Analytic Representation of Envelope Surfaces Generated by Motion of Surfaces of Revolution Ivana Linkeová

International Journal of Scientific & Engineering Research Volume 8, Issue 8, August-2017 1577 ISSN 2229-5518 Analytic Representation of Envelope Surfaces Generated by Motion of Surfaces of Revolution Ivana Linkeová

Abstract—A modified DG/K (Differential Geometry/Kinematics) approach to analytical solution of envelope surfaces generated by continuous motion of a generating surface – general surface of revolution – is presented in this paper. This approach is based on graphical representation of an envelope surface in parametric space of a solid generated by continuous motion of the generating surface. Based on graphical analysis, it is possible to decide whether the envelope surface exists and recognize the expected form of unknown analytical representation of the envelope surface. The obtained results can be used in application of envelope surfaces in mechanical engineering, especially in 3-axis and 5-axis point and flank milling of freeform surfaces.

Index Terms—Characteristic curve, Eenvelope surface, Explicit surface, Flank milling, Parametric curve, Parametric surface, Surface of revolution, Tangent plane, Tangent vector. ————————————————————

1INTRODUCTION ETERMINATION of analytic representation of an enve- analysis of the problem in parametric space of the generated D lope surface generated by continuous motion of a gener- solid. Based on graphical analysis, it is possible to decide ating surface in sufficiently general conceptionof the whether the envelope surface exists or not. If the envelope generating surface as well as the trajectory of motion is a very surface exists, it is possible to recognize expected form of ana- challenging problem of kinematic geometry and its applica- lytical representation of the envelope surface and determine tions in mechanical engineering [1], [2]. From application its concrete expression. point of view, the generating surface corresponds to the mill- The paper is organized as follows. The specification of mo- ing tool in CNC (Computer Numeric Control) milling, the tion is given in Section 2. Section 3 deals with the definition trajectory of motion corresponds to the milling path and the and basic properties of an envelope surface generated by uni- resulting envelope surface corresponds to the machined sur- variate motion of a generating surface. The method developed face. for determination of analytical representation of the envelope Academic textbooks describe primarily synthetical solution surface is described in Section 4. Examples of envelope sur- of envelope surface problem based on pointwise construction faces generated by motion of general surface of revolution are of characteristic curve of envelope surface, i.e. the curve along given in Section 5. which the generating and envelope surfaces are touching at each instant during the wholeIJSER motion. If an analytical solution OTION is published, it is either restricted to implicit expression of the 2 M figures or such a motion (linear motion, rotation, helical mo- Envelope surface is a figure generated by motion of another tion) and such a generating surface (plane [3] or spherical surface. To be able to investigate the generated figure, it is surface [4], [5]) are chosen to obtain envelope surface its ana- necessary to specify the motion, first. Let ( ), ( ), ( ), ( ) lytical representation is known in advance of solution. So be Cartesian local coordinate system of moving figure. Ana- called DG/K (Differential Geometry/Kinematics) approach to lytic representation of motion along a trajectory�Ω 𝑡𝑡 𝜉𝜉 𝑡𝑡 (𝜂𝜂 )𝑡𝑡 in𝜁𝜁 h𝑡𝑡o-� find analytical representation of envelope surface in suffi- mogenous coordinates of projective extension of Euclidean ciently general conception can be find in [2], [6]. In DG/K three-dimensional space (3D space) given by 𝐓𝐓 𝑡𝑡 approach, the envelope surface is considered to be the surface ( ) = ( ( ), ( ), ( ), 1), [ 1, 2] (1) which is touching every surface of the univariate family of is transformation matrix generating surfaces along a characteristic curve. Then, the 𝐓𝐓 𝑡𝑡 𝑥𝑥 𝑡𝑡( )𝑦𝑦 𝑡𝑡 𝑧𝑧( 𝑡𝑡) (𝑡𝑡) ∈ 𝑡𝑡0 𝑡𝑡 envelope surface is represented by a set of characteristic 𝑥𝑥 ( ) 𝑦𝑦 ( ) 𝑧𝑧 ( ) 0 curves. ( ) = 𝜉𝜉 𝑡𝑡 𝜉𝜉 𝑡𝑡 𝜉𝜉 𝑡𝑡 , [ 1, 2], (2) ( ) ( ) ( ) 0 In this paper, a modified DG/K approach is described. The ⎛𝜂𝜂𝑥𝑥 𝑡𝑡 𝜂𝜂𝑦𝑦 𝑡𝑡 𝜂𝜂𝑧𝑧 𝑡𝑡 ⎞ 𝐆𝐆 𝑡𝑡 ( ) ( ) ( ) 1 𝑡𝑡 ∈ 𝑡𝑡 𝑡𝑡 envelope surface is considered to be the surface in the solid ⎜𝜁𝜁𝑥𝑥 𝑡𝑡 𝜁𝜁𝑦𝑦 𝑡𝑡 𝜁𝜁𝑧𝑧 𝑡𝑡 ⎟ generated by univariate motion of the generating surface at where ( ), ( ), ( ) and ( ), ( ), ( ) and ( ), ( ), ( ) are cosine𝑥𝑥 𝑡𝑡 of𝑦𝑦 angles𝑡𝑡 𝑧𝑧formed𝑡𝑡 by axes ( ), ( ) and ( ) of whose every pointall three tangent vectors to all three para- ⎝𝑥𝑥 𝑦𝑦 𝑧𝑧 𝑥𝑥 ⎠ 𝑦𝑦 𝑧𝑧 𝑥𝑥 𝑦𝑦 local coordinate𝜉𝜉 𝑡𝑡 𝜉𝜉 𝑡𝑡 system𝜉𝜉 𝑡𝑡 of moving𝜂𝜂 𝑡𝑡 𝜂𝜂 figure𝑡𝑡 𝜂𝜂 and𝑡𝑡 axes𝜁𝜁 of𝑡𝑡 world𝜁𝜁 𝑡𝑡 metric curves of this solid are coplanar. The main contribution 𝑧𝑧 of this approach consists in implementation of a graphical 𝜁𝜁coordinate𝑡𝑡 system , and in the given order𝜉𝜉 𝑡𝑡 𝜂𝜂 (provided𝑡𝑡 𝜁𝜁 𝑡𝑡 that ( ), ( ) and ( ) are univariate real functions of real variable ———————————————— , defined, continuous𝑥𝑥 𝑦𝑦 and𝑧𝑧 at least once differentiable on • Ivana Linkeová is an Associate Professor in Department of Technical 𝜉𝜉[ 1𝑡𝑡, 2𝜂𝜂]). 𝑡𝑡 𝜁𝜁 𝑡𝑡 Mathematics of Faculty of Mechanical Engineering at Czech Technical 𝑡𝑡 If the trajectory is a freeform curve, the motion is called University in Prague, Czech Republic, e-mail: [email protected] curvilinear.𝑡𝑡 𝑡𝑡 An example of curvilinear motion is motion of IJSER © 2017 http://www.ijser.org International Journal of Scientific & Engineering Research Volume 8, Issue 8, August-2017 1578 ISSN 2229-5518 milling tool in CNC 5-axis milling. Two types of motion can be of tangent vectors ( , , ), ( , , ) and ( , , ) can be distinguished: general motion when the mutual position of the calculated as the determinant𝑢𝑢 𝑣𝑣 ( , , ) of matrixhaving𝑡𝑡 the local coordinate system of moving figure with respect to the three vectors as its rows𝐁𝐁 𝑢𝑢 (or𝑣𝑣 𝑡𝑡 columns),𝐁𝐁 𝑢𝑢 𝑣𝑣 i.e.𝑡𝑡 𝐁𝐁 𝑢𝑢 𝑣𝑣 𝑡𝑡 natural coordinate system of trajectory (given by tangent, ( , , ) ( , 𝐷𝐷, 𝑢𝑢) 𝑣𝑣 𝑡𝑡 ( , , ) normal and binormal line of the trajectory) is preserved dur- ( , , ) = 𝑢𝑢( , , ) 𝑢𝑢( , , ) 𝑢𝑢( , , ) 𝐵𝐵 𝐵𝐵 𝐵𝐵 . (8) ing the whole motion and translational motion when the mu- 𝑥𝑥𝑣𝑣(𝑢𝑢, 𝑣𝑣, 𝑡𝑡) 𝑦𝑦𝑣𝑣(𝑢𝑢, 𝑣𝑣, 𝑡𝑡) 𝑧𝑧𝑣𝑣(𝑢𝑢, 𝑣𝑣, 𝑡𝑡) 𝐵𝐵 𝐵𝐵 𝐵𝐵 tual position of the local coordinate system of moving figure 𝐷𝐷 𝑢𝑢If there𝑣𝑣 𝑡𝑡 exists�𝑥𝑥𝑡𝑡 𝑢𝑢a solution𝑣𝑣 𝑡𝑡 𝑦𝑦 of𝑡𝑡 𝑢𝑢equation𝑣𝑣 𝑡𝑡 𝑧𝑧 𝑡𝑡 𝑢𝑢 𝑣𝑣 𝑡𝑡 � 𝐵𝐵 𝐵𝐵 𝐵𝐵 with respect to the world coordinate system is preserved dur- ( , , ) = 0𝑥𝑥, 𝑢𝑢 𝑣𝑣 𝑡𝑡 𝑦𝑦 𝑢𝑢 𝑣𝑣 𝑡𝑡 𝑧𝑧 𝑢𝑢 𝑣𝑣 𝑡𝑡 (9) ing the whole motion. then this solution is analytical representation of unknown envelope𝐷𝐷 𝑢𝑢 𝑣𝑣 𝑡𝑡 surface in parametric space ( , , ) of the solid 3ENVELOE SURFACE ( , , ) generated by motion of the generating surface. Example of the whole procedure of envelope𝑢𝑢 𝑣𝑣 𝑡𝑡 surface gen- To derive generally analytical representation of an envelope 𝐁𝐁eration𝑢𝑢 𝑣𝑣 𝑡𝑡 is given in fig. 1. The generating surface is a general surface generated by motion of a generating surface, let us surface of revolution ( , ) given by axis of revolution and consider the generating surface in 3D space with analytic rep- meridian , see fig. 1 (a). In the case of general surface of resentation revolution the meridian𝐒𝐒 𝑢𝑢 is𝑣𝑣 a freeform curve. To simplify𝑜𝑜 the ( , ) = ( ( , ), ( , ), ( , ), 1), picture, the𝑚𝑚 trajectory of motion is planar freeform curve ( ) [ , ], [ , ], (3) 1 2 1 2 located in the plane perpendicular to the axis . Several posi- and𝐒𝐒 𝑢𝑢 𝑣𝑣transformation𝑥𝑥 𝑢𝑢 𝑣𝑣 𝑦𝑦 matrix𝑢𝑢 𝑣𝑣 𝑧𝑧 (2)𝑢𝑢 𝑣𝑣of continuous motion. Then a tions of the generating surface along the trajectory is drawn𝐓𝐓 in𝑡𝑡 𝑢𝑢solid ∈ 𝑢𝑢 𝑢𝑢 𝑣𝑣 ∈ 𝑣𝑣 𝑣𝑣 fig. 1 (b). The generating surface moves by general𝑜𝑜 motion ( , , ) = ( , ) ( ) , because its position with respect to the trajectory is preserved. [ ] [ ] [ ] 1, 2 , 1, 2 , 1, 2 (4) In particular, the meridian plane of the generating surface and 𝐁𝐁 𝑢𝑢 𝑣𝑣 𝑡𝑡 𝐒𝐒 𝑢𝑢 𝑣𝑣 ∙ 𝐆𝐆 𝑡𝑡 is generated by motion of the generating surface (3) along the the normal plane of the trajectory are identical. The solid 𝑢𝑢 ∈ 𝑢𝑢 𝑢𝑢 𝑣𝑣 ∈ 𝑣𝑣 𝑣𝑣 𝑡𝑡 ∈ 𝑡𝑡 𝑡𝑡 trajectory (1). A part ( , , ) generated by motion of generating surface along the ( , ), [ 1, 2], [ 1, 2] (5) trajectory is drawn in fig. 1 (c). The resulting envelope surface of superficies of this solid is called envelope surface if the 𝐁𝐁(𝑢𝑢, 𝑣𝑣) together𝑡𝑡 with characteristic curve ( ) is shown in fig. 1 𝐄𝐄following𝑠𝑠 𝑡𝑡 𝑠𝑠 conditions ∈ 𝑠𝑠 𝑠𝑠 𝑡𝑡 are ∈ 𝑡𝑡 satisfied.𝑡𝑡 (d). Note that in practical applications, this type of motion and 1. Surface (5) and any -parametric surface 𝐄𝐄envelope𝑠𝑠 𝑡𝑡 surface generation corresponds𝐂𝐂 𝑠𝑠 to 2.5-axis flank ( , , ), [ 1, 2], = 0,1, , (6) milling. of solid (4) are tangent𝑢𝑢𝑢𝑢 along -parametric curve ( , ) of 𝑖𝑖 𝑖𝑖 𝐁𝐁surface𝑢𝑢 𝑣𝑣 𝛾𝛾 (5) 𝛾𝛾called∈ 𝑡𝑡 characteristic𝑡𝑡 𝑖𝑖 ⋯ curve𝑛𝑛 of envelope surface. 𝑖𝑖 2. At each point of surface (5) exist𝑠𝑠 s common tangent𝐄𝐄 𝑠𝑠 plane𝛾𝛾 and normal line of surface (5) and one -parametric surface (6) of solid (4). 3. A surface which is simultaneouslyIJSER a part𝑢𝑢 𝑢𝑢of surface (5) and any -parametric surface (6) does not exist. As the characteristic curve is located on the generating surface, each𝑢𝑢𝑢𝑢 point of characteristic curve moves together with the generating surface. To satisfy the contact of the generating and envelope surfaces along the characteristic curve, the common tangent plane of the generating and envelope surface has to exist. The tangent plane at any point of the generating surface is given by tangent vectors to the parametric - and - curves of the generating surface. Considering the motion of the generating surface along the trajectory, it is necessary𝑢𝑢 𝑣𝑣to exclude points of the generating surface located inside the solid (4). To exclude these points, the tangent plane has to Fig. 1: Process of envelope surface generation contain the tangent vector of parametric -curve of the solid, too. Obviously, this condition is fulfilled in case the tangent vectors of all three parametric curves of solid𝑡𝑡 (4) are coplanar, i.e. their mixed product is equal to zero. 4 ANALYTIC REPRESENTATION OF ENVELOPE SURFACE Tangent vectors of parametric curves of solid (4) are given There are two basic situations when solving envelope surface by first partial derivatives of cordinate functions of the solid problem: the shape and analytical representation of envelope (4) with respect to individual varialbes, i.e. surface is known in advance, and for the given generating ( , , ) ( , , ) = = ( ( , , ), ( , , ), ( , , ), 0), surface and the given motion we know neither wheatear the 𝜕𝜕𝐁𝐁 𝑢𝑢 𝑣𝑣 𝑡𝑡 envelope surface exists nor its possible shape and analytical 𝑤𝑤= , , . 𝑤𝑤 𝑤𝑤 𝑤𝑤 (7) 𝐁𝐁 𝑢𝑢 𝑣𝑣 𝑡𝑡 𝜕𝜕𝜕𝜕 𝑥𝑥𝐁𝐁 𝑢𝑢 𝑣𝑣 𝑡𝑡 𝑦𝑦𝐁𝐁 𝑢𝑢 𝑣𝑣 𝑡𝑡 𝑧𝑧𝐁𝐁 𝑢𝑢 𝑣𝑣 𝑡𝑡 representation. Envelope surfaces generated by motion of a The mixed product sphere belong to the first category [8], see examples in fig. 2. 𝑤𝑤 𝑢𝑢 𝑣𝑣 𝑡𝑡 ( , , ) [ ( , , ) × ( , , )] 𝑢𝑢 𝑣𝑣 𝑡𝑡 IJSER © 2017 𝐁𝐁 𝑢𝑢 𝑣𝑣 𝑡𝑡 ∙ 𝐁𝐁 𝑢𝑢 𝑣𝑣 𝑡𝑡 𝐁𝐁 𝑢𝑢 𝑣𝑣 𝑡𝑡 http://www.ijser.org International Journal of Scientific & Engineering Research Volume 8, Issue 8, August-2017 1579 ISSN 2229-5518 lution: envelope surface is - or -parametric surface of the solid ( , , ), envelope surface is an explicit surface of the solid and envelope surface𝑣𝑣𝑣𝑣 is a𝑢𝑢 𝑢𝑢surface in the solid expressed parametrically,𝐁𝐁 𝑢𝑢 𝑣𝑣 𝑡𝑡 see Sections 4.1, 4.2 and 4.3. 4. 1 Envelope Surface Is Parametric Surface of the Solid Envelope surface is - or -parametric surface of the solid ( , , ) in case the intersection curve is straight line = , [ 1, 2]or 𝑣𝑣=𝑣𝑣 , 𝑢𝑢𝑢𝑢[ 1, 2] 𝐁𝐁of 𝑢𝑢non𝑣𝑣 -𝑡𝑡variable shape during the whole animation. Then it is possible𝑢𝑢 𝛼𝛼 𝛼𝛼 to ∈ 𝑢𝑢express𝑢𝑢 𝑣𝑣 the𝛽𝛽 analytic𝛽𝛽 ∈ 𝑣𝑣 𝑣𝑣representation of envelope surface as ( , ) = ( , , )or ( , ) = ( , , ). The characteristic curve of envelope surface is - or - parametric𝐄𝐄 𝑣𝑣 𝑡𝑡 𝐁𝐁 𝛼𝛼curve𝑣𝑣 𝑡𝑡 of 𝐄𝐄the𝑢𝑢 solid𝑡𝑡 𝐁𝐁 (𝑢𝑢 𝛽𝛽, ,𝑡𝑡 ) in this case. Envelope surfaces of this type are generated by motion of a sphere𝑣𝑣 (fig.𝑢𝑢

Fig. 2: Example of envelope surfaces generated by motion of a sphere (a) 2), general motion of a surface𝐁𝐁 𝑢𝑢of 𝑣𝑣revolution𝑡𝑡 along a planar canal surface (generated by general motion of a sphere), (b) torus (gener- trajectory (fig. 1) and by rotary motion of a surface of revolu- ated by rotary motion of a sphere), (c) serpentine of Archimedes (gener- tion when axis of rotary motion and axis of the generating ated by helical motion of a sphere), (d) cylinder of revolution (generated by linear motion of a sphere) surface are parallel (see example in Section 5.1) or intersecting. 4. 2 Envelope SurfaceIsExplicitSurface in the Solid If the shape and analytical representation of envelope sur- Envelope surface is an explicit surface in the solid ( , , ). face is unknown, it is possible to find it by means of the This situation occurs in case the intersection curve represents a method which has been developed to solve envelope surfaces graph of univariate functionfor each , = 0,1, , , i.e.𝐁𝐁 𝑢𝑢 𝑣𝑣 𝑡𝑡 problem in the most general conception. This method is based = ( )or = ( ) (10) 𝑖𝑖 on graphical analysis of the problem in parametric space in the case of non-variable shape and𝛾𝛾 𝑖𝑖 ⋯ 𝑛𝑛 ( , , ) of the solid (4). Individual steps of the method are as 𝑢𝑢 = 𝑢𝑢(𝑣𝑣, )or𝑣𝑣 =𝑣𝑣 𝑢𝑢( , ) (11) follows. in the case of variable shape of the intersection curve. 𝑢𝑢 𝑣𝑣 𝑡𝑡 1. Solid ( , , ) – Firstly, the analytical representation of the 𝑢𝑢 Using𝑢𝑢 𝑣𝑣 𝑡𝑡 (10),𝑣𝑣 the𝑣𝑣 𝑢𝑢analytical𝑡𝑡 representation of the envelope solid ( , , ) acc. (4) and its tangent vectors ( , , ), surface is given by 𝐁𝐁 𝑢𝑢 𝑣𝑣 𝑡𝑡 ( , , ) and ( , , ) acc. (7) is necessary to determine.𝑢𝑢 ( , ) = ( ( ), , )or ( , ) = ( , ( ), ) (12) 𝐁𝐁 𝑢𝑢 𝑣𝑣 𝑡𝑡 𝐁𝐁 𝑢𝑢 𝑣𝑣 𝑡𝑡 2. Determinant𝑣𝑣 (𝑡𝑡 , , ) – Next, the determinant ( , , ) and using (11), the analytical representation of the envelope 𝐁𝐁 𝑢𝑢 𝑣𝑣 𝑡𝑡 𝐁𝐁 𝑢𝑢 𝑣𝑣 𝑡𝑡 acc. (8) is expressed. Generally, the determinant is a func- 𝐄𝐄surface𝑣𝑣 𝑡𝑡 is𝐁𝐁 given𝑢𝑢 𝑣𝑣 by𝑣𝑣 𝑡𝑡 𝐄𝐄 𝑢𝑢 𝑡𝑡 𝐁𝐁 𝑢𝑢 𝑣𝑣 𝑢𝑢 𝑡𝑡 𝐷𝐷 𝑢𝑢 𝑣𝑣 𝑡𝑡 𝐷𝐷 𝑢𝑢 𝑣𝑣 𝑡𝑡 tion of three variables. ToIJSER be able to realize graphical rep- ( , ) = ( ( , ), , )or ( , ) = ( , ( , ), ). (13) resentation of equation ( , , ) = 0 is therefore suitable to Envelope surfaces (12) are generated by helical and linear choose sufficient number of instants = , = 0,1, , 𝐄𝐄motion𝑣𝑣 𝑡𝑡 of𝐁𝐁 a𝑢𝑢 surface𝑣𝑣 𝑡𝑡 𝑣𝑣 𝑡𝑡of revolution𝐄𝐄 𝑢𝑢 𝑡𝑡 𝐁𝐁and𝑢𝑢 𝑣𝑣by𝑢𝑢 rotary𝑡𝑡 𝑡𝑡 motion of a 𝐷𝐷 𝑢𝑢 𝑣𝑣 𝑡𝑡 (the function of three variables becomes an explicit func- surface of revolution when the axis of rotary motion and the 𝑡𝑡 𝛾𝛾𝑖𝑖 𝑖𝑖 ⋯ 𝑛𝑛 tion of two variables) and draw in three-dimensional graph axis of the generated surface are skew lines (see example in both the explicit function ( , , ) and function = 0. Section 5.2). Envelope surfaces (13) are generated by general Thus, three-dimensional graphs is obtained. or generalized translational motion of surface of revolution 𝐷𝐷 𝑢𝑢 𝑣𝑣 𝛾𝛾𝑖𝑖 𝑧𝑧 3. Animation – All three-dimensional graphs are arranged along a spatial trajectory (see example in Section 5.3). 𝑛𝑛 into animation sequence. Note that if the shape of intersection curve is so compli- 4. Existence of solution – If there exists an intersection curve cated that it cannot be described by one equation, it is possible of the surface ( , , ) and = 0 for each = 0,1, , , the to find corresponding function (10) or (11) in piecewise form envelope surface exists. The intersection curve is a repre- 𝑖𝑖 or use graphical solution described in Section 4.3. sentation of characteristic𝐷𝐷 𝑢𝑢 𝑣𝑣 𝛾𝛾 curve𝑧𝑧 of envelope𝑖𝑖 surface⋯ in𝑛𝑛 parametric space ( , , ) of the solid (4). 4. 3 Envelope SurfaceIsaSurface in the Solid Expressed 5. Non-variable or variable shape of characteristic curve – If Parametrically the intersection 𝑢𝑢curve𝑣𝑣 𝑡𝑡 has identical shape during the whole Envelope surface is a surface in the solid ( , , ) expressed animation, solution of equation ( , , ) = 0 does not de- parametrically. This situation occurs if there does not exist a pend on and, consequently, characteristic curve of enve- solution of equation ( , , ) = 0 in previously𝐁𝐁 𝑢𝑢 𝑣𝑣 𝑡𝑡 mentioned lope surface has non-variable shape.𝐷𝐷 𝑢𝑢 𝑣𝑣If 𝑡𝑡the shape of inter- forms or due to the complexity of intersecting curve it is im- section curve𝑡𝑡 is not constant during animation, the solution possible to find it. Then𝐷𝐷 𝑢𝑢 it𝑣𝑣 is𝑡𝑡 useful to express the solution of equation ( , , ) = 0 depends on and, therefore, the parametrically in the following form characteristic curve of envelope surface has variable shape. = ( ), = ( ) (14) 6. Type of solution𝐷𝐷 𝑢𝑢 𝑣𝑣 𝑡𝑡– Based on the shape𝑡𝑡 of intersection in case of non-variable shape and curve it is possible to distinguish the following type of so- 𝑢𝑢 = 𝑢𝑢(𝑠𝑠, )𝑣𝑣, =𝑣𝑣 𝑠𝑠( , ) (15)

IJSER © 2017𝑢𝑢 𝑢𝑢 𝑠𝑠 𝑡𝑡 𝑣𝑣 𝑣𝑣 𝑠𝑠 𝑡𝑡 http://www.ijser.org International Journal of Scientific & Engineering Research Volume 8, Issue 8, August-2017 1580 ISSN 2229-5518 in case of variable shape of the intersection curve. plane given by Using (14), the analytical representation of the envelope ( ) = ( cos , sin , 0 , 1), [0,2 ]. (20) surface is given by The transformation matrix (2) of rotary motion in this case ( , ) = ( ( ), ( ), ) (16) 𝐓𝐓is 𝑡𝑡 𝑟𝑟 𝑡𝑡 𝑟𝑟 𝑡𝑡 𝑡𝑡 ∈ 𝜋𝜋 and using (15), the analytical representation of the envelope cos sin 0 0 𝐄𝐄surface𝑠𝑠 𝑡𝑡 is𝐁𝐁 given𝑢𝑢 𝑠𝑠 by𝑣𝑣 𝑠𝑠 𝑡𝑡 sin cos 0 0 ( ) = . (21) ( , ) = ( ( , ), ( , ), ). (17) 0𝑡𝑡 0𝑡𝑡 1 0 − 𝑡𝑡 𝑡𝑡 Depending on the complexity of the shape of generating 𝐆𝐆 𝑡𝑡 � cos sin 0 1� surface,𝐄𝐄 𝑠𝑠 𝑡𝑡 envelope𝐁𝐁 𝑢𝑢 𝑠𝑠 𝑡𝑡 𝑣𝑣surfaces𝑠𝑠 𝑡𝑡 𝑡𝑡 (16) and (17) can be generated by The determinant (8) has the following form any motion. ( , ) =𝑟𝑟 ( 𝑡𝑡) ( 𝑟𝑟 ) sin𝑡𝑡 , Note that in the most general approach it is possible to use see fig. 4, where the spatial graph of this surface is drawn. The planar two-dimensional graph of intersection curve drawn in determinant𝐷𝐷 𝑢𝑢 𝑣𝑣 𝑟𝑟𝑟𝑟 𝑣𝑣is 𝑧𝑧′independent𝑣𝑣 𝑢𝑢 on , therefore the characteristic ( , ) plane and read parametric coordinates and of suffi- curve has non-variable shape – equal to the meridian (18). cient number of points along the intersection curve. These Solution of equation ( , ) =𝑡𝑡 0 is points𝑢𝑢 𝑣𝑣 can be consequently fitted by interpolation𝑢𝑢 or𝑣𝑣 approxi- = 0, = , (22) mationcurve to obtain mathematical model of (14) or (15). i.e. the envelope surface𝐷𝐷 is𝑢𝑢 two𝑣𝑣 -branches -parametric surface of𝑢𝑢 the generated𝑢𝑢 𝜋𝜋 solid. It is obvious from graphical representation of (22), i.e. from the mapping of envelope𝑣𝑣𝑣𝑣 surface in pa- XAMPLES 5 E rametric space ( , , ) of the generated solid shown in fig. 5. In this section, examples of envelope surfaces generated by As each branche of the envelope surface is mapped into a motion of general surface of revolution drawn in fig. 3 are plane parallel with𝑢𝑢 𝑣𝑣 ( 𝑡𝑡, ) plane, it is a parametric surface of the given. The generating surface is depicted in basic position, i.e. solid. its local coordinate system ( , , , ) is identical with the Two branches of envelope𝑢𝑢 𝑣𝑣 surface in 3D space are given by world coordinate system (0, , , ). By rotary motion of free- ( , ) = (cos [ ( ) + ], sin [ ( ) + ] , ( ), 1), form left principle half-meridianΩ given𝜉𝜉 𝜂𝜂 𝜁𝜁 by ( , ) = (cos [ ( ) + ], sin [ ( ) + ] , ( ), 1), 𝑥𝑥 𝑦𝑦 𝑧𝑧 ( ) = ( ( ), ( ), ( ), 1) = 𝐄𝐄see∗𝑣𝑣 fig.𝑡𝑡 6. 𝑡𝑡 𝑥𝑥 𝑣𝑣 𝑟𝑟 𝑡𝑡 𝑥𝑥 𝑣𝑣 𝑟𝑟 𝑧𝑧 𝑣𝑣 = (20 3 33 2 + 15 , 0,9 3 15 2 + 12 , 1), [0,1] (18) 𝐄𝐄 𝑣𝑣 𝑡𝑡 𝑡𝑡 𝑥𝑥 𝑣𝑣 𝑟𝑟 𝑡𝑡 𝑥𝑥 𝑣𝑣 𝑟𝑟 𝑧𝑧 𝑣𝑣 𝐌𝐌along𝑣𝑣 -axis,𝑥𝑥 𝑣𝑣 the𝑦𝑦 𝑣𝑣analytic𝑧𝑧 𝑣𝑣 representation of surface of revolution is𝑣𝑣 obtained− 𝑣𝑣 𝑣𝑣 𝑣𝑣 − 𝑣𝑣 𝑣𝑣 𝑣𝑣 ∈ ( , 𝑧𝑧) = ( ( , ), ( , ), ( , ), 1), [0,2 ], [0,1], (19) ( , ) = (20 3 33 2 + 15 ) cos , 𝐒𝐒(𝑢𝑢,𝑣𝑣) = (𝑥𝑥20𝑢𝑢 3𝑣𝑣 𝑦𝑦33𝑢𝑢 2𝑣𝑣+𝑧𝑧15𝑢𝑢 )𝑣𝑣sin ,𝑢𝑢 ∈ 𝜋𝜋 𝑣𝑣 ∈ 𝑥𝑥(𝑢𝑢, 𝑣𝑣) = 9 3𝑣𝑣 15− 2𝑣𝑣+ 12 . 𝑣𝑣 𝑢𝑢 𝑦𝑦 𝑢𝑢 𝑣𝑣 𝑣𝑣 − 𝑣𝑣 𝑣𝑣 𝑢𝑢 𝑧𝑧 𝑢𝑢 𝑣𝑣 𝑣𝑣 − 𝑣𝑣 𝑣𝑣 IJSER

Fig. 4: Graphical solution of equation ( , ) = 0 for envelope surface generated by rotary motion of general surface of revolution when axis of rotary motion and axis of the generating surface𝐷𝐷 𝑢𝑢 𝑣𝑣 are parallel

Fig. 3: General surface of revolution in basic position (top, front and axonometric view)

In the following sections the transformation matrices for all the considered types of motionare given. Analytical representation of the generated solid (4) is too largefor all the presented examples, therefore, it is not given here. 5.1Envelope Surface Is Parametric Surface of the Solid In this section an example of envelope surface generated by rotary motion of surface of revolution (19) is shown. The axis of rotary motion and axis of surface of revolution are parallel. The trajectory of motion is a circle of radius lying in ( , ) Fig. 5: Envelope surface (with several characteristic curves) generated by 𝑟𝑟 𝑥𝑥IJSER𝑦𝑦 © 2017 http://www.ijser.org International Journal of Scientific & Engineering Research Volume 8, Issue 8, August-2017 1581 ISSN 2229-5518 rotary motion of general surface of revolution when axis of rotary motion and axis of the generating surface are parallel (mapping in parametric space ( , , ) of the generated solid)

𝑢𝑢 𝑣𝑣 𝑡𝑡

Fig. 7: Initial position of general surface of revolution (top, front and axonometric view)

The determinant in this case given by ( , ) = ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) + Fig. 6: Envelope surface generated by rotary motion of general surface of + ( , ) ( , )𝑣𝑣 (𝑢𝑢 , ) ( , ) 𝑢𝑢 𝑣𝑣 revolution when axis of rotary motion and axis of the generating surface 𝐷𝐷 𝑢𝑢( 𝑣𝑣𝑣𝑣, ) 𝑥𝑥(𝑢𝑢𝑢𝑢, 𝑣𝑣) 𝑥𝑥 ( 𝑢𝑢, 𝑣𝑣)𝑢𝑢𝑧𝑧+ 𝑢𝑢( 𝑣𝑣, 𝑣𝑣 )− 𝑥𝑥( 𝑢𝑢, 𝑣𝑣) 𝑥𝑥 ( 𝑢𝑢, 𝑣𝑣) 𝑧𝑧 𝑢𝑢 𝑣𝑣 are parallel (top, front and axonometric view in 3D space) does𝑟𝑟𝑥𝑥 not𝑢𝑢 𝑣𝑣 depend𝑢𝑢𝑧𝑧 𝑢𝑢 𝑣𝑣 on𝑣𝑣 − 𝑟𝑟𝑥𝑥, see𝑢𝑢 fig.𝑣𝑣 𝑧𝑧8, where𝑢𝑢 𝑣𝑣𝑣𝑣 − the𝑢𝑢 spatial graph of this surface−𝑦𝑦 𝑢𝑢 𝑣𝑣 is𝑦𝑦 drawn.𝑢𝑢 𝑣𝑣 𝑧𝑧 In𝑢𝑢 this𝑣𝑣 case,𝑦𝑦 𝑢𝑢 𝑣𝑣it 𝑦𝑦is possible𝑢𝑢 𝑣𝑣 𝑧𝑧 𝑢𝑢 to𝑣𝑣 find analytical 𝑡𝑡 5.2 Envelope Surface Is ExplicitSurface of the Solid, solution of equation ( , ) = 0 in form Characterictic Curve Has Non-Variable Shape = ( ), = ( ) + , (23) 𝐷𝐷 𝑢𝑢 𝑣𝑣 Here an example of envelope surface generated by rotary i.e. the envelope∗ surface is an explicit surface in the solid with 𝑢𝑢 𝑢𝑢 𝑣𝑣 𝑢𝑢 𝑢𝑢 𝑣𝑣 𝜋𝜋 motion of surface of revolution (19) when axis of rotary mo- two branches. tion and axis of the surface of revolution are skew. The trajec- Graphical representation of solution (23) in parametric tory of motion is the circle given by (20), transformation ma- space is shown in fig. 9. Note that each branche of envelope trix of rotary motion is given by (21). surface in parametric space is a ruled surface ( -parametric The difference from the previous example is the initial posi- curves of this surface are straight lines), i.e. the characteristic 𝑡𝑡 tion of surface of revolution (19) given by rotation of surface of curve has non-variable shape. revolution around axis by angle , 0, , see fig. 6, to Neither the analytical expression of the solution (23) nor obtain skew position of axesIJSER. Transformation matrix of this envelope surface is given here because it is too large.Envelope rotation is 𝜉𝜉 𝜃𝜃 𝜃𝜃 ≠ 𝜃𝜃 ≠𝜋𝜋 surface in 3D space is drawn in fig. 10. 1 0 0 0 𝐆𝐆� 0 cos sin 0 = 0 sin cos 0 𝜃𝜃 𝜃𝜃 𝐆𝐆� �0 0 0 1� and the solid generated by− rotary𝜃𝜃 motion𝜃𝜃 of surface of revolution in this case is than given by ( , , ) = ( , ) ( ), [0,2 ], [0,1], [0,2 ].

𝐁𝐁 𝑢𝑢 𝑣𝑣 𝑡𝑡 𝐒𝐒 𝑢𝑢 𝑣𝑣 ∙ 𝐆𝐆� ∙ 𝐆𝐆 𝑡𝑡 𝑢𝑢 ∈ 𝜋𝜋 𝑣𝑣 ∈ 𝑡𝑡 ∈ 𝜋𝜋

Fig. 8: Graphical solution of equation ( , ) = 0for envelope surface generated by rotary motion of general surface of revolution when axis of rotary motion and axis of the generating surface𝐷𝐷 𝑢𝑢 𝑣𝑣 are skew

IJSER © 2017 http://www.ijser.org International Journal of Scientific & Engineering Research Volume 8, Issue 8, August-2017 1582 ISSN 2229-5518 0 𝑐𝑐 𝑑𝑑 𝑒𝑒 7 2+4 8 6 13 2+14 3 10 (3 2) 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 0 ( ) = − 𝑡𝑡 𝑡𝑡− �− 𝑡𝑡 𝑡𝑡− � 𝑡𝑡 𝑡𝑡− , 30 8 2 9 +3 5 2+24 8 3 23 2+83 50 ⎛ 𝑏𝑏 𝑏𝑏 𝑏𝑏 0⎞ 𝐆𝐆 𝑡𝑡 ⎜ � 𝑡𝑡 − 𝑡𝑡 � �𝑡𝑡 𝑡𝑡− � � 𝑡𝑡 𝑡𝑡− � ⎟ ⎜ ( ) ( ) ( ) 1⎟ 𝑎𝑎 𝑎𝑎 𝑎𝑎 where ⎝ 𝑥𝑥 𝑡𝑡 𝑦𝑦 𝑡𝑡 𝑧𝑧 𝑡𝑡 ⎠ = 62386 4 94038 3 + 17140 2 132900 + 32200, = 7033 4 14360 3 + 10392 2 3099 + 388 � 𝑎𝑎 = 2( 2766𝑡𝑡 4− 856 3𝑡𝑡+ 17487 2𝑡𝑡+−264 400𝑡𝑡 ), 𝑏𝑏 =� 3(2561𝑡𝑡 4− 3811 𝑡𝑡3 + 2202 2𝑡𝑡+ −264 𝑡𝑡400), 𝑐𝑐 = 5(−3737 4𝑡𝑡 −8408 𝑡𝑡3 + 6948 2𝑡𝑡+ 2708 𝑡𝑡 −388). Neither𝑑𝑑 the 𝑡𝑡concrete− 𝑡𝑡analytical𝑡𝑡 representation𝑡𝑡 − of the deter- minant𝑒𝑒 (8) nor𝑡𝑡 the− solution𝑡𝑡 of equation𝑡𝑡 𝑡𝑡( , − , ) = 0 is presented here because it is too large. The determinant ( , , ) is 𝐷𝐷 𝑢𝑢 𝑣𝑣 𝑡𝑡 function of all three variables. Therefore, the graphical analy- Fig. 9: Envelope surface (with several characteristic curves) generated by sis along with the animation sequence is suitable to𝐷𝐷 set𝑢𝑢 𝑣𝑣up𝑡𝑡 to rotary motion of general surface of revolution when axis of rotary motion solve this problem. An example of spatial graph for = 0.5 of and axis of the generating surface are skew (mapping in parametric space ( , , ) of the generated solid) surface ( , , 0.5) and plane = 0 is drawn in fig. 11. There are two intersection curves representing graphs𝑡𝑡 of 𝑢𝑢 𝑣𝑣 𝑡𝑡 functions𝐷𝐷 of𝑢𝑢 𝑣𝑣 variable. In this𝑧𝑧 case, it is possible to find solution of equation ( , , ) = 0 in explicit form = ( , ), 𝑣𝑣= ( , ) + . (24) ∗ 𝐷𝐷 𝑢𝑢 𝑣𝑣 𝑡𝑡 𝑢𝑢 Variable𝑢𝑢 𝑣𝑣 𝑡𝑡 𝑢𝑢shape𝑢𝑢 of𝑣𝑣 𝑡𝑡the two𝜋𝜋 intersection curves is depicted in fig. 12. Top views of spatial graphs of surface ( , , ) and plane = 0 for = 0,0.1, ,1 is drawn in several animation 𝑖𝑖 slides. 𝐷𝐷 𝑢𝑢 𝑣𝑣 𝛾𝛾 IJSER𝑧𝑧 𝑡𝑡 ⋯

Fig. 10: Envelope surface generated by rotary motion of general surface of revolution when axis of rotary motion and axis of the generating surface are skew (top, front and axonometric view)

Fig. 11: Graphical solution of equation ( , , ) = 0 for envelope surface generated by general motion of general surface of revolution at = 0.5 5.3Envelope Surface Is Explicit Surface of the Solid, 𝐷𝐷 𝑢𝑢 𝑣𝑣 𝑡𝑡 Characteristic Curve Has Variable Shape 𝑡𝑡 To give an example of envelope surface with variable shape of characteristic curve, the envelope surface generated by general motion of surface of revolution (19) along spatial freeform trajectory given by ( ) = ( ( ), ( ), ( ), 1), [0,1], ( ) = 7 3 + 6 2 24 + 25, 𝐓𝐓(𝑡𝑡) = 𝑥𝑥78𝑡𝑡 3𝑦𝑦+𝑡𝑡126𝑧𝑧 𝑡𝑡2 54𝑡𝑡, ∈ 𝑥𝑥(𝑡𝑡) = 30− 𝑡𝑡3 30𝑡𝑡 2− 𝑡𝑡 is𝑦𝑦 considered,𝑡𝑡 − 𝑡𝑡 here. 𝑡𝑡The− transformation𝑡𝑡 matrix of general cur- vilinear𝑧𝑧 𝑡𝑡 motion𝑡𝑡 − in𝑡𝑡 this case is given by

Fig. 14: Envelope surface generated by general motion of general surface IJSER( , , ) = 0 Fig. 12: Graphical solution of equation for envelope surface of revolution (top, front and axonometric view) generated by general motion of general surface of revolution (top views of animation slides) 𝐷𝐷 𝑢𝑢 𝑣𝑣 𝑡𝑡 Note that in case the explicit solution is impossible to find easily, these graphs can be used to find ( , ) coordinates of sufficient number of points along intersection curve. These points can be consequently fitted by suitable𝑢𝑢 approximation𝑣𝑣 or interpolation curve representing mathematical model of characteristic curve in parametric space ( , , ) of the solid (4). Graphical representation of solution (24) in parametric space is shown in fig. 13. Branches of𝑢𝑢 𝑣𝑣envelope𝑡𝑡 surface in parametric space are not ruled surface ( -parametric curves of this suface are not straight lines). It follows that the characteristic curve has variable shape. 𝑡𝑡 In 3D space the shape of both parts of envelope surface as well as the shape of both branches of characteristic curve is not identical, see fig. 14, where the envelope surface generated by general motion of general surface of revolution is drawn.

6. CONCLUSION A modified DG/K (Differential Geometry/Kinematics) approach to envelope surface solving is presented in this paper. A complex Fig. 13: Envelope surface with several characteristic curves) generated by general motion of general surface of revolution (mapping in parametric method has been developed to find the analytical representation space ( , , ) of the generated solid) IJSER © 2017 𝑢𝑢 𝑣𝑣 𝑡𝑡 http://www.ijser.org International Journal of Scientific & Engineering Research Volume 8, Issue 8, August-2017 1584 ISSN 2229-5518 of envelopesurface generated by continuous univariate motion of generating surface. The enter dataof this method is type of motion, analytic representation of generating surface and ana- lyticrepresentation of trajectory in homogeneous coordinates of extended Euclidean space. Based on mapping of envelope surface in parametric space of the solid generated by continuous motion of general surface of revolution, it is possible to recognize the expected form of analytical representation of envelope surface and choose appropriate approach to find it. The obtained results can be used in mechanical engineering applications of envelope surfaces, especially in 5-axis flank milling of freeform surfaces.

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