Computer Graphics & Geometry
V. Kovalenko, Ju. Revjakin, E. Khukhlajev
Russian Academy of Sciences, Keldysh Institute of Applied Mathematics
4, Miusskaya sq., Moscow A-47, 125047, Russia
E-mail:kvn@applmat.msk.su
Abstracts: Problems of the drawings composition in interactive CAD systems are discussed. The composition is an insertion of one drawing (a fragment) into another one. An interactive technique of preliminary fragment preparation is considered only. In current CAD systems the capabilities of fragment attachment and geometry variation are severely limited. We propose a technology to expand these capabilities. This technology is based on the individual attachment of some fragment elements and dimension driven variation of fragment geometry. It is shown that the proper positioning of attachment elements can be done only if inserted fragment elements are visualized in the stepwise manner. The implementation of proposed technology by using the tools of the interactive DDCAD system is discussed.
Key words: drawings composition; dimension driven geometry; constraints based geometric model; drawing parametrization.
The widespread approach based on construction of new product of already existing blocks may be applied to engineering design: some standard fragments are used in almost all drawings; the assembly drawings often include part drawings created earlier. The main problem can be stated as: How can a drawing (called the fragment) be inserted into another one (the basic drawing or simply the drawing)? Let us call this process a composition of drawings. An example of a drawing composition is shown in Fig. 1.
Drawing + Fragment = New drawing
Fig. 1. Drawing composition
With the development of CAD-systems the composition technique becomes available to a user. We evaluate the composition possibilities by three criteria.
A. What is the way of preliminary preparation of a fragment?
All techniques in which a user should manipulate with a program presentation of a fragment require some programming skills from the user. On the contrary, a really interactive mode of fragment creation permits a designer-nonprogrammer to take advantage of the composition technique. Later on we shall call that a fragment any drawing constructed by interactive tools of CAD system. This condition a designer-nonprogrammer can fill up the libraries of these fragments himself. The libraries can contain, for example, technological and design fragments intended to complement the drawings. The libraries of standard or nonstandard part drawings inserted into the assembly drawings are of interest too. The last two criteria characterize the composition process itself.
B. How is a fragment attached to a drawing?
C. How does the geometry of inserted fragment vary?
In most of current CAD-systems to insert a fragment into a drawing one must point out the location of some fixed fragment point (insertion point) in the drawing. This insertion point is defined at the stage of fragment construction and can not be changed later. The geometry variations are limited to fragment rotation around the insertion point and scaling of the fragment as a whole. These restrictions are stipulated by the primitive structure of a drawing model in CAD- systems. As a consequence, the possibilities of a drawing composition are sufficiently reduced.
2. Dimension Driven Drawings Composition with Individual
Attachment of Elements
Unlike the attachment and the geometry variation of a fragment as a whole, we propose a composition technology based on the individual attachment of some fragment elements and dimension driven geometry.
The term Dimension Driven Geometry (DDG) was proposed in [1]. This term means automatic computation of drawing geometry in accordance with changes in dimension values. DDG implies that a drawing shape is kept. Later on we'll consider only the drawings consisted of segments, circles and arcs. To keep the fragment shape constant means to keep obvious geometric relations between elements (parallelism, perpendicularity, tangency, incidence, symmetry).
The scenario to implement the proposed technology looks as follows.
- First, the user can change fragment dimension values.
- On each attachment step the user picks out some fragment element (an attachment element) and sets this element in the drawing. If one chooses only points as attachment elements positioning is reduced to picking an arbitrary point of the drawing (in particular, the point can be a certain characteristic point: a segment end, a circle centre or a line intersection). Other fragment elements that can be calculated at this step are inserted into the drawing and are immediately
visualized. Necessity of this stepwise visualization for proper placing of the subsequent attachment elements will be demonstrated later.
- After all the fragment elements have been inserted, the fragment dimensions are transferred into the drawing with assigned values and the insertion is completed. Some dimensions may prove to be unnecessary for fragment computing. In this case their values are determined by real location of inserted elements in the drawing.
3. An Example of the Composition Scenario
Let's consider the composition scenario for the above example (Fig. 2.). Let it is necessary to attach segment CD of the fragment (Fig. 2.1) to A'B' in the basic drawing (Fig. 2.2). At the first step point C' corresponding to point C is located inside A'B' (Fig. 2.3). At the second step point D' corresponding to point D is inside A'B' (Fig. 2.4). Segments CE and DF can't be yet calculated (and not visualized), but it is possible to calculate lines l1' and l2' (CE and DF are based on l1 and l2) because l1 and l2 are normal to CD. Therefore, lines l1' and l2' corresponding to l1 and l2 are shown in the drawing (Fig. 2.5). At the third step (Fig. 2.6) we must locate point E' corresponding to point E certainly at l1' in order to keep the fragment shape (i.e. perpendicularity of CD and CE). The third step shows a necessity of stepwise visualization for the proper placing of the following attachment elements in the drawing while keeping the fragment shape constant. In general, if an attachment fragment element is connected with earlier computed elements by any geometric relations then it is clear that this element must be located in the drawing so that these relations are satisfied. There is a single geometric relation for a point (an incidence); therefore if the attachment
fragment point belongs to the earlier computed line, then this point should be located on a respective inserted line in the drawing.
2.1 Fragment 2.2 Part of basic drawing 2.3 After first step
2.4 At second step 2.5 After second step 2.6 After third step
Fig. 2. An example of the composition scenario
After the third step the fragment is calculated completely. Radius of circle c is superfluous and so its value is determined by the real radius of inserted circle c'. One can see that if the length of CE segment is determined by any dimension then the third step is omitted.
4. Constraints Based Geometric Modelling System DDCAD
The proposed composition technology can be implemented in such a CAD-system only which is able to vary drawing geometry. The DDCAD [2] system just possesses such possibilities. Here we describe in short the basic notions and possibilities of DDCAD. Details are described in [2].
The drawing model in DDCAD consists of parametric and topological models. The parametric model describes the geometric base of a drawing and consists of: a set of parametric elements (points, lines and circles); a set of geometric relations between parametric elements (incidence, parallelism, perpendicularity, tangency and symmetry); a set of dimensional relations corresponding to the drawing dimensions (distance, angle and radius).
The topological model consists of basic contour elements of the drawing supported by parametric elements - arcs and segments are based on the supporting lines and two end points. In DDCAD a drawing is constructed in the interactive manner with simultaneous generation of the drawing model without any additional operation for drawing parametrization. An overview of various approaches to drawing parametrization and comparison of their technological
efficiency is given in [3]. To construct a drawing element it is necessary to give a sufficient number of constraints.
Geometric constraints are kept in the parametric model as geometric relations. The drawing is dimensioned in a general way. Dimensions are kept in the parametric model as dimensional relations. The order of constructing a drawing is not essential. A drawing can be edited interactively and the drawing model is changed accordingly. The constructed drawing is a sample from a class of drawings with the same shape but different dimension values.
The basic method of drawing geometry calculation in DDCAD is the method of consecutive calculation of elements - one of the variants of the propagation.
This method consists of the following:
- An initial pair of parametric elements is fixed at the first step.
- An elementary task of element calculation is solved at every next step. This task is formed from the relations between an element to be calculated and the earlier calculated elements. The single solution is chosen among several solutions so that a similar one to be chosen for the drawing
sample.
The parametric model may be presented as the nonoriented graph (the model graph); graph nodes are elements and graph arcs are relations between elements. The drawing calculation sequence may be presented by the orientation of the model graph. Usage of this method allows us to implement the dimension driven parametrization (i.e. to calculate drawing exemplar with changed dimension values but the same shape).
5.Implementation of Drawings Composition in DDCAD SystemThe basic method of consecutive calculation of elements is used to calculate the fragment inserted into the drawing.
- At the beginning of every attachment step the parametric attachment element of fragment is fixed (with corresponding drawing coordinates). We can say that this attachment fragment element is equalized explicitly to the corresponding drawing element. If this drawing element is absent then it is created in the drawing.
- The parametric elements of fragment are calculated consecutively proceeding from the earlier calculated elements together with new attachment element. The coordinates of fragment elements are used as the sample to choose the solution variant.
- The topological elements based on calculated parametric elements are inserted and visualized in the drawing. If it is still not possible to visualize a topological element (not all its supporting parametric elements have been calculated) but its supporting line has already been calculated, then this supporting line is visualized temporarily instead of this topological element.
Graph presentation of composition steps for given example is shown in Fig. 3. Points C', D' and E' are created in the drawing during attachment steps. Equalized elements are joined by duplex arrows. Nonoriented incidence between attachment point E and line l requires that point E' to be located on the inserted line l1'.
Fig. 3. Elements calculation during the composition
Unification of ModelsAs a result of the composition the drawing model must be unified with the fragment model. A new problem arises: how to avoid the elements duplication during unification of models?
The fragment parametric model must be transferred to the drawing without duplication, i.e. elements possessing the same set of geometric relations and the same coordinates would not exist.
It is clear that the fragment attachment element (equalized explicitly) must not be transferred. Some nonattachment fragment elements may exist such that when transferred to the drawing they would coincide with other drawing elements. We can say that these coinciding elements are equalized implicitly. A fragment element is equalized implicitly to a drawing element if:
1) parents of this fragment element (in terms of consecutive elements calculation) are equalized (explicitly or implicitly) to some drawing elements;
2) relations between the fragment element and its parents, and relations between this drawing element and drawing elements equalized to the parents are the same;
3) calculated coordinates of this fragment element are equal to coordinates of this drawing element (the equality is necessary because there may be several different elements that satisfy conditions 1 and 2).
For example, in Fig. 3 points C, D and E are equalized explicitly to points C', D' and E', and line l is equalized implicitly to line l' (l passes through C and D, and l' passes through corresponding C' and D').
All the nonequalized parametric elements are transferred to the drawing together with all the relations between them. Relations between the nonequalized fragment elements and the equalized ones are transferred to the corresponding equalized drawing elements.
The fragment topological model is transferred to the drawing with systematic substitution of equalized supporting parametric elements.
Fig. 4. Parametric model of unified drawing
The graph presentation of unified parametric model for this example is given in Fig. 4. Fragment elements C, D, E and l are equalized and are not transferred to the drawing.
Fragment elements l1, l2, c and F are transferred to the drawing together with their relations. Relations between l1, l2, c, F and l, C, D, E are transferred to l', C', D', E' (equalized to l, C, D, E accordingly).
As a result of such a procedure the fragment is inserted into the drawing with the same shape. The unified drawing possesses the valid model as if it were constructed by interactive construction tools of the DDCAD system from the beginning to the end and the drawing is ready to further work in the DDCAD system.
[1] D.C.Gossard, R.P.Zuffante and H.Sakurai, 'Representing Dimensions, Tolerances, and Features in MCAE systems', IEEE Computer Graphics & Applications, 8, (3), 51-59, (1988).
[2] V.N.Kovalenko, Ju.G.Revjakin and E.V.Khukhlajev, 'Parametrizing of mechanical drawings by stepwise computation of geometric elements', Programming, (2), 64-78, (1992) (in Russian).
[3] W.Bouma, I.Fudos, C.Hoffmann, J.Cai and R.Paige, 'A geometric Constraint Solver', TR-93-054, Pardue University, Computer Science, (1993).
Computer Graphics & Geometry