쌍곡선

수학에서 쌍곡선(/haɪpːrbələ/(듣기), pl. 하이퍼볼라 또는 하이퍼볼라 /-li//(듣기); adj.쌍곡선 /쌍곡선 /쌍곡선 pər lyingbɒlɪk/(listen)는 평면에 놓여 있는 매끄러운 곡선의 한 종류로, 기하학적 특성이나 해집합인 방정식으로 정의됩니다.쌍곡선은 연결된 구성요소 또는 가지라고 불리는 두 개의 조각이 있는데, 이것은 서로의 거울상이고 두 개의 무한 활과 유사합니다.쌍곡선은 평면과 이중 원뿔의 교점에 의해 형성된 세 종류의 원뿔 단면 중 하나이다.(다른 원추형 부분은 포물선과 타원형입니다.원은 타원의 특수한 경우입니다.)평면이 이중 원뿔의 양쪽 반을 교차하지만 원뿔의 정점을 통과하지 않으면 원뿔은 쌍곡선입니다.

하이퍼볼라는 여러 가지 방법으로 발생합니다.

• 직교 ^[1]평면에서 $역함수{\displaystyle }$ y${\displaystyle (x)}$ $=$ /$$ {$displaystyle$ y$(x$)= $1/x}$를 나타내는 곡선으로,
• 해시계 끝의 그림자를 따라가는 길처럼
• 열린 궤도의 형태(닫힌 타원 궤도와 구별되는 것)로서, 예를 들어 행성 또는 보다 일반적으로 가장 가까운 행성 또는 다른 중력체의 탈출 속도를 초과하는 우주선(또는 천체)의 궤도,
• 균일한 매달림 케이블이 만드는 모양(배수)과 같다.
• 아원자 입자의 산란 궤적(인력이 아닌 반발력에 의해 발생하지만 원리는 동일),
• 무선 항법에서 거리 자체가 아닌 두 지점까지의 거리 간의 차이를 결정할 수 있는 경우,

기타 등등.

쌍곡선의 각 가지에는 쌍곡선의 중심에서 더 멀리 떨어진 직선(낮은 곡률)이 되는 두 개의 암이 있습니다.대각선 반대쪽 팔(각 가지에서 하나씩)은 두 팔의 점근선이라고 불리는 공통선으로 한계선에 있는 경향이 있습니다.그래서 두 개의 점근선이 있는데, 그 교차점은 쌍곡선의 대칭 중심에 있습니다. 이것은 각 가지가 다른 가지를 형성하기 위해 반사하는 거울 점이라고 생각할 수 있습니다.$곡선{\displaystyle }$ y${\displaystyle (x}$) $=$ /$$ {$displaystyle$ y$(x$)=$1/x}$의 경우 점근점은 두 개의 좌표 ^[2]축입니다.

하이퍼볼라는 편심률, 초점, 다이렉트릭스와 같은 많은 타원 분석 특성을 공유합니다.일반적으로 대응은 특정 용어의 부호 변경만으로 이루어집니다.쌍곡선 포물면(안장 표면), 쌍곡선 기하학(로바체프스키의 유명한 비유클리드 기하학), 쌍곡선 함수(sinh, cosh, tanh 등), 자이로벡터 공간(상대성과 양자 모두에서 사용하기 위해 제안된 기하학)과 같은 많은 다른 수학적 물체들은 쌍곡선에서 기원을 가지고 있다.유클리드 이외의 역학).

어원과 역사

"하이퍼볼라"라는 단어는 "지나친" 또는 "지나친"을 뜻하는 그리스어 ββ βο λ에서 유래했으며, 영어 용어인 "하이퍼볼"도 여기에서 유래했다.메네크무스에 의해 입방체를 두 배로 하는 문제에 대한 그의 연구에서 하이퍼볼레는 발견되었지만, 그 후 둔원추의 ^[3]단면이라고 불렸다.쌍곡선이라는 용어는 페르가의 아폴로니우스(기원전 262년–기원전 190년)가 원추형 부분인 ^[4]원추형 부분에 대한 그의 최종 연구에서 만든 것으로 여겨진다.타원과 포물선의 다른 두 일반적인 원추형 부분의 이름은 "부족한"과 "적용된"을 뜻하는 그리스 단어에서 유래했다; 세 가지 이름 모두 고정된 영역의 직사각형 변과 주어진 선분의 비교를 참조한 초기 피타고라스 용어에서 유래했다.직사각형을 세그먼트에 "적용"하거나(같은 길이를 가지거나), 세그먼트보다 짧거나 ^[5]세그먼트를 초과할 수 있습니다.

정의들

점의 궤적으로서

쌍곡선은 유클리드 평면에서 점 집합(점 위치)으로 기하학적으로 정의할 수 있습니다.

쌍곡선은 포인트 세트입니다.따라서 세트의 임의의 $포인트{\displaystyle }$ P$(\$$displaystyle$ P$)$에 대해 2개의 고정 $포인트{\displaystyle {}_{}}$ ${\displaystyle {F1\mathrm{}}_{}}$ $F2(\$$1}, PF_${2}$}$에 대한 ${\displaystyle 거리}$ ${\displaystyle {PF1\mathrm{}}_{},{}_{}}$ ${\displaystyle {}_{}}${1}, $PF_$${2$} $$}의 절대적인 차이는 보통 foci로 표시됩니다.$displaystyle 2a,a,a>$

${\displaystyle H=\{P: PF_{2} - PF_{1} = 2a\}\ .}$

Foci를 연결하는 선분의 $중간점{\displaystyle }$ M$(\displaystyle$ M$)$을 쌍곡선의 ^[7]중심이라고 합니다.Foci를 통과하는 선을 장축이라고 합니다.중앙까지의 거리$({displaystyle$a})를$$ 갖는 V 1, V $({$의 정점을 포함합니다.중심까지의 Foci $거리{\displaystyle }$ c$\displaystyle$c를$$ 초점 거리 또는 선형 편심이라고 합니다.${\displaystyle \frac{c}{}}$의 비율$(\$$displaystyle$ ${c}{a})$은 $편심률입니다.$

${\displaystyle {}_{}}$ F${\displaystyle {}_{}{2\mathrm{}}_{}}$ - ${\displaystyle {}_{}}$ ${\displaystyle {F\mathrm{}}_{}}$ 1 ${\displaystyle ={}_{}}$ ${\displaystyle 2}$a { $displaystyle PF$_ {2$} - PF$_ {$1}$=$2a}$ 등식은$$ 다른 방식으로 볼 수 있습니다(그림 참조).
${\displaystyle {c\mathrm{}}_{}}$ 2$({$가 ${\displaystyle {중간점\mathrm{}}_{}}$F2({$displaystyle F_{$2$$})이고 ${\displaystyle \mathrm{반지름}}$ 2({$displaystyle$2a$\})$가 중간점 F2({$displaystyle F_{1$})인 경우 $오른쪽{\displaystyle {}_{}}$ 분기의 점$$P({$displaystyle$P})에서 원 $({$2}})까지의$$ 거리는 ${$1})까지의 거리입니다.:

${\displaystyle PF_{1} = PC_{2} .}$

${\displaystyle {c\mathrm{}}_{}}$ 2$({$는 ^[8][9]쌍곡선의 원형 다이렉트릭스(${\displaystyle {초점\mathrm{}}_{}}$F2({$displaystyle F_$2}})라고 불립니다.쌍곡선의 왼쪽 가지를 얻으려면 $({$과 $관련$된 원형 직행렬을 사용해야 합니다.이 속성은 아래 직행(선)의 도움을 받는 쌍곡선의 정의와 혼동해서는 안 됩니다.

방정식이 y = A/x인 쌍곡선

xy-좌표계가 원점에 대해 각도 $+{\displaystyle }$ $(\$만큼 ${\displaystyle \mathrm{회전}}$하고 새로운 $,(\displaystyle$ \$xi,\$eta$$$})$가 $할당$되면 x ${\displaystyle =}$ ${\displaystyle =\frac{}{}}$ = = = $t$ +${\displaystyle \frac{}{\sqrt{\mathrm{}}}}$ 2$(\$ x=$tfraci$+\$xi$ +\tfraci eta eta } {\$rt$}) {rt} {rt} {rt} {rt} {rt} {r}} {rt} {rt}} {
직사각형 $쌍곡선{\displaystyle \frac{{}^{}}{}}$ ${\displaystyle \frac{{x\mathrm{}}^{}}{}}$2 - ${\displaystyle \frac{{}^{}}{}}$ ${\displaystyle \frac{{}^{}}{{}^{}}}$ ${\displaystyle \frac{{a\mathrm{}}^{}}{}}$ 2 ${\displaystyle \frac{{}^{}}{}}$ {\$displaystyle {x^{2}-y^{2$}}$=$$1$ (반각형이 동일)에는 2 ${\displaystyle \frac{{}^{}}{}}$ ${\displaystyle \frac{{a\mathrm{}}^{}}{}}$ ${\displaystyle \frac{{}^{}}{}}$ ${\displaystyle 1}$(\$displaystyle {2\xi \eta}$ {$a}}$ $=$ $1$의 $새로운{\displaystyle }$ $방정식$이 있습니다.$ystyle \eta = paramenttfrac {a^{2}/2}{\xi }}\ .}$

따라서 xy 좌표계에서 f: , A> ,\ $displaystyle$ f$:x$\$map$ to \ $tfrac${ A} ,\ ;$A>0\;,}$ (방정식 포함)

, A>, {\$displaystyle$ y =$flac {A}$ {$x}$ \ ; , $A$> $0$\ ; , } 은 1 사분원과 3 사분원에 모두 있는 직사각형 쌍곡선입니다.

• 좌표 축을 점근선으로 표시,
• $y{\displaystyle }$ ${\displaystyle =}$ {$displaystyle$ y$=x}$ 선을$$ 장축으로 한다.
• 중심${\displaystyle }$($0$ $)$ {${\displaystyle \mathrm{displaystyle}}$ (${\displaystyle \sqrt{0},}$$0)}$ 및$$ $반축{\displaystyle }$ a ${\displaystyle =}$ b ${\displaystyle =}$ 2 A$$, \$displaystyle$ a$$$=$b$$=$scrt {2$$A}}\;,}$
• 꼭지점$({\displaystyle A\sqrt{}}$ ,${\displaystyle A\sqrt{}}$)${\displaystyle }$,${\displaystyle }$( -${\displaystyle }$A${\displaystyle }$)$$, ${\displaystyle$\left ( {\$sqrt$ {A$}}$ , {\$sqrt {A}}$ \$right$} , \left ( - {\$sqrt {A$}}} , - {\$sqrt$ {A} \right } ) \ , \ ; ,$$}
• $정점{\displaystyle }$ p ${\displaystyle =}$ ${\displaystyle a}$ ${\displaystyle =}$ ${\displaystyle 2\sqrt{\mathrm{}}\sqrt{A}}$ ,$${\$displaystyle p=$a=$sqrt {2$}의 반직장 및 곡률 반지름$A}}\;,}$
• 선형 $편심{\displaystyle }$ c ${\displaystyle =}$ ${\displaystyle 2\sqrt{}}$ A$({$ c$=2sqrt {A}})$ 및$$ $편심{\displaystyle }$ e $({displaystyle$ e$=sqrt {2}}\;)$
• $접선{\displaystyle }$ y $=$ - ${\displaystyle \frac{{}_{}^{}}{}}$ ${\displaystyle \frac{{x\mathrm{}}_{}^{}}{}}$ 0 ${\displaystyle \frac{{\mathrm{}}_{}^{}}{}\frac{{}_{}^{}}{}}$ + ${\displaystyle 2}$ ${\displaystyle A\frac{}{}}$ ${\displaystyle \frac{{}_{}}{}}$ ({$displaystyle$ y=-{\$tfrac {A}$ {$x_{0}})x$+2${\tfrac {A}$ {$x_{0$$\}\}\}\}\}.{\displaystyle }$($x$0,${\displaystyle A}$ / ${\displaystyle 0)}$ . {displaystyle $(x_0}, A/x_0},$ {0}, {0}, {0}, {0}, {0}, {\}, {\}, {\}, {\}, {\}, {\}, {\}, {\}, {\$}$

원래의 쌍곡선을 $({$ 스타일 $-45^{\circ}})$ 회전시키면$$ 2사분면과 4사분면에 모두 직사각형 쌍곡선이 생성되며 +$($ 스타일 $+45$)의 경우와${\displaystyle }$같은 점근선, 중심, 반직선, 정점에서의 곡률 반지름, 선형 편심 및 편심 및 편심률$^{\contract }}$ 회전$$, 방정식 포함

${\displaystyle y=syslogfrac {-A}{x}\;,A>0\;,}$

• $반신호{\displaystyle }$ a ${\displaystyle =}$ ${\displaystyle b}$ ${\displaystyle =}$ ${\displaystyle \sqrt{2}}$A$$, {\$displaystyle$ a$=b=sembrt {2$}$A}}\;,}$
• y $=$ -$$x (\$displaystyle$ y=-$x)$ $라인$은 장축으로 한다.
• ${\displaystyle 정점\sqrt{}}{\displaystyle }$(${\displaystyle }$-${\displaystyle A\sqrt{}}$ ,${\displaystyle }$A${\displaystyle }$) ,${\displaystyle }$( ${\displaystyle A\sqrt{}}$,${\displaystyle }$- A${\displaystyle }$)$$. { $displaystyle \left$ ( - { \ $sqrt${$$A$$} , { \ $sqrt { A$} } \ $right$} , \$left$ ( { \ $sqrt$ { A$$} } , - { \$sqrt$ { $A$} } } ）$$。$}$

${\displaystyle \frac{\mathrm{방정식}}{}}$ ${\displaystyle y\frac{}{}}$ ${\displaystyle =}$ A ${\displaystyle \mathrm{x}}$ ,${\displaystyle }$A${\$ 0 , {\$displaystyle$ y =$flac {A}$ {$x}$ ,\ $A\neq 0\}$을 사용하여 쌍곡선을 이동하여 새 중심이 (${\displaystyle {}_{}}$ ${\displaystyle {0\mathrm{}}_{}}$, ${\displaystyle {d\mathrm{}}_{}}$ ${\displaystyle {}_{}}$ )$${ $displaystyle$( $c_{0$ , $d_{$0$\}\})$이 되도록 하면 새로운 방정식이 생성됩니다.

${\displaystyle y=black {A}{x-c_{0}}+d_{0}}\;,}$

새로운 점근선은 x ${\displaystyle =}$ ${\displaystyle {}_{}}$({$displaystyle$ x$=c_{0})$ $및$ y ${\displaystyle =}$ ${\displaystyle {d\mathrm{}}_{}}$0({$displaystyle$ y$=d_{0$입니다.
형상 $파라미터{\displaystyle }$ a${\displaystyle ,}$ ${\displaystyle b,}$ ${\displaystyle p,}$ ${\displaystyle c,}$(\$displaystyle$ a,$b, p, c, e)$는 변경되지 않습니다.

directrix 속성으로

중심에서 2 $\frac{{}^{}}{}$ $a$ {\$textstyle$ d$=blashfrac {a^{2}}{c}}$ $거리$에 있는 두 선을 쌍곡선의 다이렉트리스라고 합니다(그림 참조).

쌍곡선의 임의의 $점{\displaystyle }$P {\$displaystyle$ P$}$의 경우 한 초점 및 해당 직행렬에 대한 거리의 몫(그림 참조)은 편심과 같습니다.

${\displaystyle {\frac { PF_{1} } = phcfrac { PF_{2} } } = e = phcfrac {c} {a} \ }$

$쌍{\displaystyle {}_{}}$ ${\displaystyle {F1\mathrm{}}_{}}$ $({$$displaystyle$ $F_{1}, l_{1})$의 ${\displaystyle 증명}$은 ${\displaystyle {}_{}}$ ${\displaystyle {}^{}{}_{}{}^{\mathrm{}}}$ 2 ${\displaystyle =}$ (${\displaystyle x}$ ${\displaystyle -{}^{}c}$ )${\displaystyle {}^{}{}^{}}$ + $y$2, ${\displaystyle {Pl\mathrm{}}_{}}$ 1 ${\displaystyle {}^{}{}_{}{}^{\mathrm{}}}$ ${\displaystyle {}^{}}$ ($x$- ${\displaystyle {a\frac{{}^{}}{}}^{}}$ ${\displaystyle {\frac{{}^{}}{}}^{}}$c ) ${\displaystyle {\mathrm{}}^{}}$^{2$} =$ ($x-c)^{2}$ ^{1} ^{1} ^{$1} }$ } \ 。$2$ ${\displaystyle \frac{{a\mathrm{}}^{}}{}}$ ${\displaystyle \frac{{}^{}}{}{}^{}}$ x${\displaystyle {}^{}{2\mathrm{}}^{}}$ - ${\displaystyle {}^{}}$ {\$style$ y$^{2$} =$btfrac${$b^{2$} } {$a^{2}$ $x^{2}-b^{$2$$} } 2 2 2 、 2 、 2 、 2 、 2 2 、 2 2 2 。

${\displaystyle PF_{1}^{2}-{\frac {c^{2}}{a^{2}} Pl_{1}^{2}=0\.}$

두 번째 사례는 비슷하게 증명되었다.

역문도 참이며 포물선의 정의와 유사한 방식으로 쌍곡선을 정의하는 데 사용할 수 있습니다.

어떤 지점 F{F\displaystyle}(초점을 맞추), 나는{나는\displaystyle}(준선)F{F\displaystyle}과 e을과 어떤 실수 e{\displaystyle e}을 통해서가 아니라 어떤 라인을 위해 1점의 거리별 지점과 라인에에 몫을 집합(포인트의 활동 중심지),{\displaystyle e> 1}e. {\disp$레이스타일 e}$

${\displaystyle H=\left\{P},{\Biggr},{\frac {PF}{Pl }}=e\right\}}$

는 쌍곡선입니다.

($선택{\displaystyle }$ 사항 e ${\displaystyle =}$ {$display$ e$=1$}은$$ 포물선을 생성하고 e <1 {$display$ e$<1$}인 경우 타원을 생성합니다.)

증명

F ( ,) , > { $displaystyle$ F = ( $f$ , 0 , \ e$>$ 0 } 및$$ (${\displaystyle 0}$ , 0${\displaystyle }$)$${ $displaystyle ( 0$ ,$$ )이 곡선상의 점이라고${\displaystyle }$합니다.$directrix{\displaystyle }$l {\$displaystyle$ l$}$의 $방정식$은 x $=$ - ${\displaystyle \frac{f}{}}$e {\$tfrac${$f$$}{e$입니다. P ${\displaystyle =}$ (${\displaystyle x}$ ,${\displaystyle y}$) { $displaystyle$ P= ($x$ ,$y$ )$$}의 $경우$, ${\displaystyle P}$ ${\displaystyle {}^{\mathrm{}}}$ ${\displaystyle {}^{}}$ ${\displaystyle {}^{}}$ ${\displaystyle e{}^{}}$ ${\displaystyle {2\mathrm{}}^{}}$ L =$e ^$2$} ^2$ } ^2 ^2 ^2 ^2 ^2$$^2 ^2 ^2 ^2 ^2 ^2 ^2 ^2 ^2 ^2 ^2 ^2 ^2 ^2 ^2 ^2

$display$$= e^{2}\left(x+{\tfrac {f}{e}}\right$)^{2$}=(ex+f)^{$2$$}}${\displaystyle {}^{}{및\mathrm{}}^{}}$ ${\displaystyle {}^{}}$ $2{\displaystyle {}^{}{}^{}\mathrm{e}}$2${\displaystyle }$- ${\displaystyle 1}$)+${\displaystyle 2\mathrm{x}}$f ($1$+ ${\displaystyle {}^{}}$ ) -${\displaystyle {}^{}}$y $=$ 0. {$display$style $x^{2}(e^{2}-1$)+$2+e-y}^0^2$$}.$$}$

$대체{\displaystyle }$ p ${\displaystyle =f}$ ( 1${\displaystyle }$+${\displaystyle e}$) {$displaystyle$ p$=f(1$ + $e)}$의 산출량

${\displaystyle x^{2}(e^{2}-1)+2px-y^{2}=0.}$

이는 타원( < \ $displaystyle$e <$1$) 또는 포물선( ${\displaystyle }$ ${\displaystyle =}$ \ $displaystyle$ e $=$$1$) 또는 쌍곡선( > \ $displaystyle$ e >$1$)의 방정식입니다.이러한 모든 비퇴화 원뿔은 공통적으로 정점으로 원점을 가지고 있습니다(그림 참조).

${\displaystyle e}$> { $displaystyle$e $> 1$ 인 $경우{\displaystyle }$ ${\displaystyle {e\mathrm{}}^{}}$2 - ${\displaystyle 1}$ ${\displaystyle =}$ ${\displaystyle \frac{{b\mathrm{}}^{}}{}}$ 2 ${\displaystyle \frac{{a\mathrm{}}^{}}{}2}$ , p ${\displaystyle =}$ ${\displaystyle b\frac{{}^{}}{}}$ ${\displaystyle \frac{{2\mathrm{}}^{}}{}}$ { $displaystyle$ e$}$ - 1$$= b $tfrac${ $b$} { a$$} , { \ $text${$$} } \ p$$=$frac$ { b$$} { a${\displaystyle }$} , b 、 b$$、 b$$}$$、 $b{\displaystyle {}^{}}$ 、 b 。

${\displaystyle\tfrac{(x+a)^2}}{a^{2}}-{\tfrac {y^{2}}{b^{2}}=1\,}$

이것은 중심${\displaystyle (}$ -${\displaystyle }$a ,${\displaystyle 0}$ )$${$displaystyle (a,0$ {x축은 장축, 장축/$장축{\displaystyle }$ a${\displaystyle ,}$ b$${$displaystyle$ a$,b$ {displaystyle a,b}의 쌍곡선 방정식입니다.

다이렉트릭스 구축

${\displaystyle \frac{{c\mathrm{}}^{}}{}a\frac{{}^{}}{}}$ ${\displaystyle \frac{{}^{}}{}}$ 2 ${\displaystyle {}^{}}$ ${\displaystyle =}$ a$$ { { $display style$c \$cdot$ { $a$^ { c } =$a^$ {$$2} } $l$ 、 $directrix{\displaystyle {}_{}}$ ${\displaystyle {l\mathrm{}}_{}}$1$$（ $display$ $style$ l _ {$$$1}$） ${\displaystyle {l\mathrm{}}_{}}$ focus $focus{\displaystyle {}_{}}$ f f f f ${\displaystyle {F1\mathrm{}}_{}}$ $focus$ focus$$ focus focus 2 2 ${\displaystyle {22\mathrm{}}^{}}$x ${\displaystyle {circle2}^{}}$의 ${\displaystyle {}^{}}$과 반대입니다.$displaystyle x^{2}+y^{2$$}=a^{$2}}($$그림 녹색).$따라서{\displaystyle {}_{}}$ 점 ${\displaystyle {E\mathrm{}}_{}}$ 1$({$은 탈레스의 정리를 사용하여 구성할 수 있습니다(그림에는 표시되지 않음).$다이렉트릭스{\displaystyle {}_{}}$ $({$은 $({$을 통과하는 ${\displaystyle \stackrel{}{{}_{}{}_{}}}$ ${\displaystyle \stackrel{}{{F2\mathrm{}}_{}}}${\({$displaystyle {F_1})$에 수직입니다.
${\displaystyle {E1}_{}}$의$대체{\displaystyle {}_{}}$ 구성({$displaystyle E_{1$ 계산 결과 $({$과 $({$ $사이$의 수직이 교차하는 것으로 나타났습니다(그림 참조).

원뿔의 평면 섹션으로

원뿔 위의 선 기울기보다 큰 정점을 통과하지 않는 평면과 직립 이중 원뿔의 교점은 쌍곡선입니다(그림: 빨간색 곡선 참조).쌍곡선의 정의 특성을 증명하기 위해 2개의 Dandelin 구 ${\displaystyle {d1\mathrm{}}_{}}$, d2$displaystyle d_{1$}, $d_{2$를 사용합니다.이는 원 $({$ $({$}}) 및 $({$})에서 원뿔에 접하는 구입니다.$splaystyle F_{1$} $및$ $({$ 알고 $,$ $F2$({$style F_{1}, F_${2$})$는 쌍곡선의 포치입니다.

1. P$(\displaystyle$ P$)$를 교차곡선의 임의의 점으로 합니다.
2. P$(\displaystyle$ P$)$를 $포함$하는 콘의 생성기는$$A $지점$에서 원 $(\$$displaystyle$$c_{1})$과$$B $지점$에서 $2$(\$displaystyle$$c_{2})$와 교차합니다$$.
3. $선분{\displaystyle \stackrel{}{}}$ ${\displaystyle \stackrel{}{{}_{}}}$ ${\displaystyle \stackrel{}{{F\mathrm{}}_{}}}$ 1$${\({$displaystyle {PF_{1}}})$ 및$$ $P{\displaystyle \stackrel{}{}}$ ${\displaystyle \stackrel{}{A}}$¯({$displaystyle {PA})$는 $구{\displaystyle {}_{}}$ $({$}})에 접해 있으므로 길이가 같습니다.
4. $선분{\displaystyle \stackrel{}{}}$ ${\displaystyle \stackrel{}{{}_{}}}$ ${\displaystyle \stackrel{}{{F\mathrm{}}_{}}}$ 2$${\({$displaystyle {PF_{2}}})$ 및$$ ${\displaystyle \stackrel{}{P}}$ B$$¯({$displaystyle {PB})$는 $구{\displaystyle {}_{}}$ $({$}})에$$ 접선하여 길이가 같습니다.
5. ${\displaystyle 그}$ 결과: ${\displaystyle {P\mathrm{}}_{}}$ F${\displaystyle {}_{}1}$ - ${\displaystyle {P\mathrm{}}_{}}$ F ${\displaystyle 2}$ ${\displaystyle ={}_{}}$ ${\displaystyle PA}$ - ${\displaystyle P}$ ${\displaystyle B}$ ${\displaystyle =}$ ${\displaystyle A}$ B({$displaystyle$ PF_${1}$ - $PF_{2} = PA$ - $PB$=$AB$}는 쌍곡선 $점{\displaystyle }$P({$displaystyle$$$P$\})$와 독립적입니다. 왜냐하면$$P는 P({$displaystyle$ P}의 위치에 $관계없이{\displaystyle }$,$${\$displaystyle c_{1$}},$$ ${\displaystyle {c2\mathrm{}}_{}}${\$displaystyle c_{2$ 및 $(\displaystyle$ AB$)$는 정점을 통과해야 합니다$.$따라서 $P점({$displaystyle P})이 빨간색 곡선(하이퍼볼라)을 따라$$ 이동함에 따라 $선분{\displaystyle \stackrel{}{}}$ ${\displaystyle \stackrel{}{A}}$ B$${\({$displaystyle {AB})$는 길이를 변경하지 않고 정점 중심으로 회전합니다.

핀 및 스트링 구조

핀, 문자열 및 ^[10]자를 사용하여 호를 그리는 데 쌍곡선의 정의(위 참조)를 사용할 수 있습니다.

(0) ${\displaystyle {}_{}}$ ${\displaystyle {}_{}{}_{}}$ $({$}, $F_{2$ $정점{\displaystyle {}_{}}$ ${\displaystyle {V1\mathrm{}}_{}}$ $({$},$V_{$2}}) 및$$ 원형 다이렉트리스 중 하나를 선택합니다($예{\displaystyle \mathrm{}}$: ${\displaystyle \mathrm{c2}}$$(\$$displaystyle$$c_$${$$2$$\}\}).$
(1) 눈금자는 ${\displaystyle {F2\mathrm{}}_{}}지점{\displaystyle {}_{}}$({$디스플레이 스타일 F_{$2$$})에 고정되며$,$ ${\displaystyle {F2\mathrm{}}_{}}$ $지점$({$디스플레이$ 스타일$F_$2$})$을 $중심$으로 자유롭게 회전합니다. B $지점{\displaystyle }$({$디스플레이$ 스타일$B})$은 거리 2 a$({$ $스타일 2a$에 표시됩니다.
(2) ${\displaystyle 길이}$ ${\displaystyle A}$ B$displaystyle$ AB$)$의 스트링을 준비한다.
(3) 스트링의 한쪽 끝은 눈금자의$A점{\displaystyle }$(\$displaystyle$ A$)$에 고정되고 다른 한쪽 끝은 $(\$에 고정됩니다.
(4) 펜을 들고 끈을 자 가장자리에 꽉 잡는다.
(5) P ${\displaystyle {F\mathrm{}}_{}}$ 1 ${\displaystyle ={}_{}}$ ${\displaystyle P}$ B${\displaystyle }$= $PB$=$PB }$ ${\displaystyle 때문}$에 ${\displaystyle {F\mathrm{}}_{}}$ 2$({$$displaystyle$ $F_$${2})$ 주위에$$ $눈금자$를 돌리면 펜이 쌍곡선의 오른쪽 가지 호를 그리도록 지시됩니다(원형 직접 행렬에 의한 쌍곡선의 정의 참조).

쌍곡선의 스타이너 생성

쌍곡선의 단일 점을 구성하는 다음 방법은 비퇴화 원뿔 단면의 Steiner 생성에 의존합니다.

2개의 $연필{\displaystyle }$ B${\displaystyle (U),}$ ${\displaystyle B}$$(V)$ $2개$의 점${\displaystyle ,}$ V$(U$$)($각각 U$(Displaystyle$$$U$)$ 및$$${\displaystyle }$ V$($$Displaystyle$ V$)$의 투시 맵핑$(Bpi$${\displaystyle )}$이 주어진 경우${\displaystyle B}$(${\displaystyle V}$) \ $displaystyle$B ( V )$。$그러면 대응하는 라인의 교차점이 비퇴화 원뿔형 단면을 형성합니다.

$쌍곡선{\displaystyle \frac{{}^{}}{}}$ ${\displaystyle \frac{{x\mathrm{}}^{}}{}}$ ${\displaystyle \frac{{}^{}}{{}^{}}}$ ${\displaystyle a\frac{{}^{}}{}\frac{{}^{}}{}}$- y ${\displaystyle \frac{{2\mathrm{}}^{}}{}}$ ${\displaystyle \frac{{b\mathrm{}}^{}}{}}$ 2 ${\displaystyle \frac{{}^{}}{}}$ ${\displaystyle 1}$ ${\displaystyle {x^{2}}$ {$a^{2}}$ - {\$tfrac {y^{2}}$ {$b^{2$}} $=1$}의 경우 V${\displaystyle {}_{}{1\mathrm{}}_{}}$, ${\displaystyle {V\mathrm{}}_{}}$2 {\$display V_1},$ $V\_2{\displaystyle }$$}$의 연필을 사용합니다$.$쌍곡선 $및{\displaystyle }$ A ${\displaystyle =}$ ( ${\displaystyle a}$, ${\displaystyle {y\mathrm{}}_{}}$0${\displaystyle }$) , ${\displaystyle B}$ $=$ ( ${\displaystyle {}_{}{0\mathrm{}}_{}}$ ,${\displaystyle 0}$) , {$displaystyle$ A=($a,y_{0}),$ B=($x_{0$ 입니다.라인 세그먼트 ${\displaystyle \stackrel{}{B}}$ P$${\({$displaystyle {BP})$는 등간격 세그먼트 n개로 분할되어 있으며, 이 분할은 $대각선{\displaystyle }$ A$$({$displaystyle$ AB$})$와 평행하게 라인 $세그먼트{\displaystyle \stackrel{}{}}$ A ${\displaystyle \stackrel{}{P}}${\({$displaystyle {AP})$에 투영되어 있습니다(그림 참조).평행 투영법은 필요한 V $($ V_${1})$과 V $($ V_${2})$ 사이의$$ 투영 매핑의 일부입니다.관련된 두 라인 ${\displaystyle {S1\mathrm{}}_{}}$ A ${\displaystyle i_{}}$\ $displaystyle S_{1}A_{i}$ 및$$ ${\displaystyle {}_{}{}_{}}$ ${\displaystyle B_{}}$i \ $displaystyle S_{2}B_{i}$의 교차점은 고유하게 정의된 쌍곡선의 지점입니다.

비고: A(디스플레이 스타일 A$)$및 $(디스플레이 스타일$ B$)$ $지점$$$ 이상으로 확대하여 점수를 더 얻을 수 있지만 교차점 판정이 부정확해질 수 있습니다.대칭에 의해 이미 생성된 점을 확장하는 것이 더 나은 방법입니다(애니메이션 참조).

비고:

1. 스타이너 세대는 타원과 포물선에도 존재한다.
2. Steiner 생성은 직사각형 대신 평행사변형으로 시작하는 정점이 아닌 다른 점을 사용할 수 있기 때문에 평행사변형 방법이라고도 합니다.

하이퍼볼라 y = a/(x - b) + c 및 3점 형태의 내접각

$y{\displaystyle }$ ${\displaystyle =}$ ${\displaystyle a\frac{}{x}}$ -${\displaystyle \frac{}{}}$b +${\displaystyle c}$ , ${\displaystyle \text{a}0}$ 0${\displaystyle }$$0$ \ $displaystyle$ y =$btfrac${ $a$} { x - b } + $c$, \$a$ \ $neq$ 0${\displaystyle {}_{}}$ 、 ( x${\displaystyle {}_{}{1\mathrm{}}_{}}$,${\displaystyle {}_{}{y\mathrm{}}_{}}$1${\displaystyle }$)${\displaystyle }$, ( ${\displaystyle {x\mathrm{}}_{}}$3, ${\displaystyle {}_{}{3\mathrm{}}_{}}$ )$$, _ $displaystyle$ ( x _ { 1 ,$_$ 1 ) , { \ $neq$ 0 ） 。형상 $파라미터{\displaystyle }$ a${\displaystyle ,}$ ${\displaystyle b,}$ $(\displaystyle$ a,$b, c)$를 결정하는 간단한 방법은 하이퍼볼라에 대해 내접$$ 각도 정리를 사용합니다.

${\displaystyle \text{y}}$ ${\displaystyle =}$ ${\displaystyle {}_{}}$ ${\displaystyle {}_{}}$x + ${\displaystyle {}_{}{1\mathrm{}}_{}}$ , y ${\displaystyle =}$ ${\displaystyle {m\mathrm{}}_{}}$ 2${\displaystyle {}_{}}$ + ${\displaystyle {}_{}{}_{}{2\mathrm{}}_{}}$, m${\displaystyle {}_{}{1\mathrm{}}_{}}$ , ${\displaystyle {m\mathrm{}}_{}}$ 2${\displaystyle {}_{}0}$ 0$${ $displaystyle$ y =$m_{1}$ $x$ + $d_{1$} \ y$$=$m_{2}$x + $d_{2}$ \ , $m_1, m_{2$} \ $neq$} 등식을$$ 사용하여 두 라인 사이의 각도를 측정하기 위해 이 quotient에서 하나의 문맥을 사용합니다.

${\displaystyle {m_{1}} {m_{2}} \ . }$

원에 대한 내접각 정리와 유사하며 다음과 같이 구한다.

하이퍼볼라에 대한 내접 각도 정리:^[11][12]

4점 ${\displaystyle P_{}}$i ${\displaystyle {}_{}{}_{}}$( ${\displaystyle x_{}}$ ${\displaystyle {}_{}}$ , ${\displaystyle \text{y}{}_{}{i}_{}}$ )${\displaystyle }$、 ${\displaystyle \mathrm{i}=1}$,${\displaystyle 2}$, ${\displaystyle 3_{}}$, 4, x ${\displaystyle {}_{}{}_{}{}_{}}$ ${\displaystyle x_{}}$ ${\displaystyle {}_{}}$, ${\displaystyle {}_{}}$ ${\displaystyle i_{}{}_{}}$ y ${\displaystyle k_{}}$, $k$ k { $displaystyle P_{i}=(x_{i},$ y_${i$}), \ $i =$ 1,$2, 3,$ 4, $x_neq_i}\ne_i}, x_i$}, x_i}, x_i}, x_i$}$의 경우.

${\displaystyle {위}_{}}$의 측정에서 P $({$ P_${3$$}}$$및$ P 4$({$displaystyle $P_{4})$의 각도가 동일한 경우에만 $방정식{\displaystyle }$ y ${\displaystyle =\frac{a}{}}$ -${\displaystyle \frac{}{}}$b +$c$ {\$displaystyle$ y$=bfrac$ {a$}{x-b}}+$c$$})인 쌍곡선 위에 네 점이 있습니다.즉, 만약

${({(x_1})})$

(증거: 간단한 계산).점들이 쌍곡선 위에 있으면 쌍곡선의 방정식은 y ${\displaystyle =a}$ /$$ {\$displaystyle$ y$=a/x}$라고 가정할 수 있습니다.)

하이퍼볼라에 대한 내접각 정리의 결과는 다음과 같습니다.

쌍곡선 방정식의 3점 형식:

$3점{\displaystyle {}_{}}$ ${\displaystyle P_{}}$ i ${\displaystyle =}$ ( x${\displaystyle {}_{}{i}_{}}$ , ${\displaystyle y_{}}$i${\displaystyle }$) , ${\displaystyle \text{i}}$ ${\displaystyle =1}$,${\displaystyle 2}$, ${\displaystyle 3}$,${\displaystyle x_{}}$ i${\displaystyle {}_{}{}_{}}$ x${\displaystyle {}_{}{k}_{}}$ , ${\displaystyle y_{}{}_{}}$ ${\displaystyle y_{}}$k , ${\displaystyle i}$k k , i ${\displaystyle k}$ k$$、 k \ \ $displaystyle$P _ { i } = ( $x$_ {$i$ , y$_${ $i$} ) , \ i$= 1, 2$, 3$, x_i${$neq$} $x_neq$ },

${\displaystyle {{\color {red}y}-y_{1}}{{\color {green}x}-x_{2}}}{{\color {red}y}-y_{2}}}=blac {{\color {\color {\color {\color {\color {\color {red}y}y}y}y}y}_{{{{1}_1}_1}_1}_{{{{1}}}}_1}}}}}}}}}$

를 선택합니다$.$

쌍곡선 x² - y² 단위의 아핀 이미지로 = 1

쌍곡선의 또 다른 정의에서는 아핀 변환을 사용합니다.

${\displaystyle {모든\mathrm{}}^{}}$ 쌍곡선은 ${\displaystyle {x\mathrm{}}^{}}$ 2 - y 2 $=${\$displaystyle$ x$^{2}-y^{2$}=$1$인 단위 쌍곡선의 아핀 이미지입니다.

파라메트릭 표현

유클리드 평면의 아핀 $변환$은 x ${\displaystyle \to }$ ${\displaystyle {\stackrel{}{}}_{}}$ ${\displaystyle {\stackrel{}{\to }}_{}{0}_{\mathrm{}}}$ + ${\displaystyle A}$ ${\displaystyle x\stackrel{}{}}$ $→$ {$displaystyle {f}$ + A${\vec {$x}} + A {\$vec$$${x$\}\}$ $형식$이며, $여기$서 A$${\$displaystyle$ A$}$는 정규 행렬(결정식은 0이 아님)이고 f ${\displaystyle {\stackrel{}{\to }}_{}}$ $display$ {$f}$ style {$x}$이다.f ${\displaystyle {\stackrel{}{\to }}_{}{\stackrel{}{}}_{}}$, f ${\displaystyle {\stackrel{}{\to }}_{}}$ ${\displaystyle 2_{\mathrm{}}}$ ${\$$displaystyle$${f}_{1},$ {\$vec {f}}_{2}}$이 $행렬{\displaystyle }$A의 열 벡터일 $경우{\displaystyle {\stackrel{}{}}_{}}$ 단위${\displaystyle }$쌍곡선${\displaystyle (\pm <img\; alt="A"\; aria-hidden="true"\; class="mwe-math-fallback-image-inline"\; src="https://wikimedia.org/api/rest\_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3"\; style="vertical-align:\; -0.338ex;\; width:1.743ex;\; height:2.176ex;">\cosh }$ ${\displaystyle }$ (${\displaystyle t}$), ${\displaystyle \sinh }$ ($t$ ${\displaystyle \in }$ R , $displaystyle$( \$pm$ \ $cosh$\ $cosh$ ( t ) , t $sin ),$

${\displaystyle {\vec {x}}={\vec {p}}(t)={\vec {f}}_{0}\pm {\vec {f}}_{1}\cosh t+{\vec {f}}_{2}\sinh t\ .}$

${\displaystyle {\vec {f}}_{0}}$ is the center, ${\displaystyle {\vec {f}}_{0}+{\vec {f}}_{1}}$ a point of the hyperbola and ${\displaystyle {\vec {f}}_{2}}$ a tangent vector at this point.

vertices

In general the vectors ${\displaystyle {\vec {f}}_{1},{\vec {f}}_{2}}$ are not perpendicular. That means, in general ${\displaystyle {\vec {f}}_{0}\pm {\vec {f}}_{1}}$ are not the vertices of the hyperbola. But ${\displaystyle {\vec {f}}_{1}\pm {\vec {f}}_{2}}$ point into the directions of the asymptotes. The tangent vector at point ${\displaystyle {\vec {p}}(t)}$ is

${\displaystyle {\vec {p}}'(t)={\vec {f}}_{1}\sinh t+{\vec {f}}_{2}\cosh t\ .}$

Because at a vertex the tangent is perpendicular to the major axis of the hyperbola one gets the parameter ${\displaystyle t_{0}}$ of a vertex from the equation

${\displaystyle {\vec {p}}'(t)\cdot \left({\vec {p}}(t)-{\vec {f}}_{0}\right)=\left({\vec {f}}_{1}\sinh t+{\vec {f}}_{2}\cosh t\right)\cdot \left({\vec {f}}_{1}\cosh t+{\vec {f}}_{2}\sinh t\right)=0}$

and hence from

${\displaystyle \coth(2t_{0})=-{\tfrac {{\vec {f}}_{1}^{\,2}+{\vec {f}}_{2}^{\,2}}{2{\vec {f}}_{1}\cdot {\vec {f}}_{2}}}\ ,}$

which yields

${\displaystyle t_{0}={\tfrac {1}{4}}\ln {\tfrac {\left({\vec {f}}_{1}-{\vec {f}}_{2}\right)^{2}}{\left({\vec {f}}_{1}+{\vec {f}}_{2}\right)^{2}}}.}$

(The formulae ${\displaystyle \cosh ^{2}x+\sinh ^{2}x=\cosh 2x,\ 2\sinh x\cosh x=\sinh 2x,\ \operatorname {arcoth} x={\tfrac {1}{2}}\ln {\tfrac {x+1}{x-1}}}$ were used.)

The two vertices of the hyperbola are ${\displaystyle {\vec {f}}_{0}\pm \left({\vec {f}}_{1}\cosh t_{0}+{\vec {f}}_{2}\sinh t_{0}\right).}$

implicit representation

Solving the parametric representation for ${\displaystyle \;\cosh t,\sinh t\;}$ by Cramer's rule and using ${\displaystyle \;\cosh ^{2}t-\sinh ^{2}t-1=0\;}$, one gets the implicit representation

${\displaystyle \det({\vec {x}}\!-\!{\vec {f}}\!_{0},{\vec {f}}\!_{2})^{2}-\det({\vec {f}}\!_{1},{\vec {x}}\!-\!{\vec {f}}\!_{0})^{2}-\det({\vec {f}}\!_{1},{\vec {f}}\!_{2})^{2}=0}$.

hyperbola in space

The definition of a hyperbola in this section gives a parametric representation of an arbitrary hyperbola, even in space, if one allows ${\displaystyle {\vec {f}}\!_{0},{\vec {f}}\!_{1},{\vec {f}}\!_{2}}$ to be vectors in space.

As an affine image of the hyperbola y = 1/x

Because the unit hyperbola ${\displaystyle x^{2}-y^{2}=1}$ is affinely equivalent to the hyperbola ${\displaystyle y=1/x}$, an arbitrary hyperbola can be considered as the affine image (see previous section) of the hyperbola ${\displaystyle y=1/x\ :}$

${\displaystyle {\vec {x}}={\vec {p}}(t)={\vec {f}}_{0}+{\vec {f}}_{1}t+{\vec {f}}_{2}{\tfrac {1}{t}},\quad t\neq 0\ .}$

${\displaystyle M:{\vec {f}}_{0}}$ is the center of the hyperbola, the vectors ${\displaystyle {\vec {f}}_{1},{\vec {f}}_{2}}$ have the directions of the asymptotes and ${\displaystyle {\vec {f}}_{1}+{\vec {f}}_{2}}$ is a point of the hyperbola. The tangent vector is

${\displaystyle {\vec {p}}'(t)={\vec {f}}_{1}-{\vec {f}}_{2}{\tfrac {1}{t^{2}}}.}$

At a vertex the tangent is perpendicular to the major axis. Hence

${\displaystyle {\vec {p}}'(t)\cdot \left({\vec {p}}(t)-{\vec {f}}_{0}\right)=\left({\vec {f}}_{1}-{\vec {f}}_{2}{\tfrac {1}{t^{2}}}\right)\cdot \left({\vec {f}}_{1}t+{\vec {f}}_{2}{\tfrac {1}{t}}\right)={\vec {f}}_{1}^{2}t-{\vec {f}}_{2}^{2}{\tfrac {1}{t^{3}}}=0}$

and the parameter of a vertex is

${\displaystyle t_{0}=\pm {\sqrt[{4}]{\tfrac {{\vec {f}}_{2}^{2}}{{\vec {f}}_{1}^{2}}}}.}$

${\displaystyle {\vec {f}}_{1} = {\vec {f}}_{2} }$ is equivalent to ${\displaystyle t_{0}=\pm 1}$ and ${\displaystyle {\vec {f}}_{0}\pm ({\vec {f}}_{1}+{\vec {f}}_{2})}$ are the vertices of the hyperbola.

The following properties of a hyperbola are easily proven using the representation of a hyperbola introduced in this section.

Tangent construction

The tangent vector can be rewritten by factorization:

${\displaystyle {\vec {p}}'(t)={\tfrac {1}{t}}\left({\vec {f}}_{1}t-{\vec {f}}_{2}{\tfrac {1}{t}}\right)\ .}$

This means that

the diagonal ${\displaystyle AB}$ of the parallelogram ${\displaystyle M:\ {\vec {f}}_{0},\ A={\vec {f}}_{0}+{\vec {f}}_{1}t,\ B:\ {\vec {f}}_{0}+{\vec {f}}_{2}{\tfrac {1}{t}},\ P:\ {\vec {f}}_{0}+{\vec {f}}_{1}t+{\vec {f}}_{2}{\tfrac {1}{t}}}$ is parallel to the tangent at the hyperbola point ${\displaystyle P}$ (see diagram).

This property provides a way to construct the tangent at a point on the hyperbola.

This property of a hyperbola is an affine version of the 3-point-degeneration of Pascal's theorem.^[13]

Area of the grey parallelogram

The area of the grey parallelogram ${\displaystyle MAPB}$ in the above diagram is

${\displaystyle {\text{Area}}={\Big }\det \left(t{\vec {f}}_{1},{\tfrac {1}{t}}{\vec {f}}_{2}\right){\Big }={\Big }\det \left({\vec {f}}_{1},{\vec {f}}_{2}\right){\Big }=\cdots ={\frac {a^{2}+b^{2}}{4}}}$

and hence independent of point ${\displaystyle P}$. The last equation follows from a calculation for the case, where ${\displaystyle P}$ is a vertex and the hyperbola in its canonical form ${\displaystyle {\tfrac {x^{2}}{a^{2}}}-{\tfrac {y^{2}}{b^{2}}}=1\ .}$

Point construction

For a hyperbola with parametric representation ${\displaystyle {\vec {x}}={\vec {p}}(t)={\vec {f}}_{1}t+{\vec {f}}_{2}{\tfrac {1}{t}}}$ (for simplicity the center is the origin) the following is true:

For any two points ${\displaystyle P_{1}:\ {\vec {f}}_{1}t_{1}+{\vec {f}}_{2}{\tfrac {1}{t_{1}}},\ P_{2}:\ {\vec {f}}_{1}t_{2}+{\vec {f}}_{2}{\tfrac {1}{t_{2}}}}$ the points

${\displaystyle A:\ {\vec {a}}={\vec {f}}_{1}t_{1}+{\vec {f}}_{2}{\tfrac {1}{t_{2}}},\ B:\ {\vec {b}}={\vec {f}}_{1}t_{2}+{\vec {f}}_{2}{\tfrac {1}{t_{1}}}}$

are collinear with the center of the hyperbola (see diagram).

The simple proof is a consequence of the equation ${\displaystyle {\tfrac {1}{t_{1}}}{\vec {a}}={\tfrac {1}{t_{2}}}{\vec {b}}}$.

This property provides a possibility to construct points of a hyperbola if the asymptotes and one point are given.

This property of a hyperbola is an affine version of the 4-point-degeneration of Pascal's theorem.^[14]

Tangent-asymptotes-triangle

For simplicity the center of the hyperbola may be the origin and the vectors ${\displaystyle {\vec {f}}_{1},{\vec {f}}_{2}}$ have equal length. If the last assumption is not fulfilled one can first apply a parameter transformation (see above) in order to make the assumption true. Hence ${\displaystyle \pm ({\vec {f}}_{1}+{\vec {f}}_{2})}$ are the vertices, ${\displaystyle \pm ({\vec {f}}_{1}-{\vec {f}}_{2})}$ span the minor axis and one gets ${\displaystyle {\vec {f}}_{1}+{\vec {f}}_{2} =a}$ and ${\displaystyle {\vec {f}}_{1}-{\vec {f}}_{2} =b}$.

For the intersection points of the tangent at point ${\displaystyle {\vec {p}}(t_{0})={\vec {f}}_{1}t_{0}+{\vec {f}}_{2}{\tfrac {1}{t_{0}}}}$ with the asymptotes one gets the points

${\displaystyle C=2t_{0}{\vec {f}}_{1},\ D={\tfrac {2}{t_{0}}}{\vec {f}}_{2}.}$

The area of the triangle ${\displaystyle M,C,D}$ can be calculated by a 2 × 2 determinant:

${\displaystyle A={\tfrac {1}{2}}{\Big }\det \left(2t_{0}{\vec {f}}_{1},{\tfrac {2}{t_{0}}}{\vec {f}}_{2}\right){\Big }=2{\Big }\det \left({\vec {f}}_{1},{\vec {f}}_{2}\right){\Big }}$

(see rules for determinants). ${\displaystyle \det({\vec {f}}_{1},{\vec {f}}_{2}) }$ is the area of the rhombus generated by ${\displaystyle {\vec {f}}_{1},{\vec {f}}_{2}}$. The area of a rhombus is equal to one half of the product of its diagonals. The diagonals are the semi-axes ${\displaystyle a,b}$ of the hyperbola. Hence:

The area of the triangle ${\displaystyle MCD}$ is independent of the point of the hyperbola: ${\displaystyle A=ab.}$

Reciprocation of a circle

The reciprocation of a circle B in a circle C always yields a conic section such as a hyperbola. The process of "reciprocation in a circle C" consists of replacing every line and point in a geometrical figure with their corresponding pole and polar, respectively. The pole of a line is the inversion of its closest point to the circle C, whereas the polar of a point is the converse, namely, a line whose closest point to C is the inversion of the point.

The eccentricity of the conic section obtained by reciprocation is the ratio of the distances between the two circles' centers to the radius r of reciprocation circle C. If B and C represent the points at the centers of the corresponding circles, then

${\displaystyle e={\frac {\overline {BC}}{r}}.}$

Since the eccentricity of a hyperbola is always greater than one, the center B must lie outside of the reciprocating circle C.

This definition implies that the hyperbola is both the locus of the poles of the tangent lines to the circle B, as well as the envelope of the polar lines of the points on B. Conversely, the circle B is the envelope of polars of points on the hyperbola, and the locus of poles of tangent lines to the hyperbola. Two tangent lines to B have no (finite) poles because they pass through the center C of the reciprocation circle C; the polars of the corresponding tangent points on B are the asymptotes of the hyperbola. The two branches of the hyperbola correspond to the two parts of the circle B that are separated by these tangent points.

Quadratic equation

A hyperbola can also be defined as a second-degree equation in the Cartesian coordinates (x, y) in the plane,

${\displaystyle A_{xx}x^{2}+2A_{xy}xy+A_{yy}y^{2}+2B_{x}x+2B_{y}y+C=0,}$

provided that the constants A_xx, A_xy, A_yy, B_x, B_y, and C satisfy the determinant condition

${\displaystyle D:={\begin{vmatrix}A_{xx}&A_{xy}\\A_{xy}&A_{yy}\end{vmatrix}}<0.\,}$

This determinant is conventionally called the discriminant of the conic section.^[15]

A special case of a hyperbola—the degenerate hyperbola consisting of two intersecting lines—occurs when another determinant is zero:

${\displaystyle \Delta :={\begin{vmatrix}A_{xx}&A_{xy}&B_{x}\\A_{xy}&A_{yy}&B_{y}\\B_{x}&B_{y}&C\end{vmatrix}}=0.}$

This determinant Δ is sometimes called the discriminant of the conic section.^[16]

Given the above general parametrization of the hyperbola in Cartesian coordinates, the eccentricity can be found using the formula in Conic section#Eccentricity in terms of coefficients.

The center (x_c, y_c) of the hyperbola may be determined from the formulae

${\displaystyle x_{c}=-{\frac {1}{D}}{\begin{vmatrix}B_{x}&A_{xy}\\B_{y}&A_{yy}\end{vmatrix}};}$

${\displaystyle y_{c}=-{\frac {1}{D}}{\begin{vmatrix}A_{xx}&B_{x}\\A_{xy}&B_{y}\end{vmatrix}}.}$

In terms of new coordinates, ξ = x − x_c and η = y − y_c, the defining equation of the hyperbola can be written

${\displaystyle A_{xx}\xi ^{2}+2A_{xy}\xi \eta +A_{yy}\eta ^{2}+{\frac {\Delta }{D}}=0.}$

The principal axes of the hyperbola make an angle φ with the positive x-axis that is given by

${\displaystyle \tan 2\varphi ={\frac {2A_{xy}}{A_{xx}-A_{yy}}}.}$

Rotating the coordinate axes so that the x-axis is aligned with the transverse axis brings the equation into its canonical form

${\displaystyle {\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1.}$

The major and minor semiaxes a and b are defined by the equations

${\displaystyle a^{2}=-{\frac {\Delta }{\lambda _{1}D}}=-{\frac {\Delta }{\lambda _{1}^{2}\lambda _{2}}},}$

${\displaystyle b^{2}=-{\frac {\Delta }{\lambda _{2}D}}=-{\frac {\Delta }{\lambda _{1}\lambda _{2}^{2}}},}$

where λ₁ and λ₂ are the roots of the quadratic equation

${\displaystyle \lambda ^{2}-\left(A_{xx}+A_{yy}\right)\lambda +D=0.}$

For comparison, the corresponding equation for a degenerate hyperbola (consisting of two intersecting lines) is

${\displaystyle {\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=0.}$

The tangent line to a given point (x₀, y₀) on the hyperbola is defined by the equation

${\displaystyle Ex+Fy+G=0}$

where E, F and G are defined by

${\displaystyle E=A_{xx}x_{0}+A_{xy}y_{0}+B_{x},}$

${\displaystyle F=A_{xy}x_{0}+A_{yy}y_{0}+B_{y},}$

${\displaystyle G=B_{x}x_{0}+B_{y}y_{0}+C.}$

The normal line to the hyperbola at the same point is given by the equation

${\displaystyle F(x-x_{0})-E(y-y_{0})=0.}$

The normal line is perpendicular to the tangent line, and both pass through the same point (x₀, y₀).

From the equation

${\displaystyle {\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1,\qquad 0<b\leq a,}$

the left focus is ${\displaystyle (-ae,0)}$ and the right focus is ${\displaystyle (ae,0),}$ where e is the eccentricity. Denote the distances from a point (x, y) to the left and right foci as ${\displaystyle r_{1}\,\!}$ and ${\displaystyle r_{2}.\,\!}$ For a point on the right branch,

${\displaystyle r_{1}-r_{2}=2a,\,\!}$

and for a point on the left branch,

${\displaystyle r_{2}-r_{1}=2a.\,\!}$

This can be proved as follows:

If (x,y) is a point on the hyperbola the distance to the left focal point is

${\displaystyle r_{1}^{2}=(x+ae)^{2}+y^{2}=x^{2}+2xae+a^{2}e^{2}+\left(x^{2}-a^{2}\right)\left(e^{2}-1\right)=(ex+a)^{2}.}$

To the right focal point the distance is

${\displaystyle r_{2}^{2}=(x-ae)^{2}+y^{2}=x^{2}-2xae+a^{2}e^{2}+\left(x^{2}-a^{2}\right)\left(e^{2}-1\right)=(ex-a)^{2}.}$

If (x,y) is a point on the right branch of the hyperbola then ${\displaystyle ex>a\,\!}$ and

${\displaystyle r_{1}=ex+a,\,\!}$

${\displaystyle r_{2}=ex-a.\,\!}$

Subtracting these equations one gets

${\displaystyle r_{1}-r_{2}=2a.\,\!}$

If (x,y) is a point on the left branch of the hyperbola then ${\displaystyle ex<-a\,\!}$ and

${\displaystyle r_{1}=-ex-a,\,\!}$

${\displaystyle r_{2}=-ex+a.\,\!}$

Subtracting these equations one gets

${\displaystyle r_{2}-r_{1}=2a.\,\!}$

In Cartesian coordinates

Equation

If Cartesian coordinates are introduced such that the origin is the center of the hyperbola and the x-axis is the major axis, then the hyperbola is called east-west-opening and

the foci are the points ${\displaystyle F_{1}=(c,0),\ F_{2}=(-c,0)}$,^[17]

the vertices are ${\displaystyle V_{1}=(a,0),\ V_{2}=(-a,0)}$.^[18]

For an arbitrary point ${\displaystyle (x,y)}$ the distance to the focus ${\displaystyle (c,0)}$ is ${\displaystyle {\sqrt {(x-c)^{2}+y^{2}}}}$ and to the second focus ${\displaystyle {\sqrt {(x+c)^{2}+y^{2}}}}$. Hence the point ${\displaystyle (x,y)}$ is on the hyperbola if the following condition is fulfilled

${\displaystyle {\sqrt {(x-c)^{2}+y^{2}}}-{\sqrt {(x+c)^{2}+y^{2}}}=\pm 2a\ .}$

Remove the square roots by suitable squarings and use the relation ${\displaystyle b^{2}=c^{2}-a^{2}}$ to obtain the equation of the hyperbola:

${\displaystyle {\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1\ .}$

This equation is called the canonical form of a hyperbola, because any hyperbola, regardless of its orientation relative to the Cartesian axes and regardless of the location of its center, can be transformed to this form by a change of variables, giving a hyperbola that is congruent to the original (see below).

The axes of symmetry or principal axes are the transverse axis (containing the segment of length 2a with endpoints at the vertices) and the conjugate axis (containing the segment of length 2b perpendicular to the transverse axis and with midpoint at the hyperbola's center).^[19] As opposed to an ellipse, a hyperbola has only two vertices: ${\displaystyle (a,0),\;(-a,0)}$. The two points ${\displaystyle (0,b),\;(0,-b)}$ on the conjugate axes are not on the hyperbola.

It follows from the equation that the hyperbola is symmetric with respect to both of the coordinate axes and hence symmetric with respect to the origin.

Eccentricity

For a hyperbola in the above canonical form, the eccentricity is given by

${\textstyle e={\sqrt {1+{\frac {b^{2}}{a^{2}}}}}.}$

Two hyperbolas are geometrically similar to each other – meaning that they have the same shape, so that one can be transformed into the other by rigid left and right movements, rotation, taking a mirror image, and scaling (magnification) – if and only if they have the same eccentricity.

Asymptotes

Solving the equation (above) of the hyperbola for ${\displaystyle y}$ yields

${\displaystyle y=\pm {\frac {b}{a}}{\sqrt {x^{2}-a^{2}}}.}$

It follows from this that the hyperbola approaches the two lines

${\displaystyle y=\pm {\frac {b}{a}}x}$

for large values of ${\displaystyle x }$. These two lines intersect at the center (origin) and are called asymptotes of the hyperbola ${\displaystyle {\tfrac {x^{2}}{a^{2}}}-{\tfrac {y^{2}}{b^{2}}}=1\ .}$^[20]

With the help of the second figure one can see that

${\displaystyle {\color {blue}{(1)}}}$ The perpendicular distance from a focus to either asymptote is ${\displaystyle b}$ (the semi-minor axis).

From the Hesse normal form ${\displaystyle {\tfrac {bx\pm ay}{\sqrt {a^{2}+b^{2}}}}=0}$ of the asymptotes and the equation of the hyperbola one gets:^[21]

${\displaystyle {\color {magenta}{(2)}}}$ The product of the distances from a point on the hyperbola to both the asymptotes is the constant ${\displaystyle {\tfrac {a^{2}b^{2}}{a^{2}+b^{2}}}\ ,}$ which can also be written in terms of the eccentricity e as ${\displaystyle \left({\tfrac {b}{e}}\right)^{2}.}$

From the equation ${\displaystyle y=\pm {\frac {b}{a}}{\sqrt {x^{2}-a^{2}}}}$ of the hyperbola (above) one can derive:

${\displaystyle {\color {green}{(3)}}}$ The product of the slopes of lines from a point P to the two vertices is the constant ${\displaystyle b^{2}/a^{2}\ .}$

In addition, from (2) above it can be shown that^[21]

${\displaystyle {\color {red}{(4)}}}$ The product of the distances from a point on the hyperbola to the asymptotes along lines parallel to the asymptotes is the constant ${\displaystyle {\tfrac {a^{2}+b^{2}}{4}}.}$

Semi-latus rectum

The length of the chord through one of the foci, perpendicular to the major axis of the hyperbola, is called the latus rectum. One half of it is the semi-latus rectum ${\displaystyle p}$. A calculation shows

${\displaystyle p={\frac {b^{2}}{a}}.}$

The semi-latus rectum ${\displaystyle p}$ may also be viewed as the radius of curvature at the vertices.

Tangent

The simplest way to determine the equation of the tangent at a point ${\displaystyle (x_{0},y_{0})}$ is to implicitly differentiate the equation ${\displaystyle {\tfrac {x^{2}}{a^{2}}}-{\tfrac {y^{2}}{b^{2}}}=1}$ of the hyperbola. Denoting dy/dx as y′, this produces

${\displaystyle {\frac {2x}{a^{2}}}-{\frac {2yy'}{b^{2}}}=0\ \Rightarrow \ y'={\frac {x}{y}}{\frac {b^{2}}{a^{2}}}\ \Rightarrow \ y={\frac {x_{0}}{y_{0}}}{\frac {b^{2}}{a^{2}}}(x-x_{0})+y_{0}.}$

With respect to ${\displaystyle {\tfrac {x_{0}^{2}}{a^{2}}}-{\tfrac {y_{0}^{2}}{b^{2}}}=1}$, the equation of the tangent at point ${\displaystyle (x_{0},y_{0})}$ is

${\displaystyle {\frac {x_{0}}{a^{2}}}x-{\frac {y_{0}}{b^{2}}}y=1.}$

A particular tangent line distinguishes the hyperbola from the other conic sections.^[22] Let f be the distance from the vertex V (on both the hyperbola and its axis through the two foci) to the nearer focus. Then the distance, along a line perpendicular to that axis, from that focus to a point P on the hyperbola is greater than 2f. The tangent to the hyperbola at P intersects that axis at point Q at an angle ∠PQV of greater than 45°.

Rectangular hyperbola

In the case ${\displaystyle a=b}$ the hyperbola is called rectangular (or equilateral), because its asymptotes intersect at right angles. For this case, the linear eccentricity is ${\displaystyle c={\sqrt {2}}a}$, the eccentricity ${\displaystyle e={\sqrt {2}}}$ and the semi-latus rectum ${\displaystyle p=a}$. The graph of the equation ${\displaystyle y=1/x}$ is a rectangular hyperbola.

Parametric representation with hyperbolic sine/cosine

Using the hyperbolic sine and cosine functions ${\displaystyle \cosh ,\sinh }$, a parametric representation of the hyperbola ${\displaystyle {\tfrac {x^{2}}{a^{2}}}-{\tfrac {y^{2}}{b^{2}}}=1}$ can be obtained, which is similar to the parametric representation of an ellipse:

${\displaystyle (\pm a\cosh t,b\sinh t),\,t\in \mathbb {R} \ ,}$

which satisfies the Cartesian equation because ${\displaystyle \cosh ^{2}t-\sinh ^{2}t=1.}$

Further parametric representations are given in the section Parametric equations below.

Conjugate hyperbola

Exchange ${\displaystyle {\frac {x^{2}}{a^{2}}}}$ and ${\displaystyle {\frac {y^{2}}{b^{2}}}}$ to obtain the equation of the conjugate hyperbola (see diagram):

${\displaystyle {\frac {y^{2}}{b^{2}}}-{\frac {x^{2}}{a^{2}}}=1\ ,}$ also written as

${\displaystyle {\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=-1\ .}$

In polar coordinates

For pole = focus:

The polar coordinates used most commonly for the hyperbola are defined relative to the Cartesian coordinate system that has its origin in a focus and its x-axis pointing towards the origin of the "canonical coordinate system" as illustrated in the first diagram.
In this case the angle ${\displaystyle \varphi }$ is called true anomaly.

Relative to this coordinate system one has that

${\displaystyle r={\frac {p}{1\mp e\cos \varphi }},\quad p={\tfrac {b^{2}}{a}}}$

and

${\displaystyle -\arccos \left(-{\frac {1}{e}}\right)<\varphi <\arccos \left(-{\frac {1}{e}}\right).}$

for pole = center:

With polar coordinates relative to the "canonical coordinate system" (see second diagram) one has that

${\displaystyle r={\frac {b}{\sqrt {e^{2}\cos ^{2}\varphi -1}}}.\,}$

For the right branch of the hyperbola the range of ${\displaystyle \varphi }$ is

${\displaystyle -\arccos \left({\frac {1}{e}}\right)<\varphi <\arccos \left({\frac {1}{e}}\right).}$

Parametric equations

A hyperbola with equation ${\displaystyle {\tfrac {x^{2}}{a^{2}}}-{\tfrac {y^{2}}{b^{2}}}=1}$ can be described by several parametric equations:

1. ${\displaystyle {\begin{cases}x=\pm a\cosh t,\\y=b\sinh t,\end{cases}}\qquad t\in \mathbb {R} .}$
2. ${\displaystyle {\begin{cases}x=\pm a{\tfrac {t^{2}+1}{2t}},\\y=b{\tfrac {t^{2}-1}{2t}},\end{cases}}\qquad t>0}$ (rational representation).
3. ${\displaystyle {\begin{cases}x={\frac {a}{\cos t}}=a\sec t,\\y=\pm b\tan t,\end{cases}}\qquad 0\leq t<2\pi ,\ t\neq {\frac {\pi }{2}},\ t\neq {\frac {3}{2}}\pi .}$
4. Tangent slope as parameter:

   A parametric representation, which uses the slope ${\displaystyle m}$ of the tangent at a point of the hyperbola can be obtained analogously to the ellipse case: Replace in the ellipse case ${\displaystyle b^{2}}$ by ${\displaystyle -b^{2}}$ and use formulae for the hyperbolic functions. One gets

   ${\displaystyle {\vec {c}}_{\pm }(m)=\left(-{\frac {ma^{2}}{\pm {\sqrt {m^{2}a^{2}-b^{2}}}}},{\frac {-b^{2}}{\pm {\sqrt {m^{2}a^{2}-b^{2}}}}}\right),\quad m >b/a.}$

   ${\displaystyle {\vec {c}}_{-}}$ is the upper, and ${\displaystyle {\vec {c}}_{+}}$ the lower half of the hyperbola. The points with vertical tangents (vertices ${\displaystyle (\pm a,0)}$) are not covered by the representation.

   The equation of the tangent at point ${\displaystyle {\vec {c}}_{\pm }(m)}$ is

   ${\displaystyle y=mx\pm {\sqrt {m^{2}a^{2}-b^{2}}}.}$

   This description of the tangents of a hyperbola is an essential tool for the determination of the orthoptic of a hyperbola.

Hyperbolic functions

Just as the trigonometric functions are defined in terms of the unit circle, so also the hyperbolic functions are defined in terms of the unit hyperbola, as shown in this diagram. In a unit circle, the angle (in radians) is equal to twice the area of the circular sector which that angle subtends. The analogous hyperbolic angle is likewise defined as twice the area of a hyperbolic sector.

Let ${\displaystyle a}$ be twice the area between the ${\displaystyle x}$ axis and a ray through the origin intersecting the unit hyperbola, and define ${\textstyle (x,y)=(\cosh a,\sinh a)=(x,{\sqrt {x^{2}-1}})}$ as the coordinates of the intersection point. Then the area of the hyperbolic sector is the area of the triangle minus the curved region past the vertex at ${\displaystyle (1,0)}$:

${\displaystyle {\begin{aligned}{\frac {a}{2}}&={\frac {xy}{2}}-\displaystyle \int _{1}^{x}{\sqrt {t^{2}-1}}\,dt\\&={\frac {x{\sqrt {x^{2}-1}}}{2}}-{\frac {x{\sqrt {x^{2}-1}}-\ln \left(x+{\sqrt {x^{2}-1}}\right)}{2}},\end{aligned}}}$

which simplifies to the area hyperbolic cosine

${\displaystyle a=\operatorname {arcosh} x=\ln \left(x+{\sqrt {x^{2}-1}}\right).}$

Solving for ${\displaystyle x}$ yields the exponential form of the hyperbolic cosine:

${\displaystyle x=\cosh a={\frac {e^{a}+e^{-a}}{2}}.}$

From ${\displaystyle x^{2}-y^{2}=1}$ one gets

${\displaystyle y=\sinh a={\sqrt {\cosh ^{2}a-1}}={\frac {e^{a}-e^{-a}}{2}},}$

and its inverse the area hyperbolic sine:

${\displaystyle a=\operatorname {arsinh} y=\ln \left(y+{\sqrt {y^{2}+1}}\right).}$

Other hyperbolic functions are defined according to the hyperbolic cosine and hyperbolic sine, so for example

${\displaystyle \operatorname {tanh} a={\frac {\sinh a}{\cosh a}}={\frac {e^{2a}-1}{e^{2a}+1}}.}$

Properties

The tangent bisects the angle between the lines to the foci

The tangent at a point ${\displaystyle P}$ bisects the angle between the lines ${\displaystyle {\overline {PF_{1}}},{\overline {PF_{2}}}}$.

Proof

Let ${\displaystyle L}$ be the point on the line ${\displaystyle {\overline {PF_{2}}}}$ with the distance ${\displaystyle 2a}$ to the focus ${\displaystyle F_{2}}$ (see diagram, ${\displaystyle a}$ is the semi major axis of the hyperbola). Line ${\displaystyle w}$ is the bisector of the angle between the lines ${\displaystyle {\overline {PF_{1}}},{\overline {PF_{2}}}}$. In order to prove that ${\displaystyle w}$ is the tangent line at point ${\displaystyle P}$, one checks that any point ${\displaystyle Q}$ on line ${\displaystyle w}$ which is different from ${\displaystyle P}$ cannot be on the hyperbola. Hence ${\displaystyle w}$ has only point ${\displaystyle P}$ in common with the hyperbola and is, therefore, the tangent at point ${\displaystyle P}$.
From the diagram and the triangle inequality one recognizes that ${\displaystyle QF_{2} < LF_{2} + QL =2a+ QF_{1} }$ holds, which means: ${\displaystyle QF_{2} - QF_{1} <2a}$. But if ${\displaystyle Q}$ is a point of the hyperbola, the difference should be ${\displaystyle 2a}$.

Midpoints of parallel chords

The midpoints of parallel chords of a hyperbola lie on a line through the center (see diagram).

The points of any chord may lie on different branches of the hyperbola.

The proof of the property on midpoints is best done for the hyperbola ${\displaystyle y=1/x}$. Because any hyperbola is an affine image of the hyperbola ${\displaystyle y=1/x}$ (see section below) and an affine transformation preserves parallelism and midpoints of line segments, the property is true for all hyperbolas:
For two points ${\displaystyle P=\left(x_{1},{\tfrac {1}{x_{1}}}\right),\ Q=\left(x_{2},{\tfrac {1}{x_{2}}}\right)}$ of the hyperbola ${\displaystyle y=1/x}$

the midpoint of the chord is ${\displaystyle M=\left({\tfrac {x_{1}+x_{2}}{2}},\cdots \right)=\cdots ={\tfrac {x_{1}+x_{2}}{2}}\;\left(1,{\tfrac {1}{x_{1}x_{2}}}\right)\ ;}$

the slope of the chord is ${\displaystyle {\frac {{\tfrac {1}{x_{2}}}-{\tfrac {1}{x_{1}}}}{x_{2}-x_{1}}}=\cdots =-{\tfrac {1}{x_{1}x_{2}}}\ .}$

For parallel chords the slope is constant and the midpoints of the parallel chords lie on the line ${\displaystyle y={\tfrac {1}{x_{1}x_{2}}}\;x\ .}$

Consequence: for any pair of points ${\displaystyle P,Q}$ of a chord there exists a skew reflection with an axis (set of fixed points) passing through the center of the hyperbola, which exchanges the points ${\displaystyle P,Q}$ and leaves the hyperbola (as a whole) fixed. A skew reflection is a generalization of an ordinary reflection across a line ${\displaystyle m}$, where all point-image pairs are on a line perpendicular to ${\displaystyle m}$.

Because a skew reflection leaves the hyperbola fixed, the pair of asymptotes is fixed, too. Hence the midpoint ${\displaystyle M}$ of a chord ${\displaystyle PQ}$ divides the related line segment ${\displaystyle {\overline {P}}\,{\overline {Q}}}$ between the asymptotes into halves, too. This means that ${\displaystyle P{\overline {P}} = Q{\overline {Q}} }$. This property can be used for the construction of further points ${\displaystyle Q}$ of the hyperbola if a point ${\displaystyle P}$ and the asymptotes are given.

If the chord degenerates into a tangent, then the touching point divides the line segment between the asymptotes in two halves.

Orthogonal tangents – orthoptic

For a hyperbola ${\displaystyle {\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1,\,a>b}$ the intersection points of orthogonal tangents lie on the circle ${\displaystyle x^{2}+y^{2}=a^{2}-b^{2}}$.
This circle is called the orthoptic of the given hyperbola.

The tangents may belong to points on different branches of the hyperbola.

In case of ${\displaystyle a\leq b}$ there are no pairs of orthogonal tangents.

Pole-polar relation for a hyperbola

Any hyperbola can be described in a suitable coordinate system by an equation ${\displaystyle {\tfrac {x^{2}}{a^{2}}}-{\tfrac {y^{2}}{b^{2}}}=1}$. The equation of the tangent at a point ${\displaystyle P_{0}=(x_{0},y_{0})}$ of the hyperbola is ${\displaystyle {\tfrac {x_{0}x}{a^{2}}}-{\tfrac {y_{0}y}{b^{2}}}=1.}$ If one allows point ${\displaystyle P_{0}=(x_{0},y_{0})}$ to be an arbitrary point different from the origin, then

point ${\displaystyle P_{0}=(x_{0},y_{0})\neq (0,0)}$ is mapped onto the line ${\displaystyle {\frac {x_{0}x}{a^{2}}}-{\frac {y_{0}y}{b^{2}}}=1}$, not through the center of the hyperbola.

This relation between points and lines is a bijection.

The inverse function maps

line ${\displaystyle y=mx+d,\ d\neq 0}$ onto the point ${\displaystyle \left(-{\frac {ma^{2}}{d}},-{\frac {b^{2}}{d}}\right)}$ and

line ${\displaystyle x=c,\ c\neq 0}$ onto the point ${\displaystyle \left({\frac {a^{2}}{c}},0\right)\ .}$

Such a relation between points and lines generated by a conic is called pole-polar relation or just polarity. The pole is the point, the polar the line. See Pole and polar.

By calculation one checks the following properties of the pole-polar relation of the hyperbola:

• For a point (pole) on the hyperbola the polar is the tangent at this point (see diagram: ${\displaystyle P_{1},\ p_{1}}$).
• For a pole ${\displaystyle P}$ outside the hyperbola the intersection points of its polar with the hyperbola are the tangency points of the two tangents passing ${\displaystyle P}$ (see diagram: ${\displaystyle P_{2},\ p_{2},\ P_{3},\ p_{3}}$).
• For a point within the hyperbola the polar has no point with the hyperbola in common. (see diagram: ${\displaystyle P_{4},\ p_{4}}$).

Remarks:

1. The intersection point of two polars (for example: ${\displaystyle p_{2},p_{3}}$) is the pole of the line through their poles (here: ${\displaystyle P_{2},P_{3}}$).
2. The foci ${\displaystyle (c,0),}$ and ${\displaystyle (-c,0)}$ respectively and the directrices ${\displaystyle x={\tfrac {a^{2}}{c}}}$ and ${\displaystyle x=-{\tfrac {a^{2}}{c}}}$ respectively belong to pairs of pole and polar.

Pole-polar relations exist for ellipses and parabolas, too.

Other properties

• The following are concurrent: (1) a circle passing through the hyperbola's foci and centered at the hyperbola's center; (2) either of the lines that are tangent to the hyperbola at the vertices; and (3) either of the asymptotes of the hyperbola.^[23][24]
• The following are also concurrent: (1) the circle that is centered at the hyperbola's center and that passes through the hyperbola's vertices; (2) either directrix; and (3) either of the asymptotes.^[24]

Arc length

The arc length of a hyperbola does not have an elementary expression. The upper half of a hyperbola can be parameterized as

${\displaystyle y=b{\sqrt {{\frac {x^{2}}{a^{2}}}-1}}.}$

Then the integral giving the arc length ${\displaystyle s}$ from ${\displaystyle x_{1}}$ to ${\displaystyle x_{2}}$ can be computed as:

${\displaystyle s=b\int _{\operatorname {arcosh} {\frac {x_{1}}{a}}}^{\operatorname {arcosh} {\frac {x_{2}}{a}}}{\sqrt {1+\left(1+{\frac {a^{2}}{b^{2}}}\right)\sinh ^{2}v}}\,\mathrm {d} v.}$

After using the substitution ${\displaystyle z=iv}$, this can also be represented using the incomplete elliptic integral of the second kind ${\displaystyle E}$ with parameter ${\displaystyle m=k^{2}}$:

${\displaystyle s=ib{\Biggr [}E\left(iv\,{\Biggr }\,1+{\frac {a^{2}}{b^{2}}}\right){\Biggr ]}_{\operatorname {arcosh} {\frac {x_{2}}{a}}}^{\operatorname {arcosh} {\frac {x_{1}}{a}}}.}$

Using only real numbers, this becomes^[25]

${\displaystyle s=b\left[F\left(\operatorname {gd} v\,{\Biggr }-{\frac {a^{2}}{b^{2}}}\right)-E\left(\operatorname {gd} v\,{\Biggr }-{\frac {a^{2}}{b^{2}}}\right)+{\sqrt {1+{\frac {a^{2}}{b^{2}}}\tanh ^{2}v}}\,\sinh v\right]_{\operatorname {arcosh} {\tfrac {x_{1}}{a}}}^{\operatorname {arcosh} {\tfrac {x_{2}}{a}}}}$

where ${\displaystyle F}$ is the incomplete elliptic integral of the first kind with parameter ${\displaystyle m=k^{2}}$ and ${\displaystyle \operatorname {gd} v=\arctan \sinh v}$ is the Gudermannian function.

Derived curves

Several other curves can be derived from the hyperbola by inversion, the so-called inverse curves of the hyperbola. If the center of inversion is chosen as the hyperbola's own center, the inverse curve is the lemniscate of Bernoulli; the lemniscate is also the envelope of circles centered on a rectangular hyperbola and passing through the origin. If the center of inversion is chosen at a focus or a vertex of the hyperbola, the resulting inverse curves are a limaçon or a strophoid, respectively.

Elliptic coordinates

A family of confocal hyperbolas is the basis of the system of elliptic coordinates in two dimensions. These hyperbolas are described by the equation

${\displaystyle \left({\frac {x}{c\cos \theta }}\right)^{2}-\left({\frac {y}{c\sin \theta }}\right)^{2}=1}$

where the foci are located at a distance c from the origin on the x-axis, and where θ is the angle of the asymptotes with the x-axis. Every hyperbola in this family is orthogonal to every ellipse that shares the same foci. This orthogonality may be shown by a conformal map of the Cartesian coordinate system w = z + 1/z, where z= x + iy are the original Cartesian coordinates, and w=u + iv are those after the transformation.

Other orthogonal two-dimensional coordinate systems involving hyperbolas may be obtained by other conformal mappings. For example, the mapping w = z² transforms the Cartesian coordinate system into two families of orthogonal hyperbolas.

Conic section analysis of the hyperbolic appearance of circles

Besides providing a uniform description of circles, ellipses, parabolas, and hyperbolas, conic sections can also be understood as a natural model of the geometry of perspective in the case where the scene being viewed consists of circles, or more generally an ellipse. The viewer is typically a camera or the human eye and the image of the scene a central projection onto an image plane, that is, all projection rays pass a fixed point O, the center. The lens plane is a plane parallel to the image plane at the lens O.

The image of a circle c is

a) a circle, if circle c is in a special position, for example parallel to the image plane and others (see stereographic projection),

b) an ellipse, if c has no point with the lens plane in common,

c) a parabola, if c has one point with the lens plane in common and

d) a hyperbola, if c has two points with the lens plane in common.

(Special positions where the circle plane contains point O are omitted.)

These results can be understood if one recognizes that the projection process can be seen in two steps: 1) circle c and point O generate a cone which is 2) cut by the image plane, in order to generate the image.

One sees a hyperbola whenever catching sight of a portion of a circle cut by one's lens plane. The inability to see very much of the arms of the visible branch, combined with the complete absence of the second branch, makes it virtually impossible for the human visual system to recognize the connection with hyperbolas.

Applications

Sundials

Hyperbolas may be seen in many sundials. On any given day, the sun revolves in a circle on the celestial sphere, and its rays striking the point on a sundial traces out a cone of light. The intersection of this cone with the horizontal plane of the ground forms a conic section. At most populated latitudes and at most times of the year, this conic section is a hyperbola. In practical terms, the shadow of the tip of a pole traces out a hyperbola on the ground over the course of a day (this path is called the declination line). The shape of this hyperbola varies with the geographical latitude and with the time of the year, since those factors affect the cone of the sun's rays relative to the horizon. The collection of such hyperbolas for a whole year at a given location was called a pelekinon by the Greeks, since it resembles a double-bladed axe.

Multilateration

A hyperbola is the basis for solving multilateration problems, the task of locating a point from the differences in its distances to given points — or, equivalently, the difference in arrival times of synchronized signals between the point and the given points. Such problems are important in navigation, particularly on water; a ship can locate its position from the difference in arrival times of signals from a LORAN or GPS transmitters. Conversely, a homing beacon or any transmitter can be located by comparing the arrival times of its signals at two separate receiving stations; such techniques may be used to track objects and people. In particular, the set of possible positions of a point that has a distance difference of 2a from two given points is a hyperbola of vertex separation 2a whose foci are the two given points.

Path followed by a particle

The path followed by any particle in the classical Kepler problem is a conic section. In particular, if the total energy E of the particle is greater than zero (that is, if the particle is unbound), the path of such a particle is a hyperbola. This property is useful in studying atomic and sub-atomic forces by scattering high-energy particles; for example, the Rutherford experiment demonstrated the existence of an atomic nucleus by examining the scattering of alpha particles from gold atoms. If the short-range nuclear interactions are ignored, the atomic nucleus and the alpha particle interact only by a repulsive Coulomb force, which satisfies the inverse square law requirement for a Kepler problem.

Korteweg–de Vries equation

The hyperbolic trig function ${\displaystyle \operatorname {sech} \,x}$ appears as one solution to the Korteweg–de Vries equation which describes the motion of a soliton wave in a canal.

Angle trisection

As shown first by Apollonius of Perga, a hyperbola can be used to trisect any angle, a well studied problem of geometry. Given an angle, first draw a circle centered at its vertex O, which intersects the sides of the angle at points A and B. Next draw the line segment with endpoints A and B and its perpendicular bisector ${\displaystyle \ell }$. Construct a hyperbola of eccentricity e=2 with ${\displaystyle \ell }$ as directrix and B as a focus. Let P be the intersection (upper) of the hyperbola with the circle. Angle POB trisects angle AOB.

To prove this, reflect the line segment OP about the line ${\displaystyle \ell }$ obtaining the point P' as the image of P. Segment AP' has the same length as segment BP due to the reflection, while segment PP' has the same length as segment BP due to the eccentricity of the hyperbola. As OA, OP', OP and OB are all radii of the same circle (and so, have the same length), the triangles OAP', OPP' and OPB are all congruent. Therefore, the angle has been trisected, since 3×POB = AOB.^[26]

Efficient portfolio frontier

In portfolio theory, the locus of mean-variance efficient portfolios (called the efficient frontier) is the upper half of the east-opening branch of a hyperbola drawn with the portfolio return's standard deviation plotted horizontally and its expected value plotted vertically; according to this theory, all rational investors would choose a portfolio characterized by some point on this locus.

Biochemistry

In biochemistry and pharmacology, the Hill equation and Hill-Langmuir equation respectively describe biological responses and the formation of protein–ligand complexes as functions of ligand concentration. They are both rectangular hyperbolae.

Hyperbolas as plane sections of quadrics

Hyperbolas appear as plane sections of the following quadrics:

• Elliptic cone
• Hyperbolic cylinder
• Hyperbolic paraboloid
• Hyperboloid of one sheet
• Hyperboloid of two sheets

• 

  Elliptic cone

• 

  Hyperbolic cylinder

• 

  Hyperbolic paraboloid

• 

  Hyperboloid of one sheet

• 

  Hyperboloid of two sheets

See also

Other conic sections

Other related topics

Notes

References

• Kazarinoff, Nicholas D. (2003), Ruler and the Round, Mineola, N.Y.: Dover, ISBN 0-486-42515-0
• Oakley, C. O., Ph.D. (1944), An Outline of the Calculus, New York: Barnes & Noble
• Protter, Murray H.; Morrey, Charles B., Jr. (1970), College Calculus with Analytic Geometry (2nd ed.), Reading: Addison-Wesley, LCCN 76087042

External links

• "Hyperbola", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Apollonius' Derivation of the Hyperbola at Convergence
• Frans van Schooten: Mathematische Oeffeningen, 1659
• Weisstein, Eric W. "Hyperbola". MathWorld.