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Given a hyperbolic triangle's sides (or angles), is there an easy way to determine whether it is inscribed in...
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If I have a hyperbolic triangle, specified by edge lengths or angles, is there an easy way to determine whether it is inscribed into a circle, a horocycle, or a hypercycle?
hyperbolic-geometry
edited Jan 15 at 14:54
Blue
48.5k870154
asked Jan 15 at 14:50
Marek14Marek14
39339
$endgroup$
add a comment |
$begingroup$
If I have a hyperbolic triangle, specified by edge lengths or angles, is there an easy way to determine whether it is inscribed into a circle, a horocycle, or a hypercycle?
hyperbolic-geometry
edited Jan 15 at 14:54
Blue
48.5k870154
asked Jan 15 at 14:50
Marek14Marek14
39339
$endgroup$
$begingroup$
My go-to online reference for this kind of information is here, despite it being a bit difficult to navigate (and read). Circumcircle considerations are covered here. You might have to hunt around a bit for notation.
$endgroup$
– Blue
Jan 15 at 15:09
add a comment |
$begingroup$
If I have a hyperbolic triangle, specified by edge lengths or angles, is there an easy way to determine whether it is inscribed into a circle, a horocycle, or a hypercycle?
hyperbolic-geometry
edited Jan 15 at 14:54
Blue
48.5k870154
asked Jan 15 at 14:50
Marek14Marek14
39339
$endgroup$
If I have a hyperbolic triangle, specified by edge lengths or angles, is there an easy way to determine whether it is inscribed into a circle, a horocycle, or a hypercycle?
hyperbolic-geometry
hyperbolic-geometry
edited Jan 15 at 14:54
Blue
48.5k870154
asked Jan 15 at 14:50
Marek14Marek14
39339
edited Jan 15 at 14:54
Blue
48.5k870154
asked Jan 15 at 14:50
Marek14Marek14
39339
edited Jan 15 at 14:54
Blue
48.5k870154
edited Jan 15 at 14:54
Blue
48.5k870154
edited Jan 15 at 14:54
Blue
48.5k870154
48.5k870154
asked Jan 15 at 14:50
Marek14Marek14
39339
asked Jan 15 at 14:50
Marek14Marek14
39339
asked Jan 15 at 14:50
Marek14Marek14
39339
39339
$begingroup$
My go-to online reference for this kind of information is here, despite it being a bit difficult to navigate (and read). Circumcircle considerations are covered here. You might have to hunt around a bit for notation.
$endgroup$
– Blue
Jan 15 at 15:09
add a comment |
$begingroup$
My go-to online reference for this kind of information is here, despite it being a bit difficult to navigate (and read). Circumcircle considerations are covered here. You might have to hunt around a bit for notation.
$endgroup$
– Blue
Jan 15 at 15:09
$begingroup$
My go-to online reference for this kind of information is here, despite it being a bit difficult to navigate (and read). Circumcircle considerations are covered here. You might have to hunt around a bit for notation.
$endgroup$
– Blue
Jan 15 at 15:09
$begingroup$
My go-to online reference for this kind of information is here, despite it being a bit difficult to navigate (and read). Circumcircle considerations are covered here. You might have to hunt around a bit for notation.
$endgroup$
– Blue
Jan 15 at 15:09
add a comment |
1 Answer
1
active
oldest
votes
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Expanding on my comment, drawing on this source.
For a hyperbolic triangle with sides $a$, $b$, $c$, consider the quantity
$$h := (overline{a_2}+overline{b_2}+overline{c_2})(-overline{a_2}+overline{b_2}+overline{c_2})(overline{a_2}-overline{b_2}+overline{c_2})(overline{a_2}+overline{b_2}-overline{c_2}) tag{1}$$
where $overline{x_2} := sinh(x/2)$ (a notational convention of my own devising). We can write:
$$text{A triangle is inscribed in a};
begin{cases}
text{circumcircle} & text{if};h > 0 \
text{horocycle} &text{if};h = 0\
text{hypercycle} &text{if};h < 0
end{cases} tag{$star$}$$
Note that the circumradius, $r$, is given by
$$sinh^2 r = frac{4,overline{a_2}^2,overline{b_2}^2,overline{c_2}^2}{h} tag{2}$$
a formula that effectively re-confirms $(star)$: Certainly, a positive $h$ implies a valid (real and finite) $r$; a negative $h$ implies an invalid (imaginary) $r$; and a vanishing $h$ implies an infinite $r$.
answered Jan 15 at 22:28
BlueBlue
48.5k870154
$endgroup$
$begingroup$
I'm curious. What does a hypercycle look like?
$endgroup$
– Oscar Lanzi
Jan 15 at 23:53
1
$begingroup$
@OscarLanzi: A hypercycle is an "equidistant curve": It's the set of points at a given distance from, and on a given side of, a line.
$endgroup$
– Blue
Jan 15 at 23:59
add a comment |
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1 Answer
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1 Answer
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active
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$begingroup$
Expanding on my comment, drawing on this source.
For a hyperbolic triangle with sides $a$, $b$, $c$, consider the quantity
$$h := (overline{a_2}+overline{b_2}+overline{c_2})(-overline{a_2}+overline{b_2}+overline{c_2})(overline{a_2}-overline{b_2}+overline{c_2})(overline{a_2}+overline{b_2}-overline{c_2}) tag{1}$$
where $overline{x_2} := sinh(x/2)$ (a notational convention of my own devising). We can write:
$$text{A triangle is inscribed in a};
begin{cases}
text{circumcircle} & text{if};h > 0 \
text{horocycle} &text{if};h = 0\
text{hypercycle} &text{if};h < 0
end{cases} tag{$star$}$$
Note that the circumradius, $r$, is given by
$$sinh^2 r = frac{4,overline{a_2}^2,overline{b_2}^2,overline{c_2}^2}{h} tag{2}$$
a formula that effectively re-confirms $(star)$: Certainly, a positive $h$ implies a valid (real and finite) $r$; a negative $h$ implies an invalid (imaginary) $r$; and a vanishing $h$ implies an infinite $r$.
answered Jan 15 at 22:28
BlueBlue
48.5k870154
$endgroup$
$begingroup$
I'm curious. What does a hypercycle look like?
$endgroup$
– Oscar Lanzi
Jan 15 at 23:53
1
$begingroup$
@OscarLanzi: A hypercycle is an "equidistant curve": It's the set of points at a given distance from, and on a given side of, a line.
$endgroup$
– Blue
Jan 15 at 23:59
add a comment |
$begingroup$
Expanding on my comment, drawing on this source.
For a hyperbolic triangle with sides $a$, $b$, $c$, consider the quantity
$$h := (overline{a_2}+overline{b_2}+overline{c_2})(-overline{a_2}+overline{b_2}+overline{c_2})(overline{a_2}-overline{b_2}+overline{c_2})(overline{a_2}+overline{b_2}-overline{c_2}) tag{1}$$
where $overline{x_2} := sinh(x/2)$ (a notational convention of my own devising). We can write:
$$text{A triangle is inscribed in a};
begin{cases}
text{circumcircle} & text{if};h > 0 \
text{horocycle} &text{if};h = 0\
text{hypercycle} &text{if};h < 0
end{cases} tag{$star$}$$
Note that the circumradius, $r$, is given by
$$sinh^2 r = frac{4,overline{a_2}^2,overline{b_2}^2,overline{c_2}^2}{h} tag{2}$$
a formula that effectively re-confirms $(star)$: Certainly, a positive $h$ implies a valid (real and finite) $r$; a negative $h$ implies an invalid (imaginary) $r$; and a vanishing $h$ implies an infinite $r$.
answered Jan 15 at 22:28
BlueBlue
48.5k870154
$endgroup$
$begingroup$
I'm curious. What does a hypercycle look like?
$endgroup$
– Oscar Lanzi
Jan 15 at 23:53
1
$begingroup$
@OscarLanzi: A hypercycle is an "equidistant curve": It's the set of points at a given distance from, and on a given side of, a line.
$endgroup$
– Blue
Jan 15 at 23:59
add a comment |
$begingroup$
Expanding on my comment, drawing on this source.
For a hyperbolic triangle with sides $a$, $b$, $c$, consider the quantity
$$h := (overline{a_2}+overline{b_2}+overline{c_2})(-overline{a_2}+overline{b_2}+overline{c_2})(overline{a_2}-overline{b_2}+overline{c_2})(overline{a_2}+overline{b_2}-overline{c_2}) tag{1}$$
where $overline{x_2} := sinh(x/2)$ (a notational convention of my own devising). We can write:
$$text{A triangle is inscribed in a};
begin{cases}
text{circumcircle} & text{if};h > 0 \
text{horocycle} &text{if};h = 0\
text{hypercycle} &text{if};h < 0
end{cases} tag{$star$}$$
Note that the circumradius, $r$, is given by
$$sinh^2 r = frac{4,overline{a_2}^2,overline{b_2}^2,overline{c_2}^2}{h} tag{2}$$
a formula that effectively re-confirms $(star)$: Certainly, a positive $h$ implies a valid (real and finite) $r$; a negative $h$ implies an invalid (imaginary) $r$; and a vanishing $h$ implies an infinite $r$.
answered Jan 15 at 22:28
BlueBlue
48.5k870154
$endgroup$
Expanding on my comment, drawing on this source.
For a hyperbolic triangle with sides $a$, $b$, $c$, consider the quantity
$$h := (overline{a_2}+overline{b_2}+overline{c_2})(-overline{a_2}+overline{b_2}+overline{c_2})(overline{a_2}-overline{b_2}+overline{c_2})(overline{a_2}+overline{b_2}-overline{c_2}) tag{1}$$
where $overline{x_2} := sinh(x/2)$ (a notational convention of my own devising). We can write:
$$text{A triangle is inscribed in a};
begin{cases}
text{circumcircle} & text{if};h > 0 \
text{horocycle} &text{if};h = 0\
text{hypercycle} &text{if};h < 0
end{cases} tag{$star$}$$
Note that the circumradius, $r$, is given by
$$sinh^2 r = frac{4,overline{a_2}^2,overline{b_2}^2,overline{c_2}^2}{h} tag{2}$$
a formula that effectively re-confirms $(star)$: Certainly, a positive $h$ implies a valid (real and finite) $r$; a negative $h$ implies an invalid (imaginary) $r$; and a vanishing $h$ implies an infinite $r$.
answered Jan 15 at 22:28
BlueBlue
48.5k870154
edited Jan 16 at 0:01
answered Jan 15 at 22:28
BlueBlue
48.5k870154
answered Jan 15 at 22:28
BlueBlue
48.5k870154
answered Jan 15 at 22:28
BlueBlue
48.5k870154
48.5k870154
$begingroup$
I'm curious. What does a hypercycle look like?
$endgroup$
– Oscar Lanzi
Jan 15 at 23:53
1
$begingroup$
@OscarLanzi: A hypercycle is an "equidistant curve": It's the set of points at a given distance from, and on a given side of, a line.
$endgroup$
– Blue
Jan 15 at 23:59
add a comment |
$begingroup$
I'm curious. What does a hypercycle look like?
$endgroup$
– Oscar Lanzi
Jan 15 at 23:53
1
$begingroup$
@OscarLanzi: A hypercycle is an "equidistant curve": It's the set of points at a given distance from, and on a given side of, a line.
$endgroup$
– Blue
Jan 15 at 23:59
$begingroup$
I'm curious. What does a hypercycle look like?
$endgroup$
– Oscar Lanzi
Jan 15 at 23:53
$begingroup$
I'm curious. What does a hypercycle look like?
$endgroup$
– Oscar Lanzi
Jan 15 at 23:53
1
1
$begingroup$
@OscarLanzi: A hypercycle is an "equidistant curve": It's the set of points at a given distance from, and on a given side of, a line.
$endgroup$
– Blue
Jan 15 at 23:59
$begingroup$
@OscarLanzi: A hypercycle is an "equidistant curve": It's the set of points at a given distance from, and on a given side of, a line.
$endgroup$
– Blue
Jan 15 at 23:59
add a comment |
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$begingroup$
My go-to online reference for this kind of information is here, despite it being a bit difficult to navigate (and read). Circumcircle considerations are covered here. You might have to hunt around a bit for notation.
$endgroup$
– Blue
Jan 15 at 15:09