# A Solution to the Hidden-Line Problem for Computer-Drawn Polyhedra

TL;DR: The method presented here for solving the "hidden-line problem" for computer-drawn polyhedra is believed to be faster than previously known methods.

Abstract: The "hidden-line problem" for computer-drawn polyhedra is the problem of determining which edges, or parts of edges, of a polyhedra are visible from a given vantage point. This is an important problem in computer graphics, and its fast solution is especially critical for on-line CRT display applications. The method presented here for solving this problem is believed to be faster than previously known methods. An edge classification scheme is described that eliminates at once most of the totally invisible edges. The remaining, potentially visible edges are then tested in paths, which eventually cover the whole polyhedra. These paths are synthesized in such a way as to minimize the number of calculations. Both the case of a cluster of polyhedra and the illumination problem in which a polyhedron is illuminated from a point source of light are treated as applications of the general algorithm. Several illustrative examples are included.

##### Citations

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Ivan E. Sutherland

^{1}•TL;DR: The fundamental idea behind the three-dimensional display is to present the user with a perspective image which changes as he moves, and this display depends heavily on this "kinetic depth effect".

Abstract: The fundamental idea behind the three-dimensional display is to present the user with a perspective image which changes as he moves. The retinal image of the real objects which we see is, after all, only two-dimensional. Thus if we can place suitable two-dimensional images on the observer's retinas, we can create the illusion that he is seeing a three-dimensional object. Although stereo presentation is important to the three-dimensional illusion, it is less important than the change that takes place in the image when the observer moves his head. The image presented by the three-dimensional display must change in exactly the way that the image of a real object would change for similar motions of the user's head. Psychologists have long known that moving perspective images appear strikingly three-dimensional even without stereo presentation; the three-dimensional display described in this paper depends heavily on this "kinetic depth effect."

1,825 citations

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Herbert Freeman

^{1}•TL;DR: Various forms of line drawing representation are described, different schemes of quantization are compared, and the manner in which a line drawing can be extracted from a tracing or a photographic image is reviewed.

Abstract: This paper describes various forms of line drawing representation, compares different schemes of quantization, and reviews the manner in which a line drawing can be extracted from a tracing or a photographic image. The subjective aspects of a line drawing are examined. Different encoding schemes are compared, with emphasis on the so-called chain code which is convenient for highly irregular line drawings. The properties of chain-coded line drawings are derived, and algorithms are developed for analyzing line drawings to determine various geometric features. Procedures are described for rotating, expanding, and smoothing line structures, and for establishing the degree of similarity between two contours by a correlation technique. Three applications are described in detail: automatic assembly of jigsaw puzzles, map matching, and optimum two-dimensional template layout

1,485 citations

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TL;DR: The paper shows that the order of sorting and the types of sorting used form differences among the existing hidden-surface algorithms.

Abstract: : The paper asserts that the hidden-surface problem is mainly one of sorting. The various surfaces of an object to be shown in hidden-surface or hidden-line form must be sorted to find out which ones are visible at various places on the screen. Surfaces may be sorted by lateral position in the picture (XY), by depth (Z), or by other criteria. The paper shows that the order of sorting and the types of sorting used form differences among the existing hidden-surface algorithms. (Modified author abstract)

793 citations

### Cites background from "A Solution to the Hidden-Line Probl..."

...I .... LOUTREL 1967 TP,NP Promote VlSlblllty of a vertex to all edges at vertex Back Edge Cull 1) Edges separating back-facing planes 2) Dot product wlth normals ~ topology 3) Cull 4) LISt of edges,E s 5) 1, E t (Omitted) Inltzal Visibility I) Ray to vertex against all faces 2) Betweenness,…...

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...P Loutrel (1967) R. GaZ~mbert~ and U Montanar~ (1069) Image-Space and List-Priority LIst-Przor~ty Algorzthms R A Schumacl~er, B Brand, M Gdhland, IV Sharp (196~) M....

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Henri Gouraud

^{1}•TL;DR: The surface is approximated by small polygons in order to solve easily the hidden-parts problem, but the shading of each polygon is computed so that discontinuities of shade are eliminated across the surface and a smooth appearance is obtained.

Abstract: A procedure for computing shaded pictures of curved surfaces is presented. The surface is approximated by small polygons in order to solve easily the hidden-parts problem, but the shading of each polygon is computed so that discontinuities of shade are eliminated across the surface and a smooth appearance is obtained. In order to achieve speed efficiency, the technique developed by Watkins is used which makes possible a hardware implementation of this algorithm.

661 citations

### Cites methods from "A Solution to the Hidden-Line Probl..."

...Since then other algorithms performing the same task have been developed by Galimberty [2], Kubert [3], and Loutrel [4],...

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TL;DR: A new family of clipping algorithms is described, able to clip polygons against irregular convex plane-faced volumes in three dimensions, removing the parts of the polygon which lie outside the volume.

Abstract: A new family of clipping algorithms is described. These algorithms are able to clip polygons against irregular convex plane-faced volumes in three dimensions, removing the parts of the polygon which lie outside the volume. In two dimensions the algorithms permit clipping against irregular convex windows.Polygons to be clipped are represented as an ordered sequence of vertices without repetition of first and last, in marked contrast to representation as a collection of edges as was heretofore the common procedure. Output polygons have an identical format, with new vertices introduced in sequence to describe any newly-cut edge or edges. The algorithms easily handle the particularly difficult problem of detecting that a new vertex may be required at a corner of the clipping window.The algorithms described achieve considerable simplicity by clipping separately against each clipping plane or window boundary. Code capable of clipping the polygon against a single boundary is reentered to clip against subsequent boundaries. Each such reentrant stage of clipping need store only two vertex values and may begin its processing as soon as the first output vertex from the preceeding stage is ready. Because the same code is reentered for clipping against subsequent boundaries, clipping against very complex window shapes is practical.For perspective applications in three dimensions, a six-plane truncated pyramid is chosen as the clipping volume. The two additional planes parallel to the projection screen serve to limit the range of depth preserved through the projection. A perspective projection method which provides for arbitrary view angles and depth of field in spite of simple fixed clipping planes is described. This method is ideal for subsequent hidden-surface computations.

566 citations

##### References

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01 Jan 1967TL;DR: This paper presents a recently developed scheme for the determination of visibility in a line drawing which enables comparitively high speed calculation and excellent resolution.

Abstract: Line drawings are the most common type of rendering used to convey geometrical description. This is due to the economy of preparing such drawings and the great information density obtainable. On a pure line drawing, that is where no attempt is made to specify or suggest shadows, tone or color, the lines rendered are either the intersection curves of surfaces or the contour curves of surfaces. The nature of these curves are adequately discussed in the literature 1 and in a previous report.2 In order to convey a realistic impression of an object or an assembly of objects, the segments of lines which cannot be seen by an observer are not drawn or are drawn dashed. Without specification of visibility a drawing is ambiguous. This paper presents a recently developed scheme for the determination of visibility in a line drawing which enables comparitively high speed calculation and excellent resolution.

209 citations

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14 Nov 1967

TL;DR: In recent years, the sheer increase in demand for the graphic presentation of three-dimensional objects has almost overwhelmed conventional facilities; that is, designers, draftsmen and especially engineering artists.

Abstract: In recent years, the sheer increase in demand for the graphic presentation of three-dimensional objects has almost overwhelmed conventional facilities; that is, designers, draftsmen and especially engineering artists. For example, it is important for a designer or architect to quickly describe a three-dimensional object and view it immediately; not as an endless set of engineering drawings, but as if he were viewing the three-dimensional object itself. He should be able to take a distant look at a complicated object, and then view, in detail, any subsection of the object. In other words, he would like to quickly and cheaply simulate and view the thing he is designing.

97 citations

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Paul G. Comba

^{1}•IBM

^{1}TL;DR: A procedure has been developed for detecting intersections of convex regions in 3-space by means of a pseudocharacteristic function and a system of programs embodying these techniques is described.

Abstract: As a step toward the solution of the placement problem in engineering design, a procedure has been developed for detecting intersections of convex regions in 3-space by means of a pseudocharacteristic function. The mathematical techniques underlying the procedure are discussed, and a system of programs embodying these techniques is described. As a special case a solution is given for the hidden-line problem in graphic display.

64 citations

20 May 1968

TL;DR: The motivating philosophy of a particular approach is closely examined, and a possible implementation of this approach is described, and the advantages of the scheme are outlined.

Abstract: : The purpose of the paper is to consider the problem of producing two dimensional picture representations of objects described in three dimensional space. In discussing the main problem of the removal of hidden surfaces in a picture representation, the motivating philosophy of a particular approach is closely examined, and a possible implementation of this approach is described. The advantages of the scheme are also outlined in the paper. (Author)

62 citations

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Ruth A. Weiss

^{1}•Bell Labs

^{1}TL;DR: The various subroutines of the package and their functions are described in this paper and numerous examples of pictures that were produced by BE VISION are given.

Abstract: BE VISION is a package of FORTRAN programs for drawing orthographic views of combinations of plane and quadric surfaces. As input, the package takes rectangular coordinate equations specifying the surfaces plus a three-angle specification of the viewing direction. Output is a drawing on the Stromberg Carlson 4020 Microfilm Recorder. Many views of one scene may be obtained simply by changing the viewpoint.The various subroutines of the package and their functions are described in this paper. It also gives numerous examples of pictures that were produced by BE VISION. The package has been in use since April 1964.

60 citations