Four-dimensional geometric element definitions and interferences via five-dimensional homogeneous

Yukihito Sakai*, Shuji Hashimoto

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    7 Citations (Scopus)


    We have systematized various studies on 4-D visualization and interaction thus far, and we proposed 4-D geometric algorithms via 5-D homogeneous processing. Our framework uses 5 × 5 matrices and 5 × 5 determinants to express various types of transformations, and it simplifies geometric operations without the use of division operations. Using the proposed scheme, we developed an interactive 4-D space display system. The simplicity, generality, and duality of 5-D homogeneous processing are effective not only for 4-D geometric operations but also for interference problems among various 4-D objects. However, the processing regarding geometric characteristics of 4-D objects was not considered in our previous works. In this paper, we describe 4-D geometric elements, in general, and we discuss 4-D computational geometry via 5-D homogeneous processing. Unified geometric operations without the use of division operations constitute the most important part of 5-D homogeneous processing. We systematize the methods for 4-D geometric element definitions and interferences via 5-D homogeneous processing. In the field of 4-D visualization, the proposed algorithms can be potentially used in a user interface for feature detection of a 4-D object and collision detection of several 4-D objects. We comprehensively develop and advance the theoretical framework in the field of 4-D graphics. It is expected that this method of processing will be useful for the performance improvement of 4-D graphics systems.

    Original languageEnglish
    Pages (from-to)129-139
    Number of pages11
    JournalJournal of Visualization
    Issue number2
    Publication statusPublished - 2011


    • 4-D Euclidean space
    • 4-D geometric element definitions
    • 4-D geometric element interferences
    • 5-D homogeneous processing

    ASJC Scopus subject areas

    • Condensed Matter Physics
    • Electrical and Electronic Engineering


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