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# The Sense of Geometric Objects

IP.com Disclosure Number: IPCOM000087646D
Original Publication Date: 1977-Feb-01
Included in the Prior Art Database: 2005-Mar-03
Document File: 3 page(s) / 49K

IBM

## Related People

Murphy, AS: AUTHOR

## Abstract

In a graphics display system in which geometric objects, such as points, lines and arcs, are manipulated, ambiguity can arise. Thus, consider the following: (a) A point has two possible tangents to a circle. (b) Four possible circles can be constructed to touch three lines. (c) There are two possible bisectors of two intersecting lines. Traditionally, such ambiguities have been resolved by using modifiers, for example, XLARGE and YSMALL. However these modifiers can be difficult to use.

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The Sense of Geometric Objects

In a graphics display system in which geometric objects, such as points, lines and arcs, are manipulated, ambiguity can arise. Thus, consider the following:
(a) A point has two possible tangents to a circle.
(b) Four possible circles can be constructed to touch three lines.
(c) There are two possible bisectors of two intersecting lines. Traditionally, such ambiguities have been resolved by using modifiers, for example, XLARGE and YSMALL. However these modifiers can be difficult to use.

The ambiguity can be overcome if each geometrical object is given a "sense". Eight geometric entities, defined as follows, are shown in Fig. 1. (1) Distance - a scalar number which may be either positive or negative. (2) Angle - a scalar number in the range 0 to 2pi, anti-clockwise directions representing increasing angles. (3) Point - a two element object containing the x, y coordinates. (4) Stroke - a four element object which is the concatenation of the coordinates of the two end points (x(1) y(1), x(2) y(2)). (5) Line - an unbounded two element object (r, theta), where r is the length of the perpendicular from the origin to the line and theta is the angle this perpendicular makes with the x axis. (6) Bounded Line - a four element object (r, theta, d(1), d(2)) in which the end points are specified at distances d(1) and d(2) from the intersection of the line with the perpendicular to the origin. (7) Circle - a three element object (x, y, r), where x and y are the coordinates of the center and r is the radius which may be positive or negative. (8) Arc - a five element object (x, y, r theta(1) theta(2)) where Theta(1) and theta(2) are the angles which the radii to the end points make with the x-axis.

From these definitions, further axioms can be derived. If the unbounded line is represented by (r, theta), the line of reverse sense is (-r, theta + pi). r is positive for lines which indicate anti-clockwise rotations about the origin and negative for those which indicate clockwise rotations. For bounded lines, the distances d(1) d(2) are measured in the positive direction of the line...