Decomposition of Screw-motion Envelopes of Quadrics

Transcription

1 Decomposition of Screw-motion Envelopes of Quadrics Šárka Voráčová * Department of Applied Mathematics Faculty of Transportation, CTU in Prague Study and realization of the construction of the envelope helical and rotational surface are very important in the mechanical engineering practice. As the basic surface we take a quadrics in general position with respect to the axis of screw motion. Then the characteristic is the Algebraic curve of 4 th degree. The conditions for decomposition of characteristics in to two conic sections are derived for all types of proper quadric surfaces. Introduction. Envelope of system of surfaces Continuous movement of a basic surface crates one -parametrical system of surfaces. Envelope surface contacts every basic surfaces along the characteristic. At any point of the characteristic the tangent to the trajectory lies in the tangent plane of basic surface. The envelope of system of surface generated by elementary motion can be created as a trajectory of characteristic of the same motion, so the study of the envelope is equivalent to the study of the characteristic. The Euclidean motion in three dimensional space is ordinary represented by matrix g(t in form: gt ( = ; γ ( ( at ( γ ( t t O The trajectory X = [, xt (, yt (, zt ( ] of point X = [, xyz,, ] is given by relation X ( t = g( t X. Let the generate surface be implicitly given by Gxyz (,, =, then the system of surfaces generating by motion g(t could be obtain by substitution of expression of trajectory X(t in to the equation of surface G( x, yz, =. The envelope is given by simultaneously solving Gxt ( (, yt (, zt ( = G = t For constant value of parameter t we obtain relations defining characteristic. It s convenient to choose t=.

2 . Envelopes of quadrics in screw motion Let the basic surface be a quadric, given by a second order algebraic equation G( x, yz, =, as generating motion we take a screw motion, than the characteristic is a intersection of two quadrics. G: G( x, y, z = G H : ( t = = t The intersection curve is of 4 th degree and we will investigate the condition under which it decomposed in to two conic sections. Let us consider the pencil of quadrics generated by G, H. s = αg+ βh; α, β R The intersection curve G H factorized in to two conic section iff two planes are included in the pencil s for given value of α, β. The symmetric matrix of pencil must be of rank two for some value of α, β, thus all minors of rd degree must be zero. We obtain homogenous equations of rd degree for the parameters of pencil α, β. To satisfy all these equations there must exists some relation for a basic surface and the generating motion.. General screw motion We shall determine the relation between axis of screw motion o and the given quadric G. The quadric G is in axial form, so the axis of screw motion must be in general position with respect to the line coordinates. Let the axis of screw motions is determined by intersection point M [ mn,,] u = u, u, u ; u = (Fig. = and direction unit vector ( S = vt cost sin t sin t cost The matrix S describes screw motion with axis o e and parameter (reduced pitch v. Frame K =(, e, e, e is obtained by revolution of frame K=(, e, e, e about axis e with angle t and about axis e with angle t and translated with vector (m, n,. m cost t sin R x = ; Ry = n cost sint sint cost sint cost Matrix representing screw motion S in frame K must be in form S =(Ry Rx S (Ry Rx -.

3 Fig Now the general relation between basic quadric Q and axis of screw motion is guaranteed. To simplify further computation it s suitable to use the relations between angles t, t and coordinates of direction vector of axis u u, u,u. ( = u u sin t = ; cos t = ; u + u u + u sin t = u ; cost = u + u The basic surface is quadrics given in canonical form. There are 7 standard-form types of quadrics. To the better orientation and simplifier computation we separate them in to 4 types with common equation cylinders, cones, central regular quadrics and paraboloids. Condition for decomposition of the envelope. Cylinder The equation Ax cylinder could be in form Ax + By =. For all these types of quadric we obtain the same conditions for decomposition of the envelope.. ( u = ( uv = nu = = u. ( (. ( u ( 4. ( u ( + By = expresses an elliptic and a hyperbolic cylinder, a parabolic u u v m = v = = u =. Cone The equation Ax + By = z expresses a elliptic and a hyperbolic cone. The envelope decomposed under the following options. u = uv = nu. ( (. ( u = ( u v = mu. ( u = ( v =

4 4. ( uv = nu ( uv = mu Please note, that these relations are mostly the same as the conditions for family of cylinders.. Paraboloid The equation Ax + By = z expresses an elliptic and a hyperbolic paraboloid. The resultant conditions for decomposition of the envelope are. ( u = ( uv = nu.. ( u = ( uv = mu ( u = ( v = ( ( A= B un u m = v( u + u 4. Basic surface is the paraboloid of revolution 5. ( mu vu nu vu ( ( uu u A B uuu A B = = B Au + Bu A Au + Bu (.4 Central Regular quadrics The family of ellipsoids and one and two sheet hyperboloids could by described by implicit equation Ax + By + Cz = There are two types of conditions for decomposition of the envelope:.4. The general central quadric of revolution. ( u= ( m = ( B m u u( A C = v( B A( Cu+ Au. ( v = ( n = ( u = ( (. u ( B A = AB ( mu + vu u ( C A = AC ( nu + vu mu ( A ( BC u m ( u n u v ABC ( u n u v BC u u vn ( u n u v ( ( ( BC uumn uv un BC u n uv un + BC uuvm uv un + ( + ( + ( + ( uv ( ( ( B + BCu u vm ( B C + BC u u u ( m v ( C B BC u v u n uv u v C B B Cu u vm u n uv B Cu v u n uun C B BCum un uv BCu u vm C ( ( BCun um+ uv + BCun um+ uv The first and second obtained relations are demonstrable; the quadrics H decomposed it to two planes. At the third case the quadrics doesn t decompose, these relation could be satisfied only if one of one of the parameters A, B, C of the basic quadrics are less then zero, so the

5 generated quadrics must be hyperboloid (one or two sheeted. All other resulting expressions have axial symmetry..4.. Example Fig The generated hyperboloid and the characteristic Let the generated quadrics is one sheet hyperboloid x + y + z =. The Axis of screw motion is given by point 5 M[,,] 5 v and the unit direction vector u = 5,, 4 4. The pencil of 5 quadrics s = G+ H decomposed in to two intersecting plane. The characteristic consists of the two conic sections, the hyperbola and the parabola. Fig : The envelope of system of hyperboloids.4. The central regular quadric of revolution If the generating surface is surface of revolution, the computation becomes easier. The coordinate plane (y,z and (x,z haven t special meaning, we could suppose without loss of generality, that the axis of the screw motion is parallel to the plane (y,z, thus u =. We obtain only two simply geometric conditions for decomposition of the characteristic curve.

6 ( u = ( m = ( v = The generated motion is a rotation about axis, which intersects the axis of revolution of the quadrics. ( u = ( n = ( vu = mu Fig 4: π The angle t between axes is proportional to their distance tan ( π u m t = u = v, and the shortest transversal of axes lies in the equator s plane (Fig 4. References: J. Bureš, J. Vanžura:Algebraická geometrie, SNTL, 989 A. Karger, M. Kargerová: Základy prostorové kinematiky,vydavatelství ČVUT, Praha, Szarková D. Graphical Processing of Characteristic and Meridian Section of Helical Envelope Surface, Proceedings of Seminars on Computational Geometry, SjF STU Bratislava a MtF STU Trnava, str. 4-9, 994 * Department of Applied Mathematics, Faculty of Transportation, Czech Technical University of Prague, Na Florenci 5, voracova@fd.cvut.cz