=Paper=
{{Paper
|id=Vol-3027/paper70
|storemode=property
|title=A Constructive Algorithm for Transforming Conics in a Circle on Imaginary Laguerre Points
|pdfUrl=https://ceur-ws.org/Vol-3027/paper70.pdf
|volume=Vol-3027
|authors=Elena Boyashova,Denis Voloshinov,Tatyana Musaeva
}}
==A Constructive Algorithm for Transforming Conics in a Circle on Imaginary Laguerre Points==
https://ceur-ws.org/Vol-3027/paper70.pdf
A Constructive Algorithm for Transforming Conics in a Circle on
Imaginary Laguerre Points
Elena Boyashova 1, Denis Voloshinov 1 and Tatyana Musaeva 1
1
St. Petersburg State University of Telecommunication, Bol'shevikov, 22/1, Saint-Petersburg, 193232, Russia
Abstract
The article is devoted to the consideration of the properties of Laguerre points in their
constructive relationship with the solution of the problem of transforming a pair of conical
curves into two circles. Geometric schemes are presented that reveal the principle of specifying
a collinear transformation that establishes a correspondence between conics and their images
in the form of circles, including if the Laguerre points defining this transformation are
imaginary. In the solution of the problem, a geometric scheme for the formation of the main
conjugations of conics is given, one of which is a circle. The conclusion is substantiated that
the radical axis of the bundle of circles is a circle that has split into a pair of straight lines, one
of which is infinitely distant. An algorithm for the formation of a series of elliptic curves
induced by Laguerre points is presented.
Keywords 1
Laguerre points, involution of point series, collineation, conjugation of conical curves
1. Introduction
The development and improvement of modern information technologies in the field of geometric
modeling has provided constructive geometry with new opportunities both in the field of supporting
information-intensive scientific research and in the application of its methods for solving practical
problems [4, 8, 9]. It became possible to create and implement in drawings models of images and
transformations [1, 15, 16] not only two-dimensional and three-dimensional, but also multidimensional
spaces, as well as to implement such models in the form of automatically synthesized programs
executed on various computing devices [4, 16, 24]. The emerging opportunities aroused interest in new
studies of seemingly well-studied mathematical images: conics, quadrics and related images of
multidimensional spaces [3, 5-7, 20]. The solution of problems operating with such images becomes
extremely difficult without taking into account the presence of imaginary images in them; it requires
research related to the search for new yet unknown regularities, understanding their geometrical nature
and formulating generalizations, developing new determinants of the investigated images and their
attendant transformations. This paper presents some results of research of such kind on the example of
identifying the relationship of Laguerre points with quadratic images of a plane.
A collinear transformation defined in a plane associates the linear preimages incident with this plane
with the corresponding linear images, and maps second-order curves to second-order curves [13, 25].
Since in affine geometry the circle is a particular form of a curve of the second order, and in projective
geometry it does not differ among curves of the second order in general, then it, naturally, is transformed
by any non-identical collineation into some conical section. Collinear transformations form a group;
therefore, there is also an inverse transformation that takes an arbitrary non-degenerate conic to a circle.
The method for determining the frame of such a collineation is to select a homology (a particular
form of a collinear transformation) in which two points of the preimage and the image, respectively,
coincided, and the remaining two points of the second-order curve would translate a pair of cyclic points
of the plane at which all circles of the plane have common crossection. Cyclic points by their nature are
GraphiCon 2021: 31st International Conference on Computer Graphics and Vision, September 27-30, 2021, Nizhny Novgorod, Russia
EMAIL: helen.glass@mail.ru (E. Boyashova); denis.voloshinov@yandex.ru (D. Voloshinov); neli_6868@mail.ru (T. Musaeva)
ORCID: 0000-0002-6539-5336 (E. Boyashova); 0000-0001-5248-4359 (D. Voloshinov); 0000-0003-0717-0507 (T. Musaeva)
©️ 2021 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
imaginary [10-12, 14, 17, 19, 21, 23] and they are incident to the line of infinity. Algorithms for
performing such a transformation are well known [18, 22], and, based on the incidence with the line at
infinity, they do not require reference to the imaginary values of cyclic points.
A natural continuation of this problem is a search for transformation of two arbitrary non-degenerate
conics into two circles by means of a collinear conversion. The presence of such an algorithm makes it
possible, for example, to solve the problem of smooth conjugation of two conics by the third conic. In
[18, 22], it was shown how the search for the frame of such a collinear transformation is carried out. A
distinctive feature of this algorithm is the use of Laguerre points [2, 13, 25] originating from equal
involutions induced by both conics on some straight line.
However, despite the legitimacy and general correctness of this approach to solving the problem, in
its practical solution, certain instrumental difficulties may arise due to the fact that Laguerre points may
turn out to be imaginary, and it becomes difficult to perform a collinear transformation by conventional
means.
This article is devoted to the consideration of a set of algorithms and related issues of constructive
geometry, which make it possible to overcome the difficulties of performing a collinear transformation
that depends on the complex values of Laguerre points.
2. Conic conjugation algorithms
Let two ellipses a and b having no real points of intersection be given on the plane (Fig. 1).
Figure 1: Two original conics
Let's perform the polar conic a transformation with respect to the conic b : a b a and the conic
b transformation with respect to the conic a : b a b . Let’s find the intersection points of the conics
K1 , K 2 , K3 , K 4 b a and L1 , L2 , L3 , L4 a b . The points of intersection of the diagonals of a
hexagon, built on four points K1 , K 2 , K 3 , K 4 , allow to define an auto-polar triangle with vertices P , Q
и R relative to the conic b . It is easy to show that the diagonals of a hexagon, built on four points
L1 , L2 , L3 , L4 , form an auto-polar triangle with the same vertices, i.e. coinciding with the one already
built. Thus, the triangle PQR is a common auto-polar triangle for both original conics a and b .
Let us find the intersection points of the straight line QR with the original conics a and b :
A1 , A2 QR a , B1 , B2 QR b as well as the images of the infinitely distant straight line i in the
polarities a and b : Ca a (i ) and Cb b (i ) . Draw a straight line p QR Z passing the point
P ; p ~ P . Considering the pair p P as a common tangent to the conics a and b , let’s construct
these conics by making the connection a A1 A2 Ca P p and b B1 B2 Cb P p . By
construction, these conics at the intersection have a common double point P , and also form two more
separate points U ,V a b . Straight lines PU u and PV v are those lines on which the original
a b a b
conics a and b induce the corresponding equal involutions u u and v v . In fact, these
u u v v
lines pass through the corresponding complex conjugate points formed by the intersection of the
original conics a and b .
Choosing any of the obtained lines, u , for example, we define on it an involution from any of the
a
original conics. Conic a induces an involution u . The involution of a point series on a straight line
u
that does not have real intersection points with the conic inducing involution, forms two real Laguerre
points G1 and G2 , from which each pair of corresponding points of the series is visible at right angles.
Taking as the center of homology one of the Laguerre points, say G1 , and two arbitrary non-coinciding
points S T , S ~ u T ~ u , draw lines s G1 S and t G1 T and define the corresponding infinitely
distant points S and T on them. As the axis of homology, we choose an arbitrary line x u and
project points S and T onto it from the center G1 to obtain the corresponding points S * and T * . Then
S , S * ,T ,T *
collineation will provide the transformation a* a and b* b , where a* and
S , S * ,T ,T *
b* are the essence of the circle (Fig. 2).
a
Figure 2: Converting two conics to two circles based on real Laguerre points
As it can be seen from the presented example, when executing the algorithm, there was no need to
explicitly refer to objects with complex-valued values, which means that all constructions can be
performed using traditional tools on sets of real values of objects. Drawing conjugate circles to circles
a* and b* transforming the result in reverse collineation 1 provides control over the process of
constructing conic conjugates of the two original conics a and b .
Let us now consider the same algorithm, but with different initial data. Let the initial curves a and
b are two hyperbolas located on the plane as shown in Fig. 3.
Figure 3: Initial hyperbolas in the problem of conjugation of conics
Performing the same operations as in the previous case, we will come to the conclusion that the
Laguerre points G1 and G2 will turn out to be complex-valued, which means that it becomes impossible
to perform the collinear transformation with conventional tools (Fig. 4).
Figure 4: An example of the formation of imaginary Laguerre points
To solve this problem, consider an auxiliary problem. Let there be some conic m and a straight line
n that has real-valued points N 1 and N 2 at the intersection with the conic. This means that the
m
involution n is of hyperbolic type and its Laguerre points are imaginary. Let us find the required
n
collineation by performing a number of indirect additional actions.
Consider the following construction. Let's draw a circle passing through antipodal points N 1 and
N 2 . Note that the radius of this circle is equal in absolute value to the radius of the imaginary circle,
which could be drawn on imaginary antipodal points G1 and G2 . Draw the perpendicular h n , h ~ Cd
from the center Cd ~ n of the circle d . A straight line h intersecting a circle d forms real points
J1 , J 2 h d . By analogy with the previous algorithm, we construct a collineation according to the
following scheme: draw a straight line y n , mark two arbitrary points E ~ n and F ~ n on a straight
line n , and project them from the center J 1 onto a straight line y to obtain points E* ~ y and F * ~ y
. Using straight lines e J1 E and f J1 F , let’s define the infinitely distant points E and F
E, E* , F , F *
and determine the collineation .
E , E* , F , F *
Having now performed the collinear transformation of the conic m , we obtain a conic m* m
that is not a circle, but an equilateral hyperbola. Therefore, a conic m* has two principal conjugate
conics, one of them is a circle. By denoting the collineation that transforms the hyperbola into a
circle, we obtain the formula for the required collineation ; m** m . Thus, the appearance
of complex-valued Laguerre points in the construction is not an obstacle to solving the problem, and it
can be performed without referring to imaginary images (Fig. 5).
Figure 5: Formation of two conjugate cones, one of which is a circle
Let's get back to the main task. Keeping the notation used in the additional construction, we complete
its solution. The result is shown in Fig. 6. As in the previous example, the controlled construction of
circles to circles a* and b* with the subsequent transformation of the result in reverse collineation 1
allows us to perform conjugation of hyperbolic curves.
Let's perform a collinear transformation with respect to lines u and v : u* u , v* v .
It is easy to find that one of these straight lines ( v* ) is the radical axis of the resulting circles, and the
second one becomes the infinitely distant straight line v* . This construction confirms once again that
the radical axis of a bundle of circles cannot be considered in isolation from the infinitely distant straight
line. In essence, both of these straight lines are items of a degenerate radical circle that split into two
linear parts, and the addition of the radical axis with a second component, an infinitely distant straight
line, eliminates the problem of intersecting a bundle of circles with a straight line from exception. The
proposed extension of the concept of the radical axis provides a single constructive commonality of its
elements, as a result of which, without any restrictions, it can be assumed that any component of such
bundle has two points of intersection with a straight line.
Figure 6: Performing conic transforms using imaginary Laguerre points
3. Laguerre points and geometric images induced by them
a
It has already been noted that the complex-valued Laguerre points produced by involution u and
u
considered as antipodal define an imaginary circle d with a radius equal in magnitude to the radius of
the circle defined at the antipodal points N 1 and N 2 (Fig. 5). The four focal points of the ellipse have
a similar property (two focuses are real, and the remaining two are complex conjugate). Thus, a number
of elliptic curves are associated with Laguerre points, the method of constructing representatives of
which is shown in Fig. 7.
Let there be given four focal points F1 , F2 and F3 , F4 , satisfying the above conditions. Let's
choose some control point A and draw a circle w A F3 F4 . Let’s find the center of the circle w and
draw a straight line through it: m F3 F4 , m ~ Cd . Let's define the points M 1 , M 2 w m . Let's
construct a circle z M1 F1 F2 and find its center C z . The point O that is the center of the circle
d F1 F2 and considered as the center of the desired ellipse e , as well as the points C d and C z allows
you to build an ellipse e O, Cd , Cz controlled by the point A . Thus, we have obtained a one-parameter
series of confocal conics induced by Laguerre points in the problem under consideration (Fig. 8). Note
also that the focal circle d and its conjugate paired imaginary circle d F3 F4 are the inversion circles
for circles t1 and t2 with centers at a point O and are conjugate to the circles w and z , defining a
system of confocal conics and form two conjugate salinons t1 w t2 and t1 z t2 .
A feature of this series is the tendency of the shape of ellipses to the shape of a circle with an increase
in the size of their semiaxes. It can be seen from the drawing that the set of points M 1 defines an
isosceles hyperbola g conjugate to a circle d , which implies the existence in this system of one more
isosceles hyperbola, also conjugate to these curves.
Figure 7: Elliptic curves induced by Laguerre points
Figure 8: Set of curves induced by Laguerre points
4. Conclusion
As a result of the study, the following results were obtained:
the properties of imaginary Laguerre points in their application to the problem of transforming
two arbitrary non-degenerate conics into two circles have been studied;
structural geometric schemes for solving the corresponding problems are presented;
the proof of the fact, that the radical axis is a circle of a special kind, split into a pair of straight
lines; one of them is the infinitely distant straight line of the plane is given;
a new constructive algorithm for the formation of a series of curved lines induced by Laguerre
points is presented.
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