=Paper=
{{Paper
|id=Vol-2744/short27
|storemode=property
|title=Constructive Geometric Models with Imaginary Objects (short paper)
|pdfUrl=https://ceur-ws.org/Vol-2744/short27.pdf
|volume=Vol-2744
|authors=Denis Voloshinov,Alexandra Solovjeva
}}
==Constructive Geometric Models with Imaginary Objects (short paper)==
https://ceur-ws.org/Vol-2744/short27.pdf
Constructive Geometric Models with Imaginary Objects *
Denis Voloshinov1, Alexandra Solovjeva1
1 St. Petersburg State University of Telecommunication, Saint-Petersburg, Russia
denis.voloshinov@yandex.ru
alexis.soloveva@gmail.com
Abstract. The article is devoted to the consideration of a number of theoretical
questions of projective geometry related to specifying and displaying imaginary
objects, especially, conics. The lack of development of appropriate constructive
schemes is a significant obstacle to the study of quadratic images in three-dimen-
sional space and spaces of higher order. The relationship between the two circles,
established by the inversion operation with respect to the other two circles, in
particular, one of which is imaginary, allows obtain a simple and effective
method for indirect setting of imaginary circles in a planar drawing. The applica-
tion of the collinear transformation to circles with an imaginary radius also makes
it possible to obtain unified algorithms for specifying and controlling imaginary
conics along with usual real second-order curves. As a result, it allows eliminate
exceptional situations that arise while solving problems with quadratic images in
spaces of second and higher order.
Keywords: Scientific Visualization, Constructive Geometric Modeling, Geo-
metric Experiment, Projective Geometry, Quadric, Imaginary Geometric Ob-
jects.
1 Introduction
Nowadays the attention of many researchers working in the field of constructive geom-
etry is directed to solving problems related to modeling quadrics. Until recently, the
well-known, but almost forgotten constructive schemes for constructing quadrics given
with nine points located in three-dimensional space could not be implemented practi-
cally due to the instrumental complexity of such schemes in the planar drawing [1]. The
development of automation tools for geometric constructions based on the theoretical
principles of projective geometry, the beginning of introducing imaginary objects into
constructive geometry make it possible to remove from the agenda the issue of instru-
mental limitations of the geometric method [2]. However, the elimination of this prob-
lem revealed a number of other problems, primarily related to the insufficient develop-
ment of constructive geometric algorithms when working with imaginary objects that
appear in abundance, for example, when solving problems with curves or surfaces of
Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons License
Attribution 4.0 International (CC BY 4.0).
2 D. Voloshinov, A. Solovjeva
the second order. One of the simple but illustrative examples of such problem is the
projective algorithm for determination the intersection of a quadric surface with a plane
according to the initial data, represented by three conics located on its surface. Since
the choice of the plane position in the general case can be arbitrary, the probability of
the absence of an explicit (real) intersection of a straight line with the projections of
conical sections is very high. This situation, however, should not led to the algorithm
failure in general problem solving, because it may have real result. For obvious reasons,
the “manual” solution of such problems is very difficult, but in the case of computer
simulation, solution can be obtained. Of course, corresponding effective project algo-
rithms, developed on effective structural geometric schemes, should present. These
considerations explain the relevance of the study, some of obtained results will be dis-
cussed below.
2 The imaginary circle construction
Diagnostic methods for assessment of impairment of conscious level are still widely
used in the treatment in an Intensive Сare Unit. It includes tests of orientation, attention,
memory, language and visual-spatial skills. At first, J. Bennett invented Glasgow Coma
Let two visibly intersecting circles a and b be given on the plane: U ,V = a b . Let’s
find out the central points A and B of the circles a and b and draw a straight line
l = A B passing them. Our goal is to construct an inversion circle that transmits the
circle a into a circle b . It is easy to understand that this problem has two solutions.
l ; A1 ; A2
We define an involution 1 | and find its dual points P1 and P2 , where
l ; B2 ; B1
A1 , A2 = a l , B1 , B2 = b l . Taking them as diametrical ones, we construct a circle
p = P1 P2 (See Fig. 1). Then, b = p ( a ) where p ( a ) is the inversion transformation
induced in the plane by a circle p . Due to the involution properties of inversion it
follows that a = p ( b ) .
l ; A1 ; B2
Now we define an involution 2 | and determine its dual points Q1 and Q2
l ; B1 ; A2
. Taking them as diametrical ones, we construct a circle q = Q1 Q2 (See Fig. 2). Then
b = q ( a ) , where q ( a ) is the inversion transformation induced in the plane by a circle
q . Due to the involution properties of inversion it follows that a = q ( b ) .
Thus, two circles are detected that can equally map the original circles onto each
other. This, however, does not mean the equality of the point series transformation, i.e.
at 𝑇 ∼ 𝑎, Tp = p (T ) , Tq = q (T ) – Tp Tq . Moreover, the circle 𝑑 = 𝑇 ∘ 𝑇𝑝 ∘ 𝑇𝑞
drawn through the inverse image and its images on the circle b will be orthogonal to
the circles of the bundle (U , V ) and have a center lying on its radical axis. In other
Constructive Geometric Models with Imaginary Objects 3
words, we can say that the points T , T p and Tq model each other and the circle d
performs a communication line in the internal apparatus of modeling the transformation
under consideration.
Fig. 1. Demonstration of first circle of inversion definition
Fig. 2. Demonstration of first circle of inversion definition
The resulting circles p and q are orthogonal: p ⊥ q . They form an orthogonal pair
in a bundle of circles defined by the centers U ,V = a b and induced by a pair of
4 D. Voloshinov, A. Solovjeva
circles ( a, b ) . The following statements are true for this pair: p = q ( p ) and
q = p ( q ) (See Fig. 3).
Fig. 3. Demonstration of non-equality of point sequences
Carrying out further arguments in the context of the considered problem, we can for-
mulate the task of constructing inversion circles for the resulting pair of circles p and
q . Acting by analogy with the previous case, we obtain two circles r and s :
q = r ( p ) p = r ( q ) q = s ( p ) p = s ( q )
, , , . The circles r and s form the second
orthogonal pair in the bundle defined by the centers U ,V = a b and induced by the
pair of circles
( a, b ) . Continuing the construction, we find that r = p ( s ) , s = p ( r ) ,
r = q ( s ) s = q ( r ) ( p, q ) are inversion circles for the
, i.e. the components of the pair
circles of the opposite pair
( r , s ) and vice versa (See Fig. 4).
Let’s shift one of the circles a or b so that their intersection points U ,V = a b
become implicit, i.e. imaginary. This action will lead to the fact that the points of one
of the pairs of dual points of involutions 1 and 2 either take imaginary values, and
the explicit construction of one of the circles p or q becomes instrumentally impos-
sible. However, from a mathematical point of view, nothing fatal happens. If we con-
sider the analytical equivalent of the corresponding state of the obtained geometric sys-
tem, we can find that in this case we are dealing with intersections of circles, one of
which has a real center but an imaginary radius.
Constructive Geometric Models with Imaginary Objects 5
Fig. 4. The second pair of orthogonal circles in a common bundle
Now we have to answer the next question: since the problem of finding inversion
circles can lead to the appearance of circles with imaginary radii, can such circles play
the role of source data in geometrical problems, and what result will be obtained? It
should be answered in the affirmative. So, for example, constructing the inversion cir-
cles for one real circle and the other – the imaginary circle with a real center and imag-
inary radius, in the general case we get two inversion circles having not only an imag-
inary radius, but also an imaginary center.
Using only traditional tools for performing geometric models in the form of draw-
ings, the imaginary circle cannot be imagined explicitly. However, relying on the de-
pendences just obtained, it is possible to interact indirectly with such an object and
control it by influencing existing real objects, which led to the formation of an imagi-
nary object. It should be recognized that despite the purely formal and typical nature of
the actions performed for this, such a technique entails an increase in the complexity of
the practical application of the model. Therefore, when developing systems for auto-
mating geometric constructions, it is necessary to provide not only methods for the
intrasystem representation of imaginary objects, but also to create algorithmic tools for
interactive-graphic control of these data to provide comprehensive functionality of con-
structive geometry algorithms with images of a real and imaginary mathematical na-
ture.
In relation to the problem under consideration, we can assume that the imaginary
circle that is the result of an operation on real circles can be fully replaced by them,
while the operations performed with this circle and the images generated as a result of
these actions can be represented in real and complex-valued values, matched with
model circles in a constructive relationship.
Let some imaginary circle a be given on the plane. We set ourselves the goal of
indicating a point incident with this circle. If such a problem were formulated for a real
6 D. Voloshinov, A. Solovjeva
circle, then its solution did not cause any difficulties. However, under the condition of
the imaginary circle a , it becomes not quite clear how to perform such an appointment.
To solve the problem we draw an additional real circle b (See Fig. 5). Through the
centers of both circles, draw a straight line l and find the points of intersection of the
circles a and b with it: A1 , A2 = l a , B1 , B2 = l b . We define an involution
l ; A1 ; A2
1 | , find its double points P1 and P2 , through which, as diametrical, we draw
l ; B1 ; B2
a circle p = P1 P2 . A circle p is one of the inversion circles that translate the circle a
into b and vice versa: a = p ( b ) , b = p ( a ) . The second inversion circle is con-
structed similarly. We should pay attention to the fact that the circles of inversion p
and q are essentially imaginary. Choosing any of them, for example p , and transfer-
ring an arbitrary point T of a circle b relative to it in inversion, we obtain an image of
a point a ~ Tp = p (T ) , thereby providing the ability to indirectly specify a point inci-
dent with an imaginary circle a .
Fig. 5. Demonstration of algorithm on incidental point to image circle assignment
3 Imaginary conic
Let’s consider one of the possible methods of indirect synthetic determination of imag-
inary conics. Without conducting global generalizations, in the framework of this arti-
cle, by the term imaginary conics we mean conics having a real center and imaginary
half-diameters.
It is well known that in projective geometry the conics do not differ, and any non-
degenerate conic can be transformed by a collinear transformation into another, includ-
ing case of a specific conic. Since the circle is essentially a conic, we use a combination
of inversion and collineation transformations to determine imaginary conics and control
Constructive Geometric Models with Imaginary Objects 7
them, but before that we should formulate and solve one auxiliary problem on a set of
objects that have real values to ensure the visibility of the drawing that explains it.
Let two real conics a and b be given on the plane. We find such a collinear trans-
formation by which an arbitrary real point T ~ a is mapped to a real point
T = (T ) ; T ~ b .
In order to solve this problem, we have to construct the major directions of both
conics. Consider a method of constructing these directions for one of them (See Fig. 5).
First of all we have to determine a point С – the image of an infinitely distant line
in polar transformation relative to the conic a . We draw an arbitrary line m , passing
through point С , thereby defining one of the diameters of this conic. Taking the diam-
eter as the polar, we find the pole M corresponding to this polar, and draw a line
passing it and the point C that corresponds to the diameter conjugated to the diameter
m , producing a pair of conjugated diameters ( m, n = S M ) . Let’s repeat the same
action, changing the position of the line m and marking it as m* m . Now we get a
*
(* *
)
pair of conjugate diameters m , n = S M as a result. We draw an arbitrary line t
on the plane and find the points of intersection of pairs of conjugate diameters P = m t
, Q = n t , P* = m* t , Q* = n* t on it. The formed pairs of points define an invo-
t ; P; P*
lution | on the line t with dual points R and S .
t ; Q; Q*
Then we draw a circle z of zero radius centered at a point S and define the involu-
tion induced by this circle on a straight line t . This involution distinguishes two dual
t ;V ; R
points U and V on a line t . Now we define an involution | on the line t . Its
t ;U ; S
dual points X and Y , together with the point S , indicate the directions of the major
diameters of the conic a , and they, in turn, highlight the diametric points on it
X 1a , X 2a = ( S X ) a , Y1a , Y2a = ( S Y ) a (See Fig. 5).
Fig. 6. Demonstration of new synthetic algorithm on major diameters of conic definition
8 D. Voloshinov, A. Solovjeva
Having completed similar constructions with respect to the conic b , we find four points
X 1a , Y1a , X 2a , Y2a
of the same nature: X 1b , X 2b , Y1b , Y2b . Defining collineation at the
X 1b , Y1b , X 2b , Y2b
obtained points, we can get a solution of the original problem, i.e. provided T ~ a we
are able to set the point T ~ b by performing found collinear transformation :
T = (T ) (See Fig. 6).
Fig. 7. Demonstration of unified collinear transformation of one conic to another
Despite the fact that the solution of the problem was carried out using real images of
the plane, the presented algorithm is applicable to imaginary objects. Now let the conic
a be real and the conic b be imaginary. Having set a point T on the conic a , as a
result of performing the collinear transformation, we obtain an imaginary point
(T ~ b ) = (T ~ a ) that is clearly incident to the conic b ; therefore, the method of
indirectly indicating the point incident to the imaginary conic is defined.
We now consider some issues of mapping a pair of imaginary conics onto a pair of
real conics.
Let two imaginary conics а and b be given on the plane. Find the points of their
intersection A, B, C , D = a b . We also indicate on this plane four arbitrary real points
A , B , C and D ' , no triple of points of which lie on one straight line. We establish
A, B, C , D
a collinear transformation . Applying this transformation with respect
A, B , C , D
to the initial conics, we obtain two real explicitly intersecting conics a = ( a ) and
b = ( b ) . The resulting conics are elements of a conic bundle given by the centers A
, B , C and D ' . Thus, a correspondence between two bundles of conics through col-
lineation is established and objects of one bundle are unambiguously interpreted by
the real images of the other bundle (See Fig. 7).
Constructive Geometric Models with Imaginary Objects 9
Fig. 8. Demonstration of unified collinear transformation of one conic to another
Having the opportunity to build the center and the main directions of the imaginary
conic, it is no longer difficult to carry out a constructive solution to problems that are
practically inaccessible with the traditional method of implementing geometric algo-
rithms. Among them are the problems of finding focal points of imaginary second-order
curves based on the general constructive method [3], determining imaginary quadratic
involutions, and many others that have well-known real analogues.
4 Conclusion
As a result of the study, the following tasks were solved:
─ A constructive algorithm for constructing an imaginary circle based on the inversion
transformation has been developed;
─ A method of indirect constructive control of imaginary quadrics through real images
is presented;
─ A method for organizing a visual-graphic interface of an information environment
that executes geometric algorithms is proposed;
─ A constructive complex of algorithms has been developed that is equally suitable for
operations with real and imaginary conics, which allowed expanding the capabilities
of geometric methods in solving problems associated with modeling quadrics and
their multidimensional analogues using planar drawing methods.
References
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tizacija [Constructive geometric modeling. Theory, Practice, Automation]. Saarbrücken:
Lambert Academic Publ., 2010. 355 p.
3. Voloshinov, D. Unified constructive algorithm for foci of second order curves determina-
tion. Geometriya i grafika [Geometry and graphics], 2018, vol. 6, i. 2, pp. 47–54.