Photogrammetry V: Dr. Gamal Hassan Seedahmed Dept. of Surveying Eng.

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Presentation transcript:

Photogrammetry V: Dr. Gamal Hassan Seedahmed Dept. of Surveying Eng. 3D Rotation Matrix: Euler Representation Dr. Gamal Hassan Seedahmed Dept. of Surveying Eng. Faculty of Eng. University of Khartoum Gamal.Seedahmed@gmail.com

2D Rotation Matrix It is a 2 x 2 matrix

2D Rotation Matrix: Rotation by 30 degrees MATLAB Code MATLAB Results

3D Rotation Angles

Rotation Matrix Around The X-Axis

Rx :Rotation by 30 deg MATLAB Code MATLAB Results

3D Point Rotated By Rx

3D Point Rotated By Rx (30 deg)

Derivative for Rx with respect to Omega

Rotation Matrix Around The Y-Axis

Rotation Matrix Around The Z-Axis

Closer Look to the structure of the 3 rotation matrices

Simultaneous Rotation By The 3 Angles

3D Point Rotation by R

Properties of R It is true for the transpose of R

Orthogonality Property of R And similarly true for the other rows and columns

Rotation by an Angle Theta About an arbitrary Axis Containing the Origin and having Unit Length Direction W=(wo , w1 , w2 )

Example No. 1

Example No. 1: Results

Example No. 2

Example No. 2: Results

Example No. 3

Example No. 3: Results

Example No. 4

Example No. 4: Results

Major Properties of Rotation Matrix The rotation matrix depends only on: The rotation axis direction (w). And the rotation angle.

General Representation of the 3D Rotation Matrix

How to compute theta from R?

How to compute S from R?