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# Thickened Arc ALGORITHM

IP.com Disclosure Number: IPCOM000060856D
Original Publication Date: 1986-May-01
Included in the Prior Art Database: 2005-Mar-09
Document File: 2 page(s) / 14K

IBM

## Related People

Niblett, PD: AUTHOR

## Abstract

This article outlines a method of rastering a thickened arc to an all-points-addressable buffer. It is applicable to a graphics display device or to a plotter or printer. The standard method for drawing a thickened arc is to split it into a number of short line segments as if it were unthickened and then to thicken each line individually by drawing another line parallel to it, using some convention based on the angle in which the line is travelling. This approach has the following disadvantages: 1. Wedge-shaped holes may appear where two lines meet, unless the thickening line is carefully extended. 2. Some pels may be drawn more than once. If the arc is drawn in exclusive OR mode (e.g., for dragging across a fixed picture) this causes the pels (picture elements) to disappear and the arc looks ragged (and thinner in places).

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Thickened Arc ALGORITHM

This article outlines a method of rastering a thickened arc to an all-points- addressable buffer. It is applicable to a graphics display device or to a plotter or printer. The standard method for drawing a thickened arc is to split it into a number of short line segments as if it were unthickened and then to thicken each line individually by drawing another line parallel to it, using some convention based on the angle in which the line is travelling. This approach has the following disadvantages: 1. Wedge-shaped holes may appear where two lines meet, unless the thickening line is carefully extended. 2. Some pels may be drawn more than once. If the arc is drawn in exclusive OR mode (e.g., for dragging across a fixed picture) this causes the pels (picture elements) to disappear and the arc looks ragged (and thinner in places). 3. The points drawn do not necessarily form a perfect envelope of the ideal ellipse to which they approximate. In the following it is recognized that a full elliptic arc divides the 2D plane in which it lies into two regions: its INTERIOR and its EXTERIOR. We may choose which region the arc's circumference is in. Let us say that it is included in the interior. It is assumed that the arc is centered about the origin of our coordinate space. The ellipse may be described by an analytic equation Ax2 + 2Hxy + By2 - D = 0 for some A, B, H, D, where A, B, D >= 0 If we define a function f(x,y) = Ax2 + 2Hxy + By2 - D then f(x,y) <= 0 for all points in the interior and f(x,y) > 0 for the exterior. The set of all interior points with at least one neighboring point in the exterior is termed the INTERNAL CONTOUR of the ellipse; the set of all exterior points having at least one neighboring interior point is termed its EXTERNAL CONTOUR. Provided that the interior points themselves form a connected set, it can be shown that the internal contour so defined is a connected path of points in the plane. The external contour is always a connected path. Theo Pavlidis [* section 7.5.1] describes a contour tracing algorithm called TRACER, which can traverse either contour. If this algorithm is used in two passes to draw both contours, then the set of points drawn will produce a thickened ellipse overcoming the problems mentioned abo...