<\/p>\n\n\n\n
The orientation of body in space is defined\n3 by angles. The step by step rotation process and chain of three-dots multiplication\ngive an easy way to compute pile of rotations in 3D and high dimensional space\nand give a general orientation system. A visualization of quaternion is proposed.<\/p>\n\n\n\n
The orientation of a rigid body\nin space is defined by 3 angles, for\nexample the angles\nof pitch, roll and yaw<\/a> of an airplane.\nThe most commonly used orientation systems are Euler angles and Tait\u2013Bryan angles<\/a>. Quaternion<\/a> is often used\nfor computing rotation of body. <\/p>\n\n\n\n In this article we will present a new\nsystem for computing rotation in space that consists of rotating the proper frame of\nreference of the\nrigid body with respect to that of the space. This\nsystem is more general than Euler angles and Tait\u2013Bryan angles and easier to use than rotation matrix and quaternion.<\/p>\n\n\n\n Below we will refer 2-dimensional space as 2D and 3-dimensional space as 3D. First, we will explain our approach using 2D body, then\napply it to 3D body\nand body of even higher dimensions. We will use our system\nto give a visualization of the geometrical meaning of\nquaternion in 3D space.<\/p>\n\n\n\n Let us begin with defining\nthe frame of reference of a rotated body and that of the space. A frame\nis defined by its basis that\nis a set of orthogonal unit vectors. The frame of the space is fixed. The frame of the rotated body is solidary to the body and rotates with it. The orientation in space of\nthe body is defined by the angles its\nframe makes with that of the space. <\/p>\n\n\n\n We refer the basis of the frame of the space as space\nbasis and that of the body the proper basis. The\nset of unit vectors of the space basis is (e<\/strong>x<\/sub>, e<\/strong>y<\/sub>) for 2D space and (e<\/strong>x<\/sub>, e<\/strong>y<\/sub>, e<\/strong>z<\/sub>) for 3D space.\nThe set of unit vectors of the proper basis is (e<\/strong>x<\/sub>i<\/sup>, e<\/strong>y<\/sub>i<\/sup>) and (e<\/strong>x<\/sub>i<\/sup>, e<\/strong>y<\/sub>i<\/sup>, e<\/strong>z<\/sub>i<\/sup>) when it has gone i<\/em> rotations. i=0<\/em> at the\nstart. <\/p>\n\n\n\n \u2026<\/p>\n\n\n\n Figures and equations are in the article\nbelow:<\/p>\n\n\n\n \u00abStep\nby step rotation<\/a> in normal and high\ndimensional space and meaning of quaternion<\/a>\u00bb<\/p>\n\n\n\n https:\/\/www.academia.edu\/52628458\/Step_by_step_rotation_in_normal_and_high_dimensional_space_and_meaning_of_quaternion<\/a><\/p>\n\n\n\n https:\/\/pengkuanonmaths.blogspot.com\/2021\/09\/step-by-step-rotation-in-normal-and.html<\/a><\/p>\n","protected":false},"excerpt":{"rendered":" The orientation of body in space is defined 3 by angles. The step by step rotation process and chain of three-dots multiplication give an easy way to compute pile of rotations in 3D and high dimensional space and give a … Continue reading