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Let's start with Atlanta, a function that I have mentioned in the last lesson Atlanta function is one

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of the most important functions during programming, and you will see its applications in robotics quite

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frequently.

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Its purpose is the same with architectures or inverse tangent function, namely finding and from the

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value that has been obtained by tangent function.

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Let's be more clear.

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Imagine a right angle triangle with sides of A and B and Angle Tito.

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As we know from Trigonometry Tangent Tito is given by this formula.

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We can find data from this formula by finding inverse of it, namely inverse tangent.

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While it's very simple this method, Holmes has some serious problems.

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Let's assume that we have a circle that represents all the angles we can have from zero to two PI.

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Let's try to calculate tangent of PI over four, which is nothing but division of Sun Pi over four and

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Cosine Pi or four, which will make one.

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Now let's take three pi or four radians and calculate its own tangent.

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You will see that its tangent is also one.

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So as you can see, we lost information here.

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Namely, one can be result of inverse tangent of PI or four PI or four and three PI or four, and with

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simple arc tangent function, we cannot know that.

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So the range of upon function or a tangent function is minus PI or two and plus PI or two exclusive

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exclusive because cosine plus or minus PI over two is zero, which causes discontinuities, which is

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another problem of our tangent function.

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However, with a ton to function or acting is to function.

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We don't have any of these problems.

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Its range is zero two to a point, which is whole circle, and there is no discontinuity at plus or

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minus PI over two.

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If you think carefully, the reason that Atom two forks is that in the case of atom function or Klingons

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function by division, we lost sine information because two sons signs from sine and cosine functions

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collapse into one sine either a plus or minus.

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That creates uncertainty, which is not the case in order to function because we don't divide.

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We give two inputs and let a time to do the rest of the work.

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OK, it's not about Atlanta function, let's continue to our last topic.

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Now we will see another two methods of representing rotation.

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Let's start with axis angle representation.

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Rotations are not always how to be informed about the principle axis, namely X, Y and Z.

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Indeed, we can represent in the rotation by a single rotation around some axis by a specific angle

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of Keita.

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We can find this rotation axis by analyzing eigenvalues and eigenvectors over rotation matrix.

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Let's take this rotation matrix as an example and try to find eigenvalues and eigenvectors of it.

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If you analyze its eigenvalues, you will see eigenvalue with real value of one and its corresponding

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eigenvectors from eigenvalue and eigenvectors relation.

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We know that matrix times eigen vector equals the eigenvalue times eigenvectors.

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In our case, eigenvalue is one, so formula becomes like that.

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This means that we multiply a vector with a matrix and we don't get any change in its magnitude.

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So this matrix can be on a rotation matrix because rotation matrix only cause rotation, no change in

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the absolute value of the vector or length of the vector.

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So I can vector that corresponds to the eigenvalue of one is the axis of rotation.

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We can find the rotation angle by comparing eigenvalues with this formula.

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Don't worry, we will see this clearly in our next lesson, which will be practical applications of

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these.

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So this is our angle and rotation axis that represents observer rotation matrix.

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However, this answer is not unique.

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Namely, the rotation of data around minus v axis results the same rotation.

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So this angle?

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This is the representation we have for quantities to represent a rotation, namely one quantity for

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representing the rotation angle and three quantities for representing the rotation axis.

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However, not all of these are independent as our rotation axis is a unit vector.

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Two of its values Unit Vector two of its values are independent.

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The third one is defined based on the other two, so we have three independent quantities.

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We can describe axis angle representation by this formula.

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TITA is the rotation angle, and the key is our rotation axis.

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As Kate is a unit vector, the model of Vector R will be angle tita to and its direction will give rotation

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axis of K.

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However, it's important to note that we cannot do a vector algebra or vector are named.

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We cannot combine to axis angular representations using standard rules of vector algebra because this

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would mean that rotations compute, which is not correct.

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Let's continue with unit quarter news, which is very important, and once the user representation of

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rotations in robotics quaternary in our extension of the complex number name, the hyper complex number,

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and it can be written as a scalar plus a vector parts.

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So this represents the scalar part and V represents the vector part.

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I would like to mention you also that I, J K are orthogonal complex numbers and they're square and

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their multiplication with each other gives negative one.

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We can additionally represent Kryptonians in this way.

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Quote and support addition and subtraction operations, which is performed Element Y.

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As you can see below, the example of addition of two quarter new addition is associative and commutative.

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Apart from the addition and subtraction, Cortana's also support continuum, or Hamiltonian product.

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This represents a relative rotation, as in the case of rotation mattresses.

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Here is the Hamiltonian product formula as in the case of rotation mattresses, post multiplication

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gives rotation with respect to the current frame, while print multiplication give rotation with respect

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to the fixed frame to represent rotations.

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We use unit quarter news unit continues our subcategory of continuous and they represent rotation of

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T to about a unit axis of V.

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We generally refer to them as simply quaternary news.

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They are represented in this way.

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From this form love, we can get unit quarter in use that represents rotation around principle axis.

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For unit quarter use, their magnitude is equal to one.

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So why do you turn use instead of roll PTO or a lower angle representations?

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Even if the visualization of unit quaternary use is complex because they are compact?

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They are computational efficient.

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Also the most important advantage that they don't have Gimbel look that Aler and.

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All pitch your angle representations are prone to.

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We will see this problem in the next lesson.