=Paper=
{{Paper
|id=Vol-2744/paper59
|storemode=property
|title=Plane Tangent to Quasi-Rotation Surface
|pdfUrl=https://ceur-ws.org/Vol-2744/paper59.pdf
|volume=Vol-2744
|authors=Ivan Beglov,Konstantin Panchuk
}}
==Plane Tangent to Quasi-Rotation Surface==
https://ceur-ws.org/Vol-2744/paper59.pdf
Plane Tangent to Quasi-Rotation Surface*
Ivan Beglov 1[0000−0003−3795−8119] and Konstantin Panchuk 2[0000−0001−9302−8560]
1 MIREA – Russian Technological University, Vernadskogo Ave., 78, 119454,
Moscow, Russia
beglov_ivan@mail.ru
2 Omsk State Technical University, Mira Ave., 11, 644050, Omsk, Russia
panchuk_kl@mail.ru
Abstract. The analysis of a surface generated by quasi-rotation of a straight line
around a circle is provided in the present paper. The considered case features a
straight generatrix belonging to the plane of a circular axis of quasi-rotation and
intersecting it in two points. A geometric method of determination of a point be-
longing to a surface given its projection on the axis plane is demonstrated. Geo-
metric construction of the curves of intersection between the considered surface
and a conic surface is presented. A method of determination of points belonging
to the considered surface as points belonging to the curve of intersection of two
conic surfaces is acquired. Step-by-step constructions illustrating the solution of
the problem of determination of a plane tangent to the considered surface in a
given point are provided. The problem is solved through the methods of descrip-
tive geometry. Every construction is performed according to an analytic algo-
rithm, not involving approximate methods of determination of the sought points.
The construction is carried out in a CAD system through the use of tools “straight
line by two points” and “circle by center and point”. The presented solution to
the defined problem is connected to the solution to the problem of determination
of the rays reflected from the considered surface. The results of the paper expose
the geometric properties of surfaces of quasi-rotation. The provided constructions
can serve as the basis for the research of optical properties of the considered sur-
faces.
Keywords: Quasi-Rotation, Surface of Quasi-Rotation, Tangent Plane.
1 Introduction
Quasi-rotation is a geometric correspondence between a point located in common plane
with a conic, which is the axis of quasi-rotation, and, in general case, four circles lo-
cated in planes perpendicular to the plane of the conic. If the axis of quasi-rotation
constitutes a circle, then the number of circles corresponding to a point is reduced to
Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons License
Attribution 4.0 International (CC BY 4.0).
* Publication financially supported by RFBR grant №20-01-00547
2 I. Beglov and K. Panchuk
two. This degeneracy is due to the fact that a circle, unlike other conics, has only one
real focus. Fig. 1 depicts the initial conditions and the solution to the task of quasi-
rotation of a point L around an axis i.
Fig. 1. Quasi-rotation of a point L around an axis i: the initial conditions, construction
of quasi-rotation of the point L around i on angle φ and on 180° angle.
The geometric objects on Fig. 1 are designated according to the system of designation
accepted in paper [1] and applied in description of quasi-rotation. For example, the
point L’ constitutes the result of quasi-rotation of the point L around the axis i on 180°
angle around the closest center of rotation S’; k’ is the circle correspondent to this quasi-
rotation centered at the point S’. The quasi-rotation correspondence is detailed in papers
[1, 2, 3, 4, 5]. Quasi-rotation, just as regular rotation, can be applied in solution to var-
ious problems of formation. The geometric properties of surfaces of quasi-rotation were
considered in paper [5]. Further to the research of geometrical properties of the surfaces
of quasi-rotation and their possible practical applications, the solution to the task of
construction of a plane tangent to a surface of quasi-rotation in an arbitrary point is
considered in the present paper.
2 Problem Definition
The determinant of a surface of quasi-rotation is of the following general form:
𝛼(𝑙, 𝑖 )[𝑙𝑖 = 𝑄𝑅𝑇𝑖 (𝑙 )], (1)
where α represents the surface of quasi-rotation, l represents the generatrix, i represents
the axis of quasi-rotation, QRTi represents the Quasi-Rotation Transformation appa-
ratus. A surface generated upon quasi-rotation of a straight line around a circle as the
Plane Tangent to Quasi-Rotation Surface 3
axis is considered in the present paper. The determinant (1) of such surface is of the
following form:
𝛼(𝑙(𝑙1 , 𝑙2 ), 𝑖(𝑖1 , 𝑖2 ))[𝑙𝑖 = 𝑄𝑅𝑇𝑖 (𝑙 )], (2)
where l1 and l2 represent projections of the line, i1 and i2 represent projections of the
axis. Fig. 2 depicts two projections of the generatrix l and axis i belonging to a common
plane and the horizontal projection of a point A.
Fig. 2. The initial condition to the problem of generation of a plane tangent to the surface
of quasi-rotation in a point A.
The problem is to construct a plane tangent to the surface of quasi-rotation in its point
A. The geometric part of the determinant of the surface α (2) is defined by its two pro-
jections (see Fig. 2). The point A is defined by its projection and the condition of be-
longing to the surface α. Let us put the initial conditions of the problem into symbolic
form:
Given: 𝑙(𝑙1 , 𝑙2 ), 𝑖(𝑖1 , 𝑖2 ), 𝐴(𝐴1 , 𝐴𝜖𝛼 ), 𝛼 (𝑙(𝑙1 , 𝑙2 ), 𝑖 (𝑖1 , 𝑖2 ))[𝑙𝑖 = 𝑄𝑅𝑇𝑖 (𝑙 )],
Find: Г =? , Г ⩃ 𝛼 = 𝐴,
where Г represents the sought tangent plane, symbol «⩃» represents tangency of the
geometric objects. Therefore, «Г ⩃ 𝛼 = 𝐴» means that the plane Г is tangent to the
surface α in the point A.
3 Theory
The apparatus QRTi provides two circles correspondent to any point of generatrix l (see
Fig. 1). Therefore, the surface α defined by the initial condition of the problem consists
of two sheets β’ and β’’ tangent to each other along the line l.
𝛼(𝛽’, 𝛽’’); 𝛽 ′ ⩃ 𝛽 ′′ = 𝑙 (3)
These sheets also intersect each other and themselves. This makes the surface structure
difficult to perceive.
In order to better understand the shape of this surface, it is convenient to consider
images of its sheets separately. Figures 3a and 3b depict halves of the sheets β’ and β’’
respectively. Judging by the top view, the point A belongs to the sheet β’. Let us find
4 I. Beglov and K. Panchuk
the front projection of the point A through the condition (А ϵ β’). The 3D models de-
picted on Fig. 3 are acquired constructively by means of a CAD system tool called
“surface by a network of curves”.
Fig. 3. Rendering of halves of the sheets of the surface α: a) sheet β’; b) sheet β’’.
A network of circles is constructed according to Fig. 4. The horizontal projections of
the circles ki are determined using the algorithm realized on Fig. 1. These acquired
models are not a part of the solution to the problem, but rather provide visual aid.
As seen from Fig. 3a, the point A can be located either on the upper, or on the lower
part of the sheet β’. Let us consider the locus of point A, at which it will be visible on
the top view. The constructions for the opposite case are symmetrical to the presented
below.
Plane Tangent to Quasi-Rotation Surface 5
Fig. 4. Construction of horizontal projections of circles required for 3D modeling of the sheets
of the surface α.
The solution to the defined problem applies construction operations logically justified
by the quasi-rotation apparatus.
Fig. 5 depicts the constructions defining the locus of the point L quasi-rotated on
angle φ around the axis i. The sequence of actions is depicted through numeration of
projection lines and indication of direction of the respective construction. Unlike Fig.1,
on Fig. 5 the point L is quasi-rotated in a plane that is not parallel to a projection plane.
Therefore, the point L is rotated until it matches the plane parallel to the projection
plane, where the center F of the circle i is located (step 1). The further construction is
performed in a similar to Fig. 1 way. Then the backwards rotation is applied (step 6).
Any geometric object belonging to the axis plane W upon quasi-rotation on angle φ is
located on the surface of the cone ωφ.
Fig. 5. Quasi-rotation of plane W and its point L on angle φ.
6 I. Beglov and K. Panchuk
4 Solution to the Problem
Let us acquire the front projection of the point A (see Fig. 6). In order to do that, let us
project the point A on plane π4 (π4⏊π1, π4||AF). The following correlations describe the
constructions presented on Fig. 6:
Ʃ ⏊π1, AF ϵ Ʃ, Ʃ ∩ i = S’, Ʃ ∩ l = Al, A ϵ k’ (4)
The projection А4 is located on projection of the circle k’4. The circle k’ is a result of
quasi-rotation of the point Аl. Therefore, by applying the algorithm presented on Fig. 1,
we acquire the second projection of the point A.
Fig. 6. Determination of the second projection of the point A and construction of the tangent n.
It follows from the definition of a plane tangent to a surface in any regular point, that
in order to solve the considered problem, it is necessary and sufficient to construct the
projections of two straight lines n and m tangent to a certain curve belonging to the
surface α and passing through the point A. Since Аl ϵ l, the circle k’ belongs to the
surface α generated through quasi-rotation of the line l, therefore, line n (n ⩃ k’=A) (see
Fig. 6) belongs to the sought tangent plane Г (n⊂Г).
Fig. 7 depicts section of the surface α by a cone λ (α∩λ=p). It is also correct that the
spatial curve p is a result of quasi-rotation of the line l on angle φ. The projection of the
Plane Tangent to Quasi-Rotation Surface 7
curve p on π1 constitutes conchoid of Nicomedes. The projection of the curve p on π3
constitutes a hyperbola with asymptotes d and q. The cone λ constitutes an image of
plane π1 upon its quasi-rotation on angle φ around the circle i. The angle φ was found
trough construction of the second projection of the point A, therefore А ϵ λ, p ϵ λ, А ϵ p.
Fig. 7. Quasi-rotation of the straight line l around axis i on angle φ.
In order to solve the problem, it is required to find line m tangent to the curve p in the
point A. The sought straight line m belongs to the plane tangent to the cone λ in the
point A.
A curve of the fourth order belonging to surface of a cone is a result of intersection
between such cone and another surface of the second order. Analysis of the images
presented on Fig. 7 allows us to assume that the curve p is a curve of intersection be-
tween two cones.
Fig. 8 depicts a pair of cones λ and ξ, curve of intersection of which visually resem-
bles the curve p. The axes of both cones belong to a common plane with points B and
G of to the curve p. Generatrix b of the cone ξ passes through the points B and G.
Generatrix u of the cone ξ intersects the curve p in the vertex C of the cone λ. The
projections of the generatrix u and the curve p on plane CGB are tangent to each other
in point C. The sought straight line m belongs to the plane tangent to the cone ξ in the
point A. Therefore, the sought line m is the result of intersection between two planes
that are tangent to respective cones λ and ξ in the point A.
8 I. Beglov and K. Panchuk
Fig. 8. A pair of cones intersecting each other along the curve p.
Let us find the cone ξ (see Fig. 9). In order to do that, it is sufficient to find projections
of its outline generatrices b and u. The straight line b contains the points G and B. Its
construction is not a problem of concern. The curve u is constructed with the condition
that its profile projection is tangent to profile projection of the curve p in the point С3
(u3⩃p3=C3). In other words, it is required to construct a straight line tangent to p3 in
point С3. In order to do that, let us interpret the curve p3 as a projection of a flat hyper-
bola pτ located on the surface of the cone τ. Outline generatrices of the cone τ are as-
ymptotes d and q of hyperbola pτ, axis of the cone τ is the line s. The plane of base v of
the cone τ contains a point T (T ϵ p, T ϵ v) that was precisely found through quasi-rota-
tion of the point Тl. The radius of the base of the cone τ is equal to the distance between
the point J and the line s. Let us now find the projection of the cone τ on plane π5
(π5⏊π3, π5⏊s). The line tangent to the hyperbola pτ belongs to the plane W (pτϵW) as
well as the plane CVQ (CVQ ⩃ τ=QC). Construction of a plane tangent to a cone is a
problem of basic course of descriptive geometry. Intersection between the planes W and
CVQ results in a straight line that is tangent to the hyperbola pτ in the point С
(W∩CVQ=CU). The line CU is the second outline generatrix u of the cone ξ.
As a result, the problem of construction of the straight line m is reduced to construc-
tion of a line that passes through the point A and is tangent to both of the cones λ and ξ
(m ⩃ λ=A, m ⩃ ξ = A). Obviously, the straight line m is a result of intersection of planes
tangent to the respective cones in point A. Fig. 10 depicts the construction of such
planes (CHM ⩃ λ, ЕКМ ⩃ ξ). These planes intersect each other along the line m
(AM = CHM ∩ EKM = m). The line m belongs to the sought plane Г (m⊂Г).
Plane Tangent to Quasi-Rotation Surface 9
Fig. 9. Projectional determination of the cone ξ.
Fig. 10. Projectional determination of the straight line m.
10 I. Beglov and K. Panchuk
Fig. 11 depicts the sheet β’ of the surface α and the lines m and n defining the plane Г
tangent to it. The problem was solved from the condition that the point A is visible from
the top view. If we accept that the point A is located on the lower part of the sheet β’,
then the plane tangent to β’ in the point A is located symmetrically to the found plane
Г with respect to plane π1.
Fig. 11. The sheet β’ of the surface α and lines n and m, defining the tangent plane.
5 Conclusion
In the present paper a method of determination of a plane tangent to a surface of quasi-
rotation is acquired. The construction is based on correspondence induced by quasi-
rotation of plane π1 around a curvilinear axis belonging to said plane. A plane tangent
to one of the sheets of a two-sheet surface is determined. The proposed method can be
applied to any surface defined by determinant (2), where l represents a straight line and
i represents a circle. Construction of a plane tangent to a surface of quasi-rotation is
similar for both of its sheets. The pair of cones λ and ξ is invariable for any point of
surface α acquired through rotation of the generatrix on angle φ.
Construction of a plane tangent to a surface in a given point is a part of applied tasks
related to determination of optical properties of surfaces including analysis of images
of the surrounding objects [6]. It is known that a ray is reflected form a surface in the
same direction as if it was reflected from a surface tangent to such surface at the point
of incidence. Such optical properties of surfaces are vital in architecture [7].
Plane Tangent to Quasi-Rotation Surface 11
The solutions to the problems requiring determination of aerodynamic properties of
a surface are also tied to the tangent planes. Sections of surfaces of quasi-rotation can
find application in design of products of required aerodynamic properties.
Fig. 12. Air inlet opening. A section of the sheet β’’ of the surface α is highlighted with darker
color.
Fig. 13. Examples of air inlet openings.
The capabilities of design of technical surfaces on the basis of non-ruled transfor-
mations of plane and space are described in sources [8, 9]. Fig. 12 depicts an example
of application of the sheet β’’ of the surface α (see Fig. 3) in design of an air inlet
opening. Air inlets are common elements of constructions in various areas of engineer-
ing and technology (see Fig. 13).
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