OF ONE-SHEETED GIPERBOLOID

18 Danish Scien i ic Jou nal No100, 2025
MATHEMATICAL SCIENCES
ASIMPTOTIC LINES OF ONE-SHEETED GIPERBOLOID
Abdumajido a Sh.
h ps://doi.o g/10.5281/zenodo.17249782
In oduc ion
Le us conside a su ace 𝛷 which is gi en by
equa ion a ound poin 𝑃
𝑟=𝑟(𝑢,𝑣).
I we in e sec wi h a plane 𝛱 passing h ough a
poin 𝑃 on i , we ob ain a smoo h cu e 𝛾 passing
h ough poin 𝑃 in he in e sec ion, which we call such
a cu e a plane sec ion. A plane sec ion 𝛾 lies in he
plane 𝛱 ,so i s o sion is necessa ily ze o.
W i ing he equa ion o he plane sec ion in e ms
o he na u al pa ame e 𝑠 (i.e., a c leng h), and using
he F ene o mulas o i ( aking in o accoun ha he
o sion is equal o ze o), we w i e:
{𝜏=𝑘𝑣
𝑣=−𝑘𝜏
He e, 𝜏 is he uni angen ec o , 𝑣 is he uni p in-
cipal no mal ec o , and 𝑘 is he cu a u e o he cu e
𝛾 a poin 𝑃(𝑢0,𝑣0).. Then o second quad a ic o m
we ha e 𝐼𝐼(𝜏,𝜏)=(𝜏,𝑛
󰇍

)=(𝑘𝑣,𝑛)=𝑘cos𝜃
He e 𝜃 is he angle be ween he ec o s 𝑛
󰇍

and 𝑣.
Now, i we de ine 𝛾 by equa ion
𝜌=𝜌(𝑡)
(whe e is an a bi a y pa ame e ), hen since 𝑡 is
a unc ion o 𝑠, and conside ing he ollowing equali-
ies, 𝑑𝜌
𝑑𝑡=𝜌′=𝜌𝑑𝑠
𝑑𝑡,𝜌″=𝜌″(𝑑𝑠
𝑑𝑡)2+𝜌′𝑑2𝑠
𝑑𝑡2
we ha e:
𝐼𝐼(𝜌′,𝜌′)=(𝜌″,𝑛
󰇍

)=(𝑑𝑠
𝑑𝑡)2(𝜌..,𝑛
󰇍

)=(𝑑𝑠
𝑑𝑡)2𝑘cos𝜃
We ob ain ollowing equali y
𝑘cos𝜃=𝐼𝐼(𝜌′,𝜌′)
(𝑑𝑠
𝑑𝑡)2=𝐼𝐼(𝜌′,𝜌′)
𝐼(𝜌′,𝜌′)
I can be seen ha he igh side o his equali y
depends only on he ec o 𝜌′. I we ake ano he plane
sec ion 𝛾′ o he han 𝛾 and hey ha e a common angen
(i.e., hey ha e he same di ec ion), hen he igh side
o equali y (1) is he same o hem. Now le he he
plane sec ion be pa allel o he no mal ec o . The e-
o e, equali y (1) becomes:
𝑘=±𝐼𝐼(𝜌′,𝜌′)
𝐼(𝜌′,𝜌′)
De ini ion 1. The numbe 𝐼𝐼(𝜌
󰇍
󰇍

′,𝜌
󰇍
󰇍

′)
𝐼(𝜌
󰇍
󰇍

′,𝜌
󰇍
󰇍

′) ob ained he e is
called he no mal cu a u e o he su ace 𝛷 a poin 𝑃
in he di ec ion 𝑎=ϱ′ and is deno ed by 𝑘𝑎(𝑎) .
Thus, he absolu e alue o he no mal cu a u e
o he su ace in he di ec ion 𝑎 is equal o he cu a u e
o he no mal sec ion ha de ines he no mal ec o ,
possibly di e ing in sign.
De ini ion 2. I 𝑘𝑎(𝑎)=0 in some di ec ion 𝑎,
hen such a di ec ion is called an asymp o ic di ec ion.
Fo a gi en ec o 𝑎=(𝑥,𝑦), i is necessa y and
su icien ha 𝐿𝑥2+2𝑀𝑥𝑦+𝑁𝑦=0 o he di ec ion
de ining he asymp o ic di ec ion. He e, L, M, N a e he
coe icien s o he second quad a ic o m.
De ini ion 3. I a cu e 𝛾 on a su ace is gi en by
he equa ion 𝑢=𝑢(𝑡),𝑣=𝑣(𝑡), and i s angen ec o
a any poin de ines he asymp o ic di ec ion, hen such
a cu e is called an asymp o ic cu e.
Na u ally, i a s aigh line lies on a su ace, i is
an asymp o ic line.
We ind he asymp o ic lines o a hype bolic pa-
aboloid. The hype bolic pa aboloid is a su ace o he
second o de and is gi en by he ollowing second-o -
de equa ion: 𝑧=𝑥2−𝑦2
We w i e he pa ame ic equa ions o he hype -
bolic pa aboloid:
𝑥=𝑢,𝑦=𝑣,𝑧=𝑢2−𝑣2
To calcula e he i s and second quad a ic o ms ,
we need o know he ec o s deno ed by
𝑟𝑢
󰇍
󰇍
󰇍

={1,0,2𝑢} 𝑟𝑣
󰇍
󰇍
󰇍

={1,0,−2𝑣} 𝑟𝑢𝑢
󰇍
󰇍
󰇍
󰇍
󰇍

={0,0,2}
𝑟𝑢𝑣
󰇍
󰇍
󰇍
󰇍
󰇍

={0,0,0} 𝑟𝑣𝑣
󰇍
󰇍
󰇍
󰇍
󰇍

={0,0,−2}
The coe isien s o he i s and second quad a ic
o ms a e 𝐸=1+4𝑢2,𝐹=−4𝑢𝑣,𝐺=1+4𝑣2,
𝐿= 2
√1+4𝑢2+4𝑣2,𝑀=0,𝑁= −2
√1+4𝑢2+4𝑣2
We cons uc he di e en ial equa ion o asymp-
o ic lines: 𝑑𝑢2−𝑑𝑣2=0
I s solu ions a e:
𝑢1=𝑡+𝑐1, 𝑢2=−𝑡+𝑐
Thus, he equa ions o he asymp o ic lines o he
hype bolic pa aboloid in space a e:
𝛾1:{𝑥=𝑡+𝑐1
𝑦=𝑡
𝑧=2𝑐1𝑡+𝑐12 𝛾2:{𝑥=𝑡+𝑐2
𝑦=𝑡
𝑧=2𝑐2𝑡+𝑐2
2
Asymp o ic lines o a one-shee hype boloid
A one-shee hype boloid is a quad a ic su ace o
second o de , gi en by he ollowing equa ion:
𝑥2
𝑎2+𝑦2
𝑎2−𝑧2
𝑎2=0
Pa ame ic equa ions o he one-shee hype boloid
𝑥=𝑐𝑜𝑠𝑢𝑐ℎ𝑣,𝑦=𝑏𝑠𝑖𝑛𝑢𝑐ℎ𝑣,𝑧=𝑐𝑠ℎ𝑣,
O , in ec o o m:
𝑟={𝑎𝑐𝑜𝑠𝑢𝑐ℎ𝑣,𝑎𝑠𝑖𝑛𝑢𝑐ℎ𝑣,𝑎𝑠ℎ𝑣}
To compu e he i s and second quad a ic o ms,
we calcula e he pa ial de i a i es:
𝑟𝑢={−𝑎𝑠𝑖𝑛𝑢𝑐ℎ𝑣,𝑎𝑐𝑜𝑠𝑢𝑐ℎ𝑣,0}, 𝑟𝑢𝑢
={−𝑎𝑐𝑜𝑠𝑢𝑐ℎ𝑣,−𝑎𝑠𝑖𝑛𝑢𝑐ℎ𝑣,0}, 𝑟𝑣
={𝑎𝑐𝑜𝑠𝑢𝑠ℎ𝑣,𝑎𝑠𝑖𝑛𝑢𝑠ℎ𝑣,𝑎𝑐ℎ𝑣},
𝑟𝑣𝑣 ={𝑎𝑐𝑜𝑠𝑢𝑐ℎ𝑣,𝑎𝑠𝑖𝑛𝑢𝑐ℎ𝑣,𝑎𝑠ℎ𝑣},𝑟𝑢𝑣
={−𝑎𝑠𝑖𝑛𝑢𝑠ℎ𝑣,𝑎𝑐𝑜𝑠𝑢𝑠ℎ𝑣,0}
The calcula ion o he i s quad a ic o ms yields
𝐸=𝑟𝑢2
󰇍
󰇍
󰇍
󰇍

=𝑥𝑢
2+𝑦𝑢
2+𝑧𝑢
2𝐹=𝑟𝑢𝑣
󰇍
󰇍
󰇍
󰇍
󰇍

=𝑥𝑢𝑥𝑣+𝑦𝑢𝑦𝑣+𝑧𝑢𝑧𝑣𝐺=𝑟𝑣2
󰇍
󰇍
󰇍
󰇍

=𝑥𝑣
2+𝑦𝑣2+𝑧𝑣
2
Danish Scien i ic Jou nal No100, 2025 19
𝐸=𝑎2𝑠𝑖𝑛2𝑢𝑐ℎ2𝑣+𝑎2𝑐𝑜𝑠2𝑢𝑐ℎ2𝑣=𝑎2𝑐ℎ2𝑣,𝐹
=−𝑎2𝑠𝑖𝑛𝑢𝑐𝑜𝑠𝑢𝑐ℎ𝑣𝑠ℎ𝑣
+𝑎2𝑠𝑖𝑛𝑢𝑐𝑜𝑠𝑢𝑐ℎ𝑣𝑠ℎ𝑣=0
, 𝐺=𝑎2𝑐𝑜𝑠2𝑢𝑠ℎ2𝑣+𝑎2𝑠𝑖𝑛2𝑢𝑠ℎ2𝑣+
𝑎2𝑐ℎ2𝑣=𝑎2𝑠ℎ2𝑣+𝑎2𝑐ℎ2𝑣=𝑎2𝑐ℎ2𝑣
The calcula ion o he second quad a ic o ms
yields
𝐿=
∣
∣
∣
∣
𝑥𝑢𝑢 𝑦𝑢𝑢 𝑧𝑢𝑢
𝑥𝑢𝑦𝑢𝑧𝑢
𝑥𝑣𝑦𝑣𝑧𝑣
∣
∣
∣
∣
√𝐸𝐺−𝐹2,𝑀=
∣
∣
∣
∣
𝑥𝑢𝑣 𝑦𝑢𝑣 𝑧𝑢𝑣
𝑥𝑢𝑦𝑢𝑧𝑢
𝑥𝑣𝑦𝑣𝑧𝑣
∣
∣
∣
∣
√𝐸𝐺−𝐹2,𝑁
=
∣
∣
∣
∣
𝑥𝑣𝑣 𝑦𝑣𝑣 𝑧𝑣𝑣
𝑥𝑢𝑦𝑢𝑧𝑢
𝑥𝑣𝑦𝑣𝑧𝑣
∣
∣
∣
∣
√𝐸𝐺−𝐹2
𝐿=−𝑎3𝑐ℎ3𝑣, 𝑀=0, 𝑁=2𝑎3𝑐ℎ𝑣
The di e en ial equa ion o he asymp o ic lines is
de i ed as ollows
𝐿𝑑𝑢2+2𝑀𝑑𝑢𝑑𝑣+𝑁𝑑𝑣2=0
−𝑎3𝑐ℎ3𝑣𝑑𝑢2+2𝑎2𝑐ℎ𝑣𝑑𝑣2=0
𝑐ℎ2𝑣𝑑𝑢2=2𝑑𝑣2
{𝑑𝑢
𝑑𝑣}2=2
𝑐ℎ2𝑣
∫𝑑𝑢=∫ √2
𝑐ℎ𝑣𝑑𝑣=∫ 2√2
𝑒𝑣+𝑒−𝑣𝑑𝑣=2√2∫ 𝑒𝑣𝑑𝑣
𝑒2𝑣+1
=𝑑𝑒𝑣
(𝑒𝑣)2+1
The in insic coo dina e equa ions o he asymp-
o ic lines can be exp essed in he o m
𝑢1=2√2𝑎𝑟𝑐𝑡𝑔𝑒𝑣
𝑢2=−2√2𝑎𝑟𝑐𝑡𝑔𝑒𝑣
The spa ial equa ions o he asymp o ic lines o he
one-shee hype boloid can be w i en in he o m
𝛾1:𝑥=𝑎𝑐𝑜𝑠(2√2𝑎𝑟𝑐𝑡𝑔𝑒𝑡)𝑐ℎ𝑡,𝑦
=𝑎𝑠𝑖𝑛(2√2𝑎𝑟𝑐𝑡𝑔𝑒𝑡)𝑐ℎ𝑡,𝑧=𝑎𝑠ℎ𝑡
𝛾2:𝑥=𝑎𝑐𝑜𝑠(2√2𝑎𝑟𝑐𝑡𝑔𝑒𝑡)𝑐ℎ𝑡,𝑦
=−𝑎𝑠𝑖𝑛(2√2𝑎𝑟𝑐𝑡𝑔𝑒𝑡)𝑐ℎ𝑡,𝑧
=𝑎𝑠ℎ𝑡
Re e ences:
1. A. Na mano . Di e ensial geome iya a
opologiya. (1), (2018).
2. M.A.Sobi o , A.Y. Yusupo . Di e ensial ge-
ome iya ku s., (2) (1959), 158
3. A.Ya.Na mano , A.S.Sha ipo ,
J.O.A slono . Di e ensial geome iya a opologiya
ku sidan masalala o’plami. (2014)