I think that you went to a conclusion by reading the title.
Haha! Don't Worry! Don't worry! I will explain to you the 'Four Petals Of Eight Flowers.' Ok, Before I enter into the topic, I will take you to the space. I am sitting on the top of a hill behind a banyan tree. My eyes were filled with emotions toward the sky. Suddenly, a strip of light beam scattered the dark shaded red clouds traveling from their hometown towards the Earth house. I thought to catch the falling twinkle star but, unfortunately, I slipped from the mountain and ended up beside the river. At this place, I have seen the bright natural satellite of the Earth with a smiling face floating on the river and I observed the same satellite with faded smile on the following days. I thought that I am watching the Moon in the wrong direction and tried to change the view. Now, I am excited to reverse the direction (i.e., I am on the Moon and viewing the Earth revolving around the Sun) but, the bed on which I am sleeping is not comfortable, and I woke up from the dream, and it's early morning.
I thought the Earth is rotating on its axis because I have only 24 hours and science proved it. The Earth is rotating (a 3D body) and, I am (a stationary dimension) viewing the Earth from the moon. The combination results to 4D and, they are called Quaternions that were first described by Irish mathematician William Rowan Hamilton in 1843 [1].
Yes, you heard it correctly. The Quaternion is a single vector with four values, of which it has three imaginary parts(i, j, k) and, the fourth value is the real part(r). For example, a spin (3D body) is rotating on the floor and, I am viewing with an eye ( a 4th dimension) defines quaternions. Quaternions are efficient and analyze easily in situations, where the rotations R3 are involved. The quaternion is a 4-tuple format, that represents a unique structure. Quaternions are good in obtaining the rotation axis and angle of the rigid body [1]. The following is the Quaternion (Q) equation representation.
Quaternion(Q) = w + xi + yj + zk : w; x; y; z ∈ R
A 3D body has three axes and three angles (as Euler angles) to define its rotation. But why introduce the new dimension as quaternions.
Yeah, you are correct. Everything should be 'One Man One Show.'
Now, I will answer your question 'why to introduce quaternions.'
Euler angles (three angles) define the rotation of the body in space until the Gimbal Lock phenomena occur. Gimbal Lock is when the two axes of the three gimbals are directed into a parallel structure then one degree of freedom gets locked in a 3D space. Then the visualization dips into the 2D space where it's difficult to set back orientation. So, quaternions are introduced as a 4D vector to define a 3D body in space. Ok, I hope that you are following my dream. Let's go back to my dream.
Rotation of the body is understood, with Quaternions but, what about translation.
In my dream, I said that I am watching the Earth from the Moon revolving around the Sun. I have seen that the blue ball (Earth) changes by ~1.002° every day and completes 360° for 365 days.
How can I observe both rotation and revolution of the Earth at once around the Sun to upgrade a year?
Ok! ok!
I think you are also in the same disoriented state.
Haha! Now, I will keep a full stop to your confusion.
Before going into a theoretical area, I will explain with a practical example. Let us assume that you are driving your lucky horse-powered car on a highway. Suddenly, a cat-powered car behind you crosses your horse-powered vehicle in few minutes. But you accelerated, removed the clutch, and tried to overtake the other automobile. You have observed that the other car applied brakes. Then the wheels of the cat car kissed the road with black lipstick. You have observed both circular motion (i.e., rotation) and skid (i.e., translation) of the other car's wheel. So, two quaternions are required to define the rigid body dynamics and are called Dual Quaternion.
Dual Quaternion represents two Quaternions with eight values. Out of 8, 6 are imaginary, 2 are real with a dual number(ɛ) in between the two quaternions. The first 4/8 values represent the rotation(R_d) of the 3D body in space and the next 4/8 values represent translation(T_d) multiplied to the semi rotation(R_d) of the body. The dual quaternion can also be used to represent the kinematics of a rigid body by considering it as a point, line, or surface of the body. The below Dual Quaternion (Q_d) equation defines the dual quaternion structure.
Q_d = R_d + ɛ.T_d.R_d/2 : R_d,T_d ∈ Q
So, these Dual Quaternions made me stay on the Moon and observe the Earth's rotation (R_d) on its axis and revolution i.e., translation (T_d) around the Sun. So, the current orientation and position of the body are visualized clearly through our eyes. At present, the research is on Dual Quaternions in several sectors of Artificial Intelligence to enhance the Neural Networks to train faster than before.
Now, I hope that you followed my dream correctly and observed the natural phenomena of Earth. I am happy that if you like, comment, share this article. Visit My Website for more information. Go through My CV for better understanding about me. Thank you for your valuable and precious time. The future comes with more exciting articles. References: [1] http://web.cs.iastate.edu/~cs577/handouts/quaternion.pdf
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