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 belonging to a surface given its projection on the axis plane is demonstrated. Geometric 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 descriptive geometry. Every construction is performed according to an analytic algorithm, 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 surfaces.
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 located 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 * Publication financially supported by RFBR grant №20-01-00547 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 quasirotation of a point L around an axis i.
The geometric objects on Fig. 1 are designated according to the system of designation
accepted in paper [
The determinant of a surface of quasi-rotation is of the following general form: ( , )[ 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 apparatus. 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: ( ( 1, 2), ( 1, 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.
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 projections (see Fig. 2). The point A is defined by its projection and the condition of belonging to the surface α. Let us put the initial conditions of the problem into symbolic form:
Given: ( 1, 2), ( 1, 2), ( 1, ), ( ( 1, 2), ( 1, 2))[ = ( )], Find: Г =? , Г ⩃ = , where Г represents the sought tangent plane, symbol «⩃» represents tangency of the geometric objects. Therefore, «Г ⩃ = » means that the plane Г is tangent to the surface α in the point A. 3
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.
( ’, ’’); ′ ⩃ ′′ = (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 the front projection of the point A through the condition (А ϵ β’). The 3D models depicted on Fig. 3 are acquired constructively by means of a CAD system tool called “surface by a network of curves”. 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
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. 4
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. 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.
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 between two cones.
Fig. 8 depicts a pair of cones λ and ξ, curve of intersection of which visually resembles 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.
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 hyperbola pτ located on the surface of the cone τ. Outline generatrices of the cone τ are asymptotes 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-rotation 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 construction 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 In the present paper a method of determination of a plane tangent to a surface of quasirotation is acquired. The construction is based on correspondence induced by quasirotation 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 [
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.
The capabilities of design of technical surfaces on the basis of non-ruled
transformations of plane and space are described in sources [