In the present paper the solutions based on cyclographic method are considered on the example of two applied tasks: generation of road surface forms and pocket machining process engineering. Geometric structures based on cyclographic mapping of space E3 on plane E2 and the corresponding mathematical models in the form of systems of parametric equations are provided. On the basis of the developed models, analytical solutions to the problems of shaping the surface and linear forms of the studied objects in the areas of road design and surface treatment of mechanical engineering products were obtained. The models develop the authors' previous research and are aimed at comprehensive solution to the problems of surface form generation in application to the two mentioned tasks.
Cyclography as a method of representation of space En on
space En-1 has appeared at the turn of XIX century owing to the
research of West European geometers [
In the present paper, in further development of practical application of cyclographic method, the research results summarizing the authors’ previous research in computer aided geometric modeling of road surface forms and process engineering of pocket machining are considered.
and solution of practical tasks on its basis
Formation of mathematical models of automotive road
surfaces is an essential task of their design. Geometry of these
surfaces has direct influence on traffic safety. Therefore, if the
geometry does not conform to the current norms and regulations
[
Modern mathematical modeling of road surface forms is
based on several approaches. The most widely known of them
are considered in papers [
In the present paper geometric modeling of road surface forms is considered on the basis of cyclographic mapping of road axis. Road axis is considered given in the form of a smooth spatial curve:
Pt xt , y t , z t , T0 t T , (1) where functions x(t), y(t), z(t) have continuous derivatives of up to the second order inclusive with respect to parameter t within range T0 ,T , at that, in every point within said range rank x '(t), y '(t), z '(t) 1 .
A cyclographic image of road axis as a spatial line constitutes an envelope of one-parameter multitude of bases of X (t,l) x(t) l x(1,2) (t) x(t) , Y (t,l) y(t) l y(1,2) (t) y(t) , Z (t,l) z(t) (1 l);
T0 t T , L0 l L.
Curves with vertices belonging to the given curve P(t) are
generated upon intersection of surface (3) by planes normal
with respect to orthogonal projection of road axis
P1(t) (x(t), y(t)) . The acquired profiles are different from the
conventional ones conforming to current norms and regulations
[
In order to provide required carriage way (roadside) width, initially, curves equidistant with respect to road axis orthogonal projection P1(t) are constructed. Cylindrical surfaces projecting with respect to plane (xy) are then constructed on these curves. Intersection of these surfaces and the transformed ruled surfaces results in generation of carriage way edges. The equations of these edges and road axis (1) allow us to acquire the sought ruled surfaces of carriage way. It is possible to acquire roadside surfaces bounded by carriage way edges and roadside edges in a similar fashion.
For the cases of segments straight and circular
(superelevated) in map projection, half-angle β in equations (2)
is constant. As an example, a computer rendering of straight
road segment acquired on the basis of cyclographic mapping is
depicted on fig. 1.
cones of revolution, vertices of which belong to road axis P(t) .
The equations of the envelope with half-angle of generatrix
incline at the vertex β = β(t) are of the following form [
; ; y (t) y '(t) (t) (t)
; ; (t) e(t) z '(t) e '(t) z(t); (t) x '(t)2 y '(t)2;e(t) tg ( (t)).
Simultaneous solution of the equations (1) and (2) allows us to acquire a ruled surface, parametric equations of which have the following form: (2) (3)
The transition segments are the most complex in road surface form design. The transition road segments have a particular feature of gradual change of cross profile from dual slope characteristic to straight segment to single slope characteristic to circular segment. Scheme of road surface generatrix incline variation on transition segment is presented on fig. 2.
For transition segments a function of half-angle variation β
= β(t) is applied. The plot of half-angle variation function is
depicted on fig.3; it describes law of road surface generatrix
incline variation with respect to parameter t of road axis. The
function must be continuous and single-valued within its
domain of definition. The function must be smooth with no less
than second order of smoothness in junction points in the
interest of traffic safety [
By combination of the expressions (4) and (2) and by performing certain necessary rearrangements, the equations of carriage way edges for transition segment and, correspondingly, the sought road surface form with variable generatrix incline are acquired. The result of computer aided modeling of transition segment surfaces on the basis of cyclographic mapping is presented on fig. 4.
In conclusion, the overall road surface form is divided into a number of structural segments: straight in map projection, circular (superelevated) and transition. A mathematical model taking into account essential features of road surface form generation is defined for each corresponding road segment. Calculation and rendering of road surface form on each road segement is readily realized by computer aided algebra systems due to parametric form of model representation.
Pocket machining process engineering on NC units requires
cutting tool trajectory calculation and optimization. A cutting
tool moves along equidistant trajectories called “offset curves”
(OC), gradually approaching the workpiece contour as it cuts.
An equidistant curve is a curve of fixed distance from a given
curve. The problem of optimization of OC generation includes
the following tasks: analysis and trimming of non-working OC
segments and analysis and trimming of opposing OC of
internal (island) and external boundary contours [
OC family modeling for multiply connected area bounded
by closed contours of pocket surfaces is performed on the basis
of cyclographic mapping of Euclidean space. The modeling is
considered in the view of Medial Axis Transform (MAT) [
MAT is a spatial curve reconstructed in space on the basis of cyclographic mapping of geometric information on flat area and its boundary contour. MAT is generated upon intersection of modeled α-surfaces, generatrices of which are inclined to the area on angle α = 45°.
Pocket surface with islands in plane (xy) is shaped by closed curvilinear contours consisting of arches of curves subsequently connected end-to-end with order of smoothness С2 (fig.6). External boundary contour a(a1,…,an) is described by equations of curves ai(i=1..n), of which it consists:
ai : rai (xa (ti ), ya (ti )), ti R . (5) Internal contour b(b1,…,bk) consists of arcs of curves bj(j=1...k):
bj : rbj (xb (t j ), yb (t j )), t j R . (6)
Internal contour c(c1,…,cu) consists of arcs of curves
cu(u=1...m) [
With the help of expressions (8) it is possible to acquire parametric equations of evolutes of composite curves a, b, and ec: c. For ea: (xea (ti ), yea (ti )) , eb: (xeb (t j ), yeb (t j )) , and (xec (tu ), yec (tu )) correspondingly: eai : rea (ti ) rai Rainai , ebj : reb (t j ) rbj Rbjnbj , ecu : rec (tu ) rcu Rcuncu , where Rai, Rbj и Rcu represent curvature radiuses; nai , nbj , and ncu represent unit vectors of curve normals. For evolutes eai, ebj, and ecu let us construct their spatial images - curves mai, mbj, and mcu - keeping in mind that applicate z is negative for internal contours: mai : rma (ti ) (xea, yea, zea ) , mbj : rmb (t j ) (xeb, yeb, zeb ) , mcu : rmc (tu ) (xec, yec, zec ) , where zea (xai xea )2 ( yai yea )2 , zeb (xbj xeb)2 (ybj yeb)2 , zec (xcu xec )2 ( ycu yec )2 .
α-surfaces Pai, Pbj, and Pcu are formed by pairs of curves ai and mai, bj and mbj, cu and mcu, serving as generatrices (fig. 7): Pai : rPi (ti ,l) rma (ti ) li (ra (ti ) rma (ti )) , Pbj : rPj (t j ,l j ) rmb (t j ) l j (rb (t j ) rmb (t j )) , (10)
Pcu : rPu (tu ,lu ) rmc (tu ) lu (rc (tu ) rmc (tu )) . of curve si.
Therefore, curves ai, bj, and cu constitute branches of one common curve of intersection between plane (xy) and an envelope of one-parameter multitude of α-cones with vertices on si and axes perpendicular to plane (xy). Base of each of the cones in plane (xy) is a circle of radius R tangent to curves ai, bj, cu, or ai, cu, or ai, bj, or bj, cu. Therefore, in plane (xy) a continuous multitude of points (x, y, R=z) constituting centers of circles of radiuses R=z is generated. Obviously, this multitude of points and circles represents MAT.
The acquired α-surfaces are sectioned by means of a horizontal bundle of planes along the z axis with step zi=hi==const. These lines are subject to analysis with subsequent trimming of non-working segments according to MAT. By means of orthogonal projection on plane (xy), the acquired level curves generate a family of OC (fig.9). Level lines of the internal contours Linb (i, j), Linc (i, u) level lines of the external contour Lext (i, k) are defined by the following parametric equations correspondingly: rinb(i, j) (xinb(i, j) (hi , j ), yinb(i, j) (hi , j ), zinb(i, j) (hi , j )) , where i represents the index of sectional plane; τj ,τu, and τk represent parameters of shape of segments Linb(i,j) , Linc(i,j), and Lext(i,k).
Orthogonal projection of MAT on plane (xy) is a curve, which constitutes medial axis MA.
The proposed geometric solution to the problem of OC
family generation for multiply connected area with curvilinear
boundary contours in plane (xy) is different from the known
algebraic solution [
1. The proposed solution features a vivid graphic representation of all the multitude of geometric objects and model conditions in interrelation and interconnection with each other in virtual electronic space.
2. Mathematical model of the proposed geometric model of OC family generation features parametric form of representation of its equations.
The mentioned differences make it significantly easier to acquire and analyze the solution to the task of OC family generation, which facilitates control programming for pocket machining on NC units.
The considered cyclographic solutions to two urgent and diverse applied tasks confirm practicality and unveil potential of cyclographic modeling method.