Isoptic curves of generalized conic sections in the hyperbolic plane
We recall the notion of generalized hyperbolic angle between proper and improper straight lines, which is only available in Hungarian and Esperanto. Then we summarize the generalized hyperbolic conic sections. After investigation of the real conic sections and their isoptic curves in the hyperbolic...
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| author | Csima, G. Szirmai, J. Csima, G. Szirmai, J. Csima, G. Szirmai, J. |
| author_facet | Csima, G. Szirmai, J. Csima, G. Szirmai, J. Csima, G. Szirmai, J. |
| author_sort | Csima, G. |
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| description | We recall the notion of generalized hyperbolic angle between proper and improper straight lines, which is only available in Hungarian and Esperanto. Then we summarize the generalized hyperbolic conic sections.
After investigation of the real conic sections and their isoptic curves in the hyperbolic plane $\mathbf{H}^2,$ we consider the problem of isoptic curves of generalized conic sections in the extended hyperbolic plane.
This problem is widely investigated in the Euclidean plane $\mathbf{E}^2$ but, in the hyperbolic and elliptic planes, there are few results. Furthermore, we determine and visualize the generalized isoptic curves to all hyperbolic conic sections.
For our computations, we use the classical models based on the projective interpretation of hyperbolic geometry. In this way, the isoptic curves can be visualized in the Euclidean screen of a computer.
  |
| first_indexed | 2026-03-24T02:21:51Z |
| format | Article |
| fulltext |
UDC 514
G. Csima, J. Szirmai (Budapest Univ. Technology and Economics, Inst. Math., Hungary)
ISOPTIC CURVES OF GENERALIZED CONIC SECTIONS
IN THE HYPERBOLIC PLANE
IЗООПТИЧНI КРИВI УЗАГАЛЬНЕНИХ КОНIЧНИХ ПЕРЕРIЗIВ
У ГIПЕРБОЛIЧНIЙ ПЛОЩИНI
We recall the notion of generalized hyperbolic angle between proper and improper straight lines, which is only available in
Hungarian and Esperanto. Then we summarize the generalized hyperbolic conic sections.
After investigation of the real conic sections and their isoptic curves in the hyperbolic plane \bfH 2, we consider the
problem of isoptic curves of generalized conic sections in the extended hyperbolic plane.
This problem is widely investigated in the Euclidean plane \bfE 2 but, in the hyperbolic and elliptic planes, there are few
results. Furthermore, we determine and visualize the generalized isoptic curves to all hyperbolic conic sections.
For our computations, we use the classical models based on the projective interpretation of hyperbolic geometry. In
this way, the isoptic curves can be visualized in the Euclidean screen of a computer.
Ми нагадуємо поняття узагальненого гiперболiчного кута мiж власними i невласними прямими, що на даний
момент є доступним лише угорською мовою та мовою есперанто. Крiм того, ми пiдсумовуємо данi про узагальненi
гiперболiчнi конiчнi перерiзи.
Пiсля вивчення дiйсних конiчних перерiзiв та їхнiх iзооптичних кривих у гiперболiчнiй площинi \bfH 2 ми роз-
глядаємо задачу iзооптичних кривих для узагальнених конiчних перерiзiв у розширенiй гiперболiчнiй площинi.
Ця проблема iнтенсивно вивчається для випадку евклiдової площини \bfE 2, проте є небагато результатiв для
гiперболiчних та елiптичних площин. Крiм того, ми встановлюємо та вiзуалiзуємо узагальненi iзооптичнi кривi для
всiх конiчних перерiзiв.
У наших розрахунках ми використовуємо класичнi моделi на базi проективної iнтерпретацiї гiперболiчної
геометрiї. Таким чином iзооптичнi кривi можуть бути вiзуалiзованi на евклiдовому екранi комп’ютера.
1. Introduction. Let G be one of the constant curvature plane geometries, the Euclidean \bfE 2, the
hyperbolic \bfH 2, and the elliptic \scrE 2. The isoptic curve of a given plane curve \scrC is the locus of points
P \in G, where \scrC is seen under a given fixed angle \alpha , where 0 < \alpha < \pi . An isoptic curve formed
by the locus of tangents meeting at right angle is called orthoptic curve. The name isoptic curve was
suggested by Taylor in [28].
In [2, 3], the Euclidean isoptic curves of the closed, strictly convex curves are studied, using
their support function. Papers [16, 31, 32] deal with Euclidean curves having a circle or an ellipse
for an isoptic curve. Further curves appearing as isoptic curves are well studied in Euclidean plane
geometry \bfE 2, see, e.g., [18, 30]. Isoptic curves of conic sections have been studied in [13, 26].
There are results for Bezier curves as well, see [15]. A lot of papers concentrate on the properties of
the isoptics, e.g., [19, 20, 23] and the references given there. There are some generalization of the
isoptics as well, e.g., equioptic curves in [24] or secantopics in [27].
In the case of hyperbolic plane geometry there are only few results. The isoptic curves of the
hyperbolic line segment and proper conic sections are determined by the authors in [5 – 7].
The isoptics of conic sections in elliptic geometry \scrE 2 are determined by the authors in [7].
In the papers [10] and [11], K. Fladt determined the equations of the generalized conic sections
in the hyperbolic plane using algebraic methods and, in [21], E. Molnár classified them with syn-
thetic methods. This topic has a wide literature, numerous works consider the classification of the
hyperbolic conic sections, e.g., [17].
c\bigcirc G. CSIMA, J. SZIRMAI, 2019
1684 ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 12
ISOPTIC CURVES OF GENERALIZED CONIC SECTIONS IN THE HYPERBOLIC PLANE 1685
Our goal in this paper is to generalize our method described in [7], that is based on the projec-
tive interpretation of hyperbolic plane geometry, to determine the isoptic curves of the generalized
hyperbolic conics and visualize them for some angles. Therefore we study and recall the notion of
the angle between proper and nonproper straight lines using the results of the papers [1, 12, 29].
2. The projective model. For the 2-dimensional hyperbolic plane \bfH 2 we use the projective
model in Lorentz space \bfE 2,1 of signature (2, 1), i.e., \bfE 2,1 is the real vector space \bfV 3 equipped with
the bilinear form of signature (2, 1)
\langle \bfx ,\bfy \rangle = x1y1 + x2y2 - x3y3, (1)
where the non-zero vectors \bfx = (x1, x2, x3)T and \bfy = (y1, y2, y3)T \in \bfV 3 are determined up to real
factors and they represent points X = \bfx \BbbR and Y = \bfy \BbbR of \bfH 2 in \BbbP 2(\BbbR ). The proper points of \bfH 2
are represented as the interior of the absolute conic
AC =
\bigl\{
\bfx \BbbR \in \scrP 2
\bigm| \bigm| \langle \bfx ,\bfx \rangle = 0
\bigr\}
= \partial \bfH 2 (2)
in real projective space \BbbP 2(\bfV 3, \bfitV 3). All proper interior point X \in \bfH 2 are characterized by \langle \bfx ,\bfx \rangle <
< 0. The points on the boundary \partial \bfH 2 in \scrP 2 represent the absolute points at infinity of \bfH 2. Points
Y with \langle \bfy ,\bfy \rangle > 0 are called outer or nonproper points of \bfH 2.
The point Y = \bfy \BbbR is said to be conjugate to X = \bfx \BbbR relative to AC when \langle \bfx ,\bfy \rangle = 0.
The set of all points conjugate to X = \bfx \BbbR forms a projective (polar) line
\mathrm{p}\mathrm{o}\mathrm{l}(X) :=
\bigl\{
\bfy \BbbR \in \BbbP 2
\bigm| \bigm| \langle \bfx ,\bfy \rangle = 0
\bigr\}
. (3)
Hence the bilinear form to (AC) by (1) induces a bijection (linear polarity \bfV 3 \rightarrow \bfitV 3) from the
points of \scrP 2 onto its lines (hyperplanes in general).
Point X = \bfx \BbbR and the straight line u = \BbbR \bfitu are called incident if the value of the linear form \bfitu
on the vector \bfx is equal to zero, i.e., \bfitu \bfx = 0 (\bfx \in \bfV 3 \setminus \{ \bfzero \} , \bfitu \in \bfitV 3 \setminus \{ \bfzero \} ). In this paper, we set
the sectional curvature of \bfH 2, K = - k2, to be k = 1. With this assumptions the (AC) will be the
base circle of the Cayley – Klein model.
The distance d(X,Y ) of two proper points X = \bfx \BbbR and Y = \bfy \BbbR can be calculated with
appropriate representing vectors by the formula (see, e.g., [22])
\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h} d(X,Y ) =
- \langle \bfx ,\bfy \rangle \sqrt{}
\langle \bfx ,\bfx \rangle \langle \bfy ,\bfy \rangle
. (4)
The pole of a straight line is the only point which is conjugate to all point on the line. For the
further calculations, let us denote by \bfu the pole of the straight line u = \BbbR \bfitu . It is easy to prove
that if \bfitu = (u1, u2, u3), then \bfu = (u1, u2, - u3), and follows that if u = \BbbR \bfitu and v = \BbbR \bfitv , then
\langle \bfu ,\bfv \rangle = \langle \bfitu , \bfitv \rangle .
2.1. Generalized angle of straight lines. Having regard to the fact that the majority of the
generalized conic sections have ideal and outer tangents as well, it is inevitable to introduce the
generalized concept of the hyperbolic angle. In the extended hyperbolic plane there are three classes
of lines by the number of common points with the absolute conic AC (see (2)):
1. The straight line u = \BbbR \bfitu is proper if \mathrm{c}\mathrm{a}\mathrm{r}\mathrm{d}(u \cap AC) = 2 \leftrightarrow \langle \bfitu , \bfitu \rangle > 0.
2. The straight line u = \BbbR \bfitu is nonproper if \mathrm{c}\mathrm{a}\mathrm{r}\mathrm{d}(u \cap AC) < 2:
(a) if \mathrm{c}\mathrm{a}\mathrm{r}\mathrm{d}(u \cap AC) = 1 \leftrightarrow \langle \bfitu , \bfitu \rangle = 0, then u = \BbbR \bfitu is called boundary straight line;
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 12
1686 G. CSIMA, J. SZIRMAI
(b) if \mathrm{c}\mathrm{a}\mathrm{r}\mathrm{d}(u \cap AC) = 0 \leftrightarrow \langle \bfitu , \bfitu \rangle < 0, then u = \BbbR \bfitu is called outer straight line.
We define the generalized angle between straight lines using the results of the papers [1, 12, 29] in
the projective model.
-3 -2 -1 1 2 3
-3
-2
-1
1
2
3
(a) both u and v are real line,
proper intersection
-3 -2 -1 1 2 3
-3
-2
-1
1
2
3
(b) both u and v are real line, outer
intersection
-3 -2 -1 1 2 3
-3
-2
-1
1
2
3
(c) both u and v are outer line
-3 -2 -1 1 2 3
-3
-2
-1
1
2
3
(d) line u is real but v is outer
-3 -2 -1 1 2 3
-3
-2
-1
1
2
3
(e) line u is boundary
Fig. 1. Generalized angle of straight lines u = \BbbR \bfitu and v = \BbbR \bfitv .
Definition 2.1. 1. Suppose that u = \BbbR \bfitu and v = \BbbR \bfitv are both proper lines.
(a) If \langle \bfitu , \bfitu \rangle \langle \bfitv , \bfitv \rangle - \langle \bfitu , \bfitv \rangle 2 > 0, then they intersect in a proper point (see Fig. 1 (a)) and their
angle \alpha (\bfitu , \bfitv ) can be measured by
\mathrm{c}\mathrm{o}\mathrm{s}\alpha =
\pm \langle \bfitu ,\bfitv \rangle \sqrt{}
\langle \bfitu ,\bfitu \rangle \langle \bfitv ,\bfitv \rangle
. (5)
(b) If \langle \bfitu , \bfitu \rangle \langle \bfitv , \bfitv \rangle - \langle \bfitu , \bfitv \rangle 2 < 0, then they intersect in a nonproper point (see Fig. 1 (b)) and
their angle is the length of their normal transverse and it can be calculated using the formula
\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}\alpha =
\pm \langle \bfu ,\bfv \rangle \sqrt{}
\langle \bfu ,\bfu \rangle \langle \bfv ,\bfv \rangle
. (6)
(c) If \langle \bfitu , \bfitu \rangle \langle \bfitv , \bfitv \rangle - \langle \bfitu , \bfitv \rangle 2 = 0, then they intersect in a boundary point and their angle is 0.
2. Suppose that u = \BbbR \bfitu and v = \BbbR \bfitv are both outer lines of \bfH 2 (see Fig. 1 (c)). The angle of
these lines will be the distance of their poles using the formula (6).
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 12
ISOPTIC CURVES OF GENERALIZED CONIC SECTIONS IN THE HYPERBOLIC PLANE 1687
3. Suppose that u = \BbbR \bfitu is a proper and v = \BbbR \bfitv is an outer line (see Fig. 1 (d)). Their angle is
defined as the distance of the pole of the outer line to the real line and can be computed by
\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\alpha =
\pm \langle \bfu ,\bfv \rangle \sqrt{}
- \langle \bfu ,\bfu \rangle \langle \bfv ,\bfv \rangle
. (7)
4. Suppose that at least one of the straight lines u = \BbbR \bfitu and v = \BbbR \bfitv is boundary line of \bfH 2
(see Fig. 1 (e)). If the other line fits the boundary point, the angle cannot be defined, otherwise it is
infinite.
Remark 2.1. In the previous definition we fixed that except case 1 (a) we use real distance type
values instead of complex angles which arise in other cases. The \pm on the right-hand sides are
justifiable because we consider complementary angles, i.e., \alpha and \pi - \alpha together.
3. Classification of generalized conic sections on the hyperbolic plane in dual pairs. The
literature of the hyperbolic conic section classification is very wide and it goes back until 1902 when
it was first given by Liebmann (see [17]). We also note that there is a detailed theory of conic
sections in the work of both Cooligde and Kagan (see [4, 14]). There are numerous other works in
this topic (see [9, 25]), but in this section we will summarize and extend the results of K. Fladt (see
[10, 11]) about the generalized conic sections on the extended hyperbolic plane.
Let us denote a point with \bfx and a line with \bfitu . Then the absolute conic (AC) can be defined as
a point conic with the \bfx T e\bfx = 0 quadratic form where e = \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\{ 1, 1, - 1\} or due to the absolute
polarity as line conic with \bfitu E\bfitu T = 0, where E = e - 1 = \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\{ 1, 1, - 1\} .
Similarly to the Euclidean geometry we use the well-known quadratic form
\bfx Ta\bfx = a11x
1x1 + a22x
2x2 + a33x
3x3 + 2a23x
2x3 + 2a13x
1x3 + 2a12x
1x2 = 0,
where \mathrm{d}\mathrm{e}\mathrm{t} a \not = 0 for a nondegenerate point conic, and
\bfitu A\bfitu T = A11u1u1 +A22u2u2 +A33u3u3 + 2A23u2u3 + 2A13u1u3 + 2A12u1u2 = 0,
where A = a - 1 for the corresponding line conic defined by the tangent lines of the previous point
conic. With the polarity \bfx = A\bfitu T and \bfitu T = a\bfx follow since \bfitu \bfx = 0.
Consider a one parameter conic family of our point conic with the (AC), defined by
\bfx T (a+ \rho e)\bfx = 0.
Since the characteristic equation \Delta (\rho ) := \mathrm{d}\mathrm{e}\mathrm{t}(a + \rho e) is an odd degree polynomial, this conic
pencil has at least one real degenerate element (\rho 1), which consists of at most two point sequences
with holding lines \bfitp 1
1 and \bfitp 2
1 called asymptotes. Therefore, we get a product
\bfx T (a+ \rho 1e)\bfx = (\bfitp 1
1\bfx )
T (\bfitp 2
1\bfx ) = \bfx T
\bigl(
(\bfitp 1
1)
T\bfitp 2
1
\bigr)
\bfx = 0
with occasional complex coordinates of the asymptotes. Each of these two asymptotes has at most
two common points with the (AC) and with the conic as well. Thus, the at most 4 common points
with at most 3 pairs of asymptotes can be determined through complex coordinates and elements
according to the at most 3 different eigenvalues \rho 1, \rho 2, and \rho 3.
In complete analogy with the previous discussion in dual formulation we get that the one param-
eter conic family of a line conic with (AC) has at least one degenerate element (\sigma 1) which contains
two line pencils at most with occasionally complex holding points \bff 11 and \bff 12 called foci
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 12
1688 G. CSIMA, J. SZIRMAI
\bfitu (A+ \sigma 1E)\bfitu T = (\bfitu \bff 11 )(\bfitu \bff
1
2 )
T = \bfitu
\bigl(
\bff 11 (\bff
1
2 )
T
\bigr)
\bfitu T = 0.
For each focus at most two common tangent line can be drawn to AC and to our line conic.
Therefore, at most four common tangent lines with at most three pairs of foci can be determined
maybe with complex coordinates to the corresponding eigenvalues \sigma 1, \sigma 2, and \sigma 3.
Combining the previous discussions with [10, 21] the classification of the conics on the extended
hyperbolic plane can be obtained in dual pairs.
First, our goal is to find an appropriate transformation, so that the resulted normal form character-
izes the conic, e.g., the straight line x1 = 0 is a symmetry axis of the conic section (a31 = a12 = 0).
Therefore we take a rotation around the origin O(0, 0, 1)T and a translation parallel with x2 = 0.
As it used before, the characteristic equation
\Delta (\rho ) = \mathrm{d}\mathrm{e}\mathrm{t}(a+ \rho e) = \mathrm{d}\mathrm{e}\mathrm{t}
\left(
a11 + \rho a12 a13
a21 a22 + \rho a23
a31 a32 a33 - \rho
\right) = 0
has at least one real root denoted by \rho 1.
This is helpful to determine the exact transformation if the equalities \rho 1 = \rho 2 = \rho 3 not hold.
That case will be covered later. With this transformations we obtain the normal form
\rho 1x
1x1 + a22x
2x2 + 2a23x
2x3 + a33x
3x3 = 0. (8)
In the following we distinguish 3 different cases according to the other two roots:
1. Two different real roots. Then the monom x2x3 can be eliminated from the equation above,
by translating the conic parallel with x1 = 0. The final form of the conic equation in this case, called
central conic section:
\rho 1x
1x1 + \rho 2x
2x2 - \rho 3x
3x3 = 0.
Because our conic is nondegenerate, therefore \rho 3x
3 \not = 0 follows and with the notations a =
\rho 1
\rho 3
and b =
\rho 2
\rho 3
our matrix can be transformed into a = \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\{ a, b, - 1\} , where a \leq b can be assumed.
The equation of the dual conic can be obtained using the polarity E respected to (AC) by E AE - 1 =
= \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}
\biggl\{
1
a
,
1
b
, - 1
\biggr\}
. By the above considerations we can give an overview of the generalized central
conics with representants:
Theorem 3.1. If the conic section has the normal form ax2+by2 = 1, then we get the following
types of central conic sections (see Figs. 2 – 4):
(a) absolute conic: a = b = 1;
(b) i) circle: 1 < a = b,
ii) circle enclosing the absolute: a = b < 1;
(c) i) hypercycle: 1 = a < b,
ii) hypercycle enclosing the absolute: 0 < a < 1 = b;
(d) hypercycle excluding the absolute: a < 0 < 1 = b;
(e) concave hyperbola: 0 < a < 1 < b;
(f) i) convex hyperbola: a < 0 < 1 < b,
ii) hyperbola excluding the absolute: a < 0 < b < 1;
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ISOPTIC CURVES OF GENERALIZED CONIC SECTIONS IN THE HYPERBOLIC PLANE 1689
(g) i) ellipse: 1 < a < b,
ii) ellipse enclosing the absolute: 0 < a < b < 1;
(h) empty: a \leq b \leq 0,
where either the conic and its dual pair lies in the same class or i) and ii) are dual pairs with a\prime =
1
a
and b\prime =
1
b
.
Fig. 2. Domains for concave hyperbola with notations of Definition 2.1 (left) and concave hyperbola (right):
a = 0.3, b = 2, \alpha =
\pi
2
.
Fig. 3. Hyperbola excluding the absolute (left): a = 0.5, b = - 2, \alpha =
\pi
3
and convex hyperbola (right):
a = 1.1, b = - 1.5, \alpha =
19\pi
36
.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 12
1690 G. CSIMA, J. SZIRMAI
Fig. 4. Ellipse (left): a = 2, b = 3, \alpha =
7\pi
18
and ellipse enclosing the absolute (right): a = 0.45, b = 0.8,
\alpha =
\pi
2
.
Fig. 5. Elliptic parabola (left): a = 2, b = 1.5, \alpha =
\pi
3
and parabola enclosing the absolute (right): a = - 2.5,
b = - 5, \alpha =
7\pi
18
.
2. Coinciding real roots. The last translation cannot be enforced but it can be proved that
\rho 2 = \rho 3 =
a33 - a22
2
follows. With some simplifications of the formulas in [10] we obtain the
normal form of the so-called generalized parabolas.
Theorem 3.2. The parabolas have the normal form ax2 + (b + 1)y2 - 2y = b - 1 and the
following cases arise (see Figs. 5 – 7):
(a) i) horocycle: 0 < a = b,
ii) horocycle enclosing the absolute: a = b < 0;
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ISOPTIC CURVES OF GENERALIZED CONIC SECTIONS IN THE HYPERBOLIC PLANE 1691
(b) i) elliptic parabola: 0 < b < a,
ii) parabola enclosing the absolute: b < a < 0;
(c) i) two sided parabola: a < b < 0,
ii) concave hyperbolic parabola: 0 < a < b;
(d) i) convex hyperbolic parabola: a < 0 < b,
ii) parabola excluding the absolute: b < 0 < a,
where all i) and ii) are dual pairs with parameters a\prime = - b2
a
and b\prime = - b.
Fig. 6. Two sided parabola (left): a = - 5, b = - 2.7, \alpha =
\pi
2
and concave hyperbolic parabola (right):
a = 1, b = 2, \alpha =
\pi
2
.
Fig. 7. Convex hyperbolic parabola (left): a = - 2, b = 1.5, \alpha =
\pi
3
and parabola excluding the absolute
(right): a = 0.8, b = - 0.4, \alpha =
\pi
3
.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 12
1692 G. CSIMA, J. SZIRMAI
Fig. 8. Semihyperbola: a = 1.4, b = 0.5, \alpha =
\pi
4
(left) and \alpha =
8\pi
18
(right).
Fig. 9. Osculating parabola: a = 0.4, \alpha =
\pi
3
(left) and \alpha =
2\pi
3
(right).
3. Two conjugate complex roots. Then the last translation cannot be performed to eliminate the
monom x2x3 but we can eliminate the monom x3x3 by an appropriate transformation described in
[10]. Shifting to inhomogeneous coordinates and simplifying the coefficients we obtain the following
theorem.
Theorem 3.3. The so-called semihyperbola has the normal form ax2 + 2by2 - 2y = 0, where
| b| < 1 and its dual pair is projectively equivalent with another semihyperbola with a\prime =
1
a
and
b\prime = - b (see Fig. 8).
4. Overviewing the above cases only one remains, when the conic has no symmetry axis at all
and \rho 1 = \rho 2 = \rho 3. Ignoring further explanations we claim the following theorem.
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ISOPTIC CURVES OF GENERALIZED CONIC SECTIONS IN THE HYPERBOLIC PLANE 1693
Theorem 3.4. If the conic has the normal form (1 - x2 - y2)+2ay(x+1) = 0, where a > 0, then
it is called osculating parabola. Its dual is also an osculating parabola by a convenient reflection
(see Fig. 9).
4. Isoptic curves of generalized hyperbolic conics. 4.1. Isoptic curves of central conics.
In this section, we will extend the algorithm for determining the isoptic curves of conic sections
described in [7] for generalized hyperbolic conic sections. We have to apply the generalized notion
of angle, therefore the equation of the isoptic curve will not be given by a implicit formula but the
isoptic curve consist of some piecewise continuous arcs that will be given by implicit equations.
First, we determine the equations of the tangent lines through a given external point P to a given
conic section \scrC . We will use not only the point conic but the corresponding line conic as well. The
algorithm do not need the points of tangency but of course they can be determined from the tangents.
This strategy provides a general simplification in the other cases as well without all details in this
work.
Let the external point P be given with homogeneous coordinates (x, y, 1)T . If a central conic
is given by a, then the corresponding line conic is defined by A, where a - 1 = A. Now, we know
that P fits on the tangent lines u = \BbbR \bfitu and v = \BbbR \bfitv where \bfitu = (u1, u2, 1) and \bfitv = (v1, v2, 1),
furthermore, u and v satisfy the equation of the line conic
u1x+ u2y + 1 = 0,
u21
a
+
u22
b
- 1 = 0,
v1x+ v2y + 1 = 0,
v21
a
+
v22
b
- 1 = 0.
Solving the above systems we obtain the coordinates of the straight lines u and v:
u1 = -
ax+
\sqrt{}
aby2 (ax2 + by2 - 1)
ax2 + by2
,
u2 =
- by2 + x
\sqrt{}
aby2 (ax2 + by2 - 1)
ax2y + by3
,
(9)
v1 =
- ax+
\sqrt{}
aby2 (ax2 + by2 - 1)
ax2 + by2
,
v2 = -
by2 + x
\sqrt{}
aby2 (ax2 + by2 - 1)
ax2y + by3
.
Of course, aby2
\bigl(
ax2 + by2 - 1
\bigr)
\geq 0 must hold otherwise P (x, y, 1)T is not an external point.
We get the exact formula of the more parted isoptic curve related to the central conic sections
(see Theorem 3.1) using the definition of the generalized angle (see Definition 2.1). The compound
isoptic can be given by classifying the straight lines u = \BbbR \bfitu and v = \BbbR \bfitv according to their poles.
We summarize our result in the following theorem.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 12
1694 G. CSIMA, J. SZIRMAI
Theorem 4.1. Let a central conic section be given by its equation ax2 + by2 = 1 (see Theo-
rem 3.1). Then the compound \alpha -isoptic curve (0 < \alpha < \pi ) of the considered conic has the equation
\bigl(
a
\bigl(
(b+ 1)x2 - 1
\bigr)
+ (a+ 1)by2 - b
\bigr) 2\bigm| \bigm| \bigm| (a - 1)2b2y4 + 2(a - 1)b (b+ a ((b - 1)x2 - 1)) y2 + (a(b - 1)x2 + a - b)2
\bigm| \bigm| \bigm| =
=
\left\{
\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}2(\alpha ), aby2
\bigl(
ax2 + by2 - 1
\bigr)
\geq 0 \wedge \bigl(
1 - u21 - u22
\bigr) \bigl(
1 - v21 - v22
\bigr)
> 0 \wedge
x2 + y2 > 1,
\mathrm{c}\mathrm{o}\mathrm{s}2(\alpha ), aby2
\bigl(
ax2 + by2 - 1
\bigr)
\geq 0 \wedge
x2 + y2 < 1,
\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}2(\alpha ), aby2
\bigl(
ax2 + by2 - 1
\bigr)
\geq 0 \wedge \bigl(
1 - u21 - u22
\bigr) \bigl(
1 - v21 - v22
\bigr)
< 0,
where u1,2 and v1,2 are derived by (9).
In Figs. 2 – 4 we visualize the isoptic curves of central conic sections. The foremost figure shows,
how different types of isoptics arise due to common tangents. We indicated the Cayley – Klein model
circle with black, the conic with dashed line and the isoptic is shaded.
4.2. Isoptic curves of parabolas. We study the isoptic curves of the generalized parabolas given
by their equations in Theorem 3.3.
The algorithm described in the previous subsection can be repeated for further conics as well.
The difference is only in the equation of the compound isoptic is because of the different conic
equation.
It is clear that the coordinates of the tangent line \bfitu = (u1, u2, 1) and \bfitv = (v1, v2, 1) will be
different from (8)
u1 = -
ax2(b+ y - 1) + y
\sqrt{}
ab2x2 (ax2 + b (y2 - 1) + (y - 1)2)
a(b - 1)x3 + b2xy2
,
u2 =
ax2 - b2y -
\sqrt{}
ab2x2 (ax2 + b (y2 - 1) + (y - 1)2)
a(b - 1)x2 + b2y2
,
v1 =
- ax2(b+ y - 1) + y
\sqrt{}
ab2x2 (ax2 + b (y2 - 1) + (y - 1)2)
a(b - 1)x3 + b2xy2
,
v2 =
ax2 - b2y +
\sqrt{}
ab2x2 (ax2 + b (y2 - 1) + (y - 1)2)
a(b - 1)x2 + b2y2
.
(10)
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 12
ISOPTIC CURVES OF GENERALIZED CONIC SECTIONS IN THE HYPERBOLIC PLANE 1695
Theorem 4.2. Let a parabola be given by its equation ax2 + (b + 1)y2 - 2y = b - 1 (see
Theorem 3.3). Then the compound \alpha -isoptic curve (0 < \alpha < \pi ) of the considered conic has the
equation \bigl(
a
\bigl(
b
\bigl(
2x2 + y2 - 1
\bigr)
+ (y - 1)2
\bigr)
+ b2
\bigl(
y2 - 1
\bigr) \bigr) 2 \bigm| \bigm| (y - 1)2
\bigl(
(y + 1)2b4 -
- 2a
\bigl(
2x2 + y2 + b(y + 1)2 - 1
\bigr)
b2 + a2
\bigl(
(y - 1)2 + b2(y + 1)2 + 2b
\bigl(
2x2 + y2 - 1
\bigr) \bigr) \bigr) \bigm| \bigm| - 1
=
=
\left\{
\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}2(\alpha ), ab2x2
\bigl(
ax2 + b
\bigl(
y2 - 1
\bigr)
+ (y - 1)2
\bigr)
\geq 0 \wedge \bigl(
1 - u21 - u22
\bigr) \bigl(
1 - v21 - v22
\bigr)
> 0\wedge
x2 + y2 > 1,
\mathrm{c}\mathrm{o}\mathrm{s}2(\alpha ), ab2x2
\bigl(
ax2 + b
\bigl(
y2 - 1
\bigr)
+ (y - 1)2
\bigr)
\geq 0 \wedge
x2 + y2 < 1,
\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}2(\alpha ), ab2x2
\bigl(
ax2 + b
\bigl(
y2 - 1
\bigr)
+ (y - 1)2
\bigr)
\geq 0 \wedge \bigl(
1 - u21 - u22
\bigr) \bigl(
1 - v21 - v22
\bigr)
< 0,
where u1,2 and v1,2 are derived by (10).
The isoptic curves of the parabolas can be seen in Figs. 5 – 7. The same convention has been
used as on the previous figures.
4.3. Isoptic curves of semihyperbola.
Theorem 4.3. Let the semihyperbola be given by its equation ax2 + 2by2 - 2y = 0, where
| b| < 1 (see Theorem 3.2). Then the compound \alpha -isoptic curve (0 < \alpha < \pi ) of the considered conic
has the equation\bigl(
2a
\bigl(
b
\bigl(
x2 + y2
\bigr)
- y
\bigr)
+ y2 - 1
\bigr) 2 \bigm| \bigm| y4 + 4a2
\bigl(
x2 + y2
\bigr) \bigl( \bigl(
b2 - 1
\bigr)
x2 + (by - 1)2
\bigr)
-
- 4a
\bigl(
y -
\bigl(
2x2 + y2
\bigr)
y + b
\bigl(
y4 +
\bigl(
x2 - 1
\bigr)
y2 + x2
\bigr) \bigr)
- 2y2 + 1
\bigm| \bigm| - 1
=
=
\left\{
\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}2(\alpha ), ax2 (a+ 2y(by - 1)) \geq 0\wedge \bigl(
1 - u21 - u22
\bigr) \bigl(
1 - v21 - v22
\bigr)
> 0\wedge
x2 + y2 > 1,
\mathrm{c}\mathrm{o}\mathrm{s}2(\alpha ), ax2 (a+ 2y(by - 1)) \geq 0\wedge
x2 + y2 < 1,
\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}2(\alpha ), ax2 (a+ 2y(by - 1)) \geq 0\wedge \bigl(
1 - u21 - u22
\bigr) \bigl(
1 - v21 - v22
\bigr)
< 0,
where
u1 = -
ax+
\sqrt{}
a (ax2 + 2y(by - 1))
y
,
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 12
1696 G. CSIMA, J. SZIRMAI
u2 =
ax2 - y + x
\sqrt{}
a (ax2 + 2y(by + 1))
y2
,
v1 =
- ax+
\sqrt{}
a (ax2 + 2y(by - 1))
y
,
v2 =
ax2 - y - x
\sqrt{}
a (ax2 + 2y(by + 1))
y2
.
The isoptic curve of the semihyperbola can be seen in Fig. 8.
4.4. Isoptic curves of the osculating parabola.
Theorem 4.4. Let the osculating parabola be given by its equation
\bigl(
1 - x2 - y2
\bigr)
+2a(x+1)y =
= 0 (see Theorem 3.4). Then the compound \alpha -isoptic curve (0 < \alpha < \pi ) of the considered conic
has the equation
\bigl(
- 2
\bigl(
x2 + y2 - 1
\bigr)
+ 2a(x+ 1)y + a2(x+ 1)2
\bigr) 2
| a2(x+ 1)3 (4(1 - x) + 4ay + a2(x+ 1))|
=
=
\left\{
\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}2(\alpha ),
\bigl(
x2 + y2 - 1 - 2a(x+ 1)y
\bigr)
\geq 0\wedge \bigl(
1 - u21 - u22
\bigr) \bigl(
1 - v21 - v22
\bigr)
> 0\wedge
x2 + y2 > 1,
\mathrm{c}\mathrm{o}\mathrm{s}2(\alpha ),
\bigl(
x2 + y2 - 1 - 2a(x+ 1)y
\bigr)
\geq 0\wedge
x2 + y2 < 1,
\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}2(\alpha ),
\bigl(
x2 + y2 - 1 - 2a(x+ 1)y
\bigr)
\geq 0\wedge \bigl(
1 - u21 - u22
\bigr) \bigl(
1 - v21 - v22
\bigr)
< 0,
where
u1 =
- (1 + ay)(x - ay) +
\sqrt{}
y2 (x2 + y2 - 1 - 2a(x+ 1)y)
(x2 + y2) - 2axy + a2y2
,
u2 = -
y2 - ax(x+ 1)y + a2(x+ 1)y2 + x
\sqrt{}
y2 (x2 + y2 - 1 - 2a(x+ 1)y)
y ((x2 + y2) - 2axy + a2y2)
,
v1 = -
(1 + ay)(x - ay) +
\sqrt{}
y2 (x2 + y2 - 1 - 2a(x+ 1)y)
(x2 + y2) - 2axy + a2y2
,
v2 = -
y2 - ax(x+ 1)y + a2(x+ 1)y2 - x
\sqrt{}
y2 (x2 + y2 - 1 - 2a(x+ 1)y)
y ((x2 + y2) - 2axy + a2y2)
.
Fig. 9 shows some cases of the isoptic curve for the osculating parabola.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 12
ISOPTIC CURVES OF GENERALIZED CONIC SECTIONS IN THE HYPERBOLIC PLANE 1697
Our method is suited for determining the isoptic curves to generalized conic sections for all
possible parameters. Moreover with this procedure above we may be able to determine the isoptics
for other curves as well. Authors now turn attention toward 3D generalization. Already, some results
in the Euclidean space can be seen in [8].
Acknowledgment. The authors would like to thank Professor Emil Molnár for his very helpful
discussions, instructions and comments to this paper, especially his constructive suggestions to the
classification.
References
1. Böhm J., Im Hof C. Flächeninhalt verallgemeinerter hyperbolischer Dreiecke // Geom. Dedicata. – 1992. – 42. –
P. 223 – 233.
2. Cieślak W., Miernowski A., Mozgawa W. Isoptics of a closed strictly convex curve // Lect. Notes Math. – 1991. –
1481. – P. 28 – 35.
3. Cieślak W., Miernowski A., Mozgawa W. Isoptics of a closed strictly convex curve. II // Rend. Semin. Mat. Univ.
Padova. – 1996. – 96. – P. 37 – 49.
4. Cooligde J. L. The elements of non-Euclidean geometry. – Oxford: Clarendon Press, 1909.
5. Csima G., Szirmai J. Isoptic curves of the conic sections in the hyperbolic and elliptic plane // Stud. Univ. Žilina.
Math. Ser. – 2010. – 24, № 1. – P. 15 – 22.
6. Csima G., Szirmai J. Isoptic curves to parabolas in the hyperbolic plane // Pollac Periodica. – 2012. – 7. – P. 55 – 64.
7. Csima G., Szirmai J. Isoptic curves of conic sections in constant curvature geometries // Math. Communs. – 2014. –
19, № 2. – P. 277 – 290.
8. Csima G., Szirmai J. On the isoptic hypersurfaces in the n-dimensional Euclidean space // KoG (Sci. and Prof. J.
Croatian Soc. for Geometry and Graphics). – 2013. – 17.
9. Epshtein I. Sh. Complete classification of real conic sections in extended hyperbolic plane // Izv. Vyssh. Uchebn.
Zaved. – 1960. – № 1. – P. 234 – 243 (in Russian).
10. Fladt K. Die allgemeine Kegelschnittgleichung in der ebenen hyperbolischen Geometrie // J. reine und angew. Math. –
1957. – 197. – S. 121 – 139.
11. Fladt K. Die allgemeine Kegelschnittgleichung in der ebenen hyperbolischen Geometrie. II // J. reine und angew.
Math. – 1958. – 199. – S. 203 – 207.
12. Horváth G. Á. Hyperbolic plane geometry revisited // J. Geom. – 2014.
13. Holzmüller G. Einführung in die Theorie der isogonalen Verwandtschaft. – Leipzig; Berlin: B.G. Teubner, 1882.
14. Kagan V. F. Foundations of geometry. – Moscow; Leningrad: Gostekhteorizdat, 1956. – Vol. 2.
15. Kunkli R., Papp I., Hoffmann M. Isoptics of Bezier curves // Computer. Aided Geom. Des. – 2013. – 30, № 1. –
P. 78 – 84.
16. Kurusa Á. Is a convex plane body determined by an isoptic? // Beitr. Algebra und Geom. – 2012. – 53. – P. 281 – 294.
17. Liebmann H. Die Kegelschnitte und die Planetenbewegung im nichteuclidischen Raum. – Berlin; Leipzig: Ges. Wiss.,
1902. – 54. – S. 244 – 260.
18. Loria G. Spezielle algebraische und traszendente ebene Kurve. – Leipzig; Berlin: B.G. Teubner, 1911. – Bd. 1, 2.
19. Michalska M. A sufficient condition for the convexity of the area of an isoptic curve of an oval // Rend. Semin. Mat.
Univ. Padova. – 2003. – 110. – P. 161 – 169.
20. Miernowski A., Mozgawa W. On some geometric condition for convexity of isoptics // Rend. Semin. Mat. Univ.
Politec. Torino. – 1997. – 55, № 2. – P. 93 – 98.
21. Molnár E. Kegelschnitte auf der metrischen Ebene // Acta Math. Acad. Sci. Hungar. – 1978. – 31, № 3-4. –
P. 317 – 343.
22. Molnár E., Szirmai J. Symmetries in the 8 homogeneous 3-geometries, symmetry // Cult. and Sci. – 2010. – 21,
№ 1-3. – P. 87 – 117.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 12
1698 G. CSIMA, J. SZIRMAI
23. Michalska M., Mozgawa W. \alpha -Isoptics of a triangle and their connection to \alpha -isoptic of an oval // Rend. Semin. Mat.
Univ. Padova (in appear).
24. Odehnal B. Equioptic curves of conic section // J. Geom. Graph. – 2010. – 14, № 1. – P. 29 – 43.
25. Rosenfeld B. A. Non-Euclidean spaces. – Moscow: Gostekhteorizdat, 1955. – 744 p.
26. Siebeck F. H. Über eine Gattung von Curven vierten Grades, welche mit den elliptischen Funktionen
zusammenhängen // J. reine und angew. Math. – 1860. – 57. – S. 359 – 370; 1861. – 59. – S. 173 – 184.
27. Skrzypiec M. A note on secantopics // Beitr. Algebra und Geom. – 2008. – 49, № 1. – S. 205 – 215.
28. Taylor C. Note on a theory of orthoptic and isoptic loci // Proc. Roy. Soc. Edinburgh. Sect. A. – 1884. – 38.
29. Vörös C. Analitikus Bólyai féle geometria. – Budapest: Első kötet, 1909.
30. Wieleitener H. Spezielle ebene Kurven. Sammlung Schubert LVI. – Leipzig: Göschen’sche Verlagshandlung, 1908.
31. Wunderlich W. Kurven mit isoptischem Kreis // Aequat. Math. – 1971. – 6. – S. 71 – 81.
32. Wunderlich W. Kurven mit isoptischer Ellipse // Monatsh. Math. – 1971. – 75. – S. 346 – 362.
Received 09.07.16,
after revision — 13.04.17
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 12
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| id | umjimathkievua-article-2287 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:21:51Z |
| publishDate | 2019 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/70/7af95b968d96dc4858ceceb8257a2370.pdf |
| spelling | umjimathkievua-article-22872020-01-20T10:36:07Z Isoptic curves of generalized conic sections in the hyperbolic plane Ізооптичнi кривi узагальнених конiчних перерiзiв у гiперболiчнiй площинi Csima, G. Szirmai, J. Csima, G. Szirmai, J. Csima, G. Szirmai, J. We recall the notion of generalized hyperbolic angle between proper and improper straight lines, which is only available in Hungarian and Esperanto. Then we summarize the generalized hyperbolic conic sections. After investigation of the real conic sections and their isoptic curves in the hyperbolic plane $\mathbf{H}^2,$ we consider the problem of isoptic curves of generalized conic sections in the extended hyperbolic plane. This problem is widely investigated in the Euclidean plane $\mathbf{E}^2$ but, in the hyperbolic and elliptic planes, there are few results.&nbsp;Furthermore, we determine and visualize the generalized isoptic curves to all hyperbolic conic sections. For our computations, we use the classical models based on the projective interpretation of hyperbolic geometry.&nbsp;In this way, the isoptic curves can be visualized in the Euclidean screen of a computer. &nbsp; Ми нагадуємо поняття узагальненого гіперболічного кута&nbsp; між власними і невласними прямими, що на даний момент є доступним лише угорською мовою та мовою есперанто.Крім того, ми підсумовуємо дані про узагальнені гіперболічні конічні перерізи Після вивчення дійсних конічних перерізів та їхніх ізооптичних кривих у гіперболічній площині $\mathbf{H}^2$ ми розглядаємо задачу ізооптичних кривих для узагальнених конічних перерізів у розширеній гіперболічній площині. Ця проблема інтенсивно вивчається для випадку евклідової площини $\mathbf{E}^2,$ проте є небагато результатів для гіперболічних та еліптичних площин.Крім того, ми встановлюємо та візуалізуємо узагальнені ізооптичні криві для всіх конічних перерізів. У наших розрахунках ми використовуємо класичні моделі на базі проективної інтерпретації гіперболічної геометрії.&nbsp;Таким чином ізооптичні криві можуть бути візуалізовані на евклідовому екрані комп'ютера. Institute of Mathematics, NAS of Ukraine 2019-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2287 Ukrains’kyi Matematychnyi Zhurnal; Vol. 71 No. 12 (2019); 1684-1698 Український математичний журнал; Том 71 № 12 (2019); 1684-1698 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2287/1527 Copyright (c) 2019 Г. Чіма,Йе. Сірмай |
| spellingShingle | Csima, G. Szirmai, J. Csima, G. Szirmai, J. Csima, G. Szirmai, J. Isoptic curves of generalized conic sections in the hyperbolic plane |
| title | Isoptic curves of generalized conic sections in the hyperbolic plane |
| title_alt | Ізооптичнi кривi узагальнених конiчних перерiзiв у гiперболiчнiй площинi |
| title_full | Isoptic curves of generalized conic sections in the hyperbolic plane |
| title_fullStr | Isoptic curves of generalized conic sections in the hyperbolic plane |
| title_full_unstemmed | Isoptic curves of generalized conic sections in the hyperbolic plane |
| title_short | Isoptic curves of generalized conic sections in the hyperbolic plane |
| title_sort | isoptic curves of generalized conic sections in the hyperbolic plane |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2287 |
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