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CONCATENATING TRANSFORMATIONS IN HIERARCHIC GRAPHICS

IP.com Disclosure Number: IPCOM000128707D
Original Publication Date: 1976-Dec-31
Included in the Prior Art Database: 2005-Sep-16
Document File: 15 page(s) / 39K

Publishing Venue

Software Patent Institute

Related People

W. R. Mallgren: AUTHOR [+4]

Abstract

Several general algorithms are described for the concatenation of geometric and clipping transformations in hierarchic graphics systems. The re-sults are derived from precise definitions of the notions of picture, (clipping) region, transformation, picture sum, and hierarchic picture structure, and may be applied to a variety of systems, including those based on either graphics data structures or display procedures. The main contributions are the clarifi-cation of the concepts used in general transformation systems and the simple algorithms that result from this clarification. Key words: graphics transformations; clipping; transformation concatenation CR category: 8.2

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THIS DOCUMENT IS AN APPROXIMATE REPRESENTATION OF THE ORIGINAL.

CONCATENATING TRANSFORMATIONS IN HIERARCHIC GRAPHICS*

W. R. Mallgren and A. C. Shaw

(Technical Report #76-02-02)

January 1976

Department of Computer Science University of Washington Seattle, Washington 98195

This work was supported in part by the National Science Foundation, Grant GJ-36273. Concatenating Transformations in Hierarchic Graphics*

William R. Mallgren and Alan C. Shaw

Department of Computer Science University of Washington Seattle, Washington 98195

Abstract:

Several general algorithms are described for the concatenation of geometric and clipping transformations in hierarchic graphics systems. The re-sults are derived from precise definitions of the notions of picture, (clipping) region, transformation, picture sum, and hierarchic picture structure, and may be applied to a variety of systems, including those based on either graphics data structures or display procedures. The main contributions are the clarifi-cation of the concepts used in general transformation systems and the simple algorithms that result from this clarification.

Key words: graphics transformations; clipping; transformation concatenation

CR category: 8.2

1. Introduction

Two common features of many graphics systems and applications are the organization of pictures in a hierarchical manner and the use of geometric and clipping transformations to alter picture subparts. The purpose of this paper is to clarify the concepts underlying transformation systems, and to present simple general algorithms for the concatenation of geometric and clipping trans-formations within and between levels in hierarchic picture structures.

The next section precisely defines the notions of picture, (clipping) region, picture sum, transformation, and hierarchic picture structure. Struc-ture traversal algorithms that concatenate transformations and perform picture

This work was supported in part by the National Science Foundation, Grant.GJ-36273. visibility (boxing) tests are then described. The last section discusses some special cases which appear frequently in practice. Our definitions and algo-rithms apply to a variety of systems, including those based on either graphics data structures or display procedures.

University of Washington Page 1 Dec 31, 1976

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CONCATENATING TRANSFORMATIONS IN HIERARCHIC GRAPHICS

2. Pictures and Transformations

Algorithms which operate on objects must be based upon precise defini-tions of both the objects and the operations which may be performed on them. In this paper the objects to be operated upon are pictures, and the operations to be performed are the transformation and combination of pictures. Each of these concepts will be defined below.

2.1 Picture Functions

We start with the abstract notion of a two-dimensional picture in the plane. Any such picture can be mathematically defined as a real-valued picture function f(x) of two real numbers

(Equation Omitted)

with th...