Euler Angles Representation of N-dimemsional Homogeneous Transformation

• Main Authors: Ti-chia Hsu, 徐蒂珈
• Other Authors: Ching-Cheng Wang
• Format: Others
• Language: zh-TW
• Published: 2007
• Online Access: http://ndltd.ncl.edu.tw/handle/17585440898516995144

碩士 === 國立成功大學 === 製造工程研究所碩博士班 === 95 === In the two-dimensional space, a Cartesian coordinate system might be transformed into another Cartesian coordinate system by rotations and translations, and their relation can be calculated via the geometric graph. Similarly, that kind of relationship in the three-dimensional space can also be calculated via the geometric graph. On the other hand, the geometric graph in the N-dimensional space cannot be visual as long as N is greater than three. Consequently, transformation relations among N-dimensional coordinate systems cannot be calculated via the geometric graph. For this reason, we set out to derive the relation between two coordinate systems by utilizing the two-dimensional transformation matrices as the basic component. When the transformation matrices in homogeneous coordinate system, the translations and rotations can be taken for an affine transformation where the dimensional size of those matrices is equal to the size of the homogeneous coordinate system. Therefore, the translations and rotations can be expressed by homogeneous matrices. Additionally, the N-dimensional homogeneous transformation matrix can be readily applied to represent 2nd-order (N-1)-dimensional space curves in the N-dimension space such as hyper-ellipses. geometric graphs which are after rotation and translation. For illustration, parametric expressions of an ellipse in various transformed coordinate systems are presented in this thesis, where the final parametric expressions of ellipses are obtained via multiplying the rotation matrix and followed by adding the position vector.