On Rotations and Orthogonal Projections in <I>n</I>-Dimensional Euclidean Space

DOI Web Site 10 References

Bibliographic Information

Other Title
  • <I>n</I>次元空間の回転と直投影について―高次元代数的図法幾何学I
  • nジゲン クウカン ノ カイテン ト チョクトウエイ ニ ツイテ コウジゲン ダイスウテキ ズホウ キカガク 1

Search this article

Abstract

One of our purpose is to arrange and to analyze a method of visual perception of an object in n-dimensional Euclidean space Rn systematically and practically.<BR>When we study the process of visual perception of an object in Rn, we have to extend the process in 3-dimensional case to the general cases. If we try to perceive the unidentified object, usually we move (rotate) the object or we change the position (turn) around it, and we look it or press the shutter. In this paper, we will generalise these “moving” (rotation) and “looking” (projection) to n-dimensional cases, and we will give them, especially for the former, a practical expression.<BR>It is well known the theoretical expression of rotation in Rn, that is to decompose it to a product of at most n/2 2-dimensional rotations, so called the canonical form of the orthogonal transformation. This expression is useful and economical in mathematical sense, but it lacks practical use especially in high dimensional cases, since the axis of the 2 -dimensional rotation in each component is change depending on the given rotation. Then this expression is not satisfied our purpose.<BR>On the other hand, from our viewpoint to treat rotation and projection together, we obtain the following expression : any rotation in Rn which fixes the origin is decomposed to a product of at most n-1 2-dimensional rotations with the fixed axis and a rotation in the projection hyperplane ( ≅ Rn-1) . Of course our expression has practical value in exchange for a little waste.

Journal

References(10)*help

See more

Keywords

Details 詳細情報について

Report a problem

Back to top