WEBVTT

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Game Basics
Mathematics for 3D Games 1 (3D Space Design, Euler Angles)

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GCC Academy

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Hello everyone

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This is Lee Deuk-woo on Game Mathematics

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Today, we will be talking about the third dimension

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Before learning about math for 3D,

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let's first learn about

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how to design the transforms in 3D space

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We will start by designing a 3D space

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What is different about setting transform in 3D than in 2D,

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and in a 3D space,

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how do we design the camera system?

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We will find out together

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And finally, we will take a look

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at a work produced with this theory

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Next, we'll learn about Euler angles,

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an expression for rotation in a 3D space

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We will first learn about the expression in an Euler angle,

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and then the strengths

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and limitations of Euler angle

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Designing a 3D Space

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When dealing with 3D space,

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Let's start with 2D space

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In 2D, the x-axis typically points to the right,
and the y-axis points upwards,

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which makes implementation relatively straightforward

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However, in 3D, the space is composed of three axes,

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and the way you design this space

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can influence the application methods

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and the mathematics we will explore

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Therefore, it is crucial to clearly define this first

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Instead of simply adopting

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a coordinate system jus because

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this application uses it,

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it is better to think about the principles

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and establish a system

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that suits your needs

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I think that is a better way to work with it

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So now, for us

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to design a 3D space,

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you must choose between two options

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Using a left-handed

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or a right-handed coordinate system

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Let me explain briefly about how to distinguish

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between the left- and right-handed system

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When you create a 3D space,

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it essentially adds

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one dimension to a 2D space

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Then, in designing a 2D space,

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to start, fix the 2D space

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As this is generally accepted, as we look at the monitor

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the x-axis points to the right

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and the y-axis points upwards

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Now, after we have fixed

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this 2D space as we see it here,

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based on the direction of the remaining dimension,

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you can distinguish

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whether the system is left- or right-handed

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When you look at the screen,

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you can see that after fixing the red

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and green axes as the x and y axes,

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the direction of the remaining z-axis

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determines whether

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the system is left-handed or right-handed

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If the z-axis points away from you,

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beyond the monitor,

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it is a left-handed coordinate system

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On the other hand, if the z-axis points toward you,

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it is a right-handed coordinate system

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That's how you can understand it

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The left-handed coordinate system is characterized as follows

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When we fix the x and the y axes,

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by the z-axis pointing forward, beyond the screen,

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stretching like this

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This makes it suitable for viewing landscapes

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or scenes stretching into the distance

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In contrast, the right-handed coordinate system,

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where the z-axis points toward you,

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can be more useful

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when observing objects up close

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Ultimately, deciding between

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the left-handed and right-handed coordinate systems

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depends on your preference,

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and it's not necessarily set by rule

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Go for what feels more comfortable

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or convenient for the system

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that you are designing

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Once you’ve chosen a coordinate system,

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the next important decision is

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determining which axis will be designated as the upward axis,

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of x, y, and z axes

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which will be designated as the upward axis?

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We have to designate one

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The Up direction, as it is commonly referred to,

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becomes particularly important later when we deal with operations

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like cross products or when calculating rotation matrices

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based on view vectors

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Knowing and setting the Up direction clearly is crucial

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Broadly, the classification involves choosing
between left-handed or right-handed systems

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and deciding the Up direction

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For example, as shown in a diagram

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shared on Twitter (X),

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systems are generally divided into
left-handed and right-handed coordinate systems,

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with an additional distinction based on

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whether the Up direction is along the y-axis or the z-axis

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That's another classification

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By classifying the coordinate system into these categories,

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just understand that these systems

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or programs use these distinctions

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In this course, when we deal with 3D graphics,

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in the program I developed,

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all examples are implemented using the specific type

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of coordinate system I have designated

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The examples will use a right-handed coordinate system

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with the Up direction along the y-axis,

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referred to as Y-up Right-Handed

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This means we will use the Right-Handed Coordinating System

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with the y-axis pointing upwards, to be very brief

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The forward and backward directions,

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the remaining directions,

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will be determined by the z-axis,

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while the left and right directions

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will be handled by the x-axis

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This is the coordinate I will be using

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Please take a look at it

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After setting up the coordinate system,

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the next step involves

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configuring 3D transforms

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When transitioning from 2D to 3D,

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additional considerations arise

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For instance, in the case of translation transforms,

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movement along the z-axis is added

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to the x and y movements already present in 2D

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However,

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since the x, y, and z axes are orthogonal,

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movements along one axis do not affect the others

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Therefore, adding an additional dimension

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to the matrix we have

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allows for straightforward solutions without much complexity

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So it's not a big problem

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For scaling transformations, enlarging along the x and y axes

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does not affect the z-axis,

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so it simply involves extending the matrix

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to accommodate the added dimension

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However, rotation transformations

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are more complex because all three axes

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influence one another

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while maintaining orthogonality

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So this is much more complex

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Translation and scaling can be easily implemented

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by simply extending a single axis

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However, in the case of rotation, the addition of one axis

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causes all three axes to influence one another,

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making it a much more complex transformation

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To understand rotation,

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consider the x, y, and z axes,

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our initial orthogonal x, y, and z axes,

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which are rotated

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to form a new set of local basis vectors

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By substituting the transformed basis vectors into the matrix,

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like this,

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you can construct a 3x3 rotation matrix,

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which consists of nine elements representing these transformations

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So that's a rotation transformation

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Using the transformed standard basis vectors, such as e_1, e_2, and e_3,

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 the rotation matrix can be established

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as mapped to x_local, y_local, and z_local

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By marking the values,

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we can then derive the corresponding matrix

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For the scaling matrix, the original 3x3 matrix

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is extended by adding one more dimension, resulting in a 4x4 matrix

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By simply adding the z-axis component to the last row,

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the matrix can be easily constructed

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However, with rotation, local axis information must

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be fully determined

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and inserted vertically

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to build the rotation matrix

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Translation matrices

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similarly involve extending the z-axis

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and follow a similar design like this

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Using these principles, we can construct the modeling matrix

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and calculate the TRS like this

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The process is not fundamentally different

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from what we explored in 2D matrices,

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but adapted for 3D space

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In this way, we have explored

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how to design transformation matrices in 3D space

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Now, let’s move on to discussing the camera and its setup

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It's not too different with the camera

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Finally, when designing the camera for 3D space,

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additional considerations arise

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or example, using the y-axis

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as the Up direction in 3D space,

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the z-axis for an object

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at the origin typically points toward the monitor,

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or towards us

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To observe an object properly,

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the camera should be positioned in the positive z-axis direction,

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facing the object

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For example, placing the camera at my position

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and pointing it toward the monitor

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means that the camera is not facing towards me

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If it is rotated 180 degrees, I towards the monitor,

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the object towards the monitor,

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it means it's facing towards me

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That's how we can face the front of the object

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If the camera is rotated 180 degrees,

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an issue arises

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where the x-axis of the camera’s coordinate system

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points to the left instead of the right

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This contradicts the standard convention

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where the x-axis should point

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to the right in the final rendered view

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Now that it's the left,

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it creates an uncomfortable experience for the viewer,

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as the output appears flipped

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To solve this,

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the camera’s coordinate system should be adjusted

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so that the x-axis points to the right,

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while the y axis stays there

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Therefore, in this camera setup,

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the final coordinate system

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must maintain the orthogonality

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of the three axes as originally defined

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So now,

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the camera is facing forward,

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but by rotating the y-axis 180 degrees,

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causing the x-axis to align to the right

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and the z-axis to point backward from the camera,

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This creates a right-handed coordinate system

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where the viewing direction

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now points in the opposite direction,

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behind the camera

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The x-axis and y-axis,

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which represent the important dimensions of the camera’s view,

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align with the standard directions we naturally perceive

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However, the z-axis points behind the camera,

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meaning the viewing direction corresponds

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to this reversed z-axis orientation

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To place our camera

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and construct a proper view coordinate system,

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the x and z axes must be flipped to match this convention

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That completes our view coordinate system

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Let's now create a view matrix

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which is responsible creating this view coordinate system

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In 2D, the view matrix was derived

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from the inverse of the camera’s transform matrix

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by first applying the translation

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and then reversing the rotation

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The same approach applies to 3D

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Translation is applied first,

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followed by the inverse rotation

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For rotation, the yaw rotation matrix is used,

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and its local axes are organized into columns,

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then transposed into rows to form the rotation’s inverse matrix

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The translation’s inverse matrix involves negating the x, y, and z components of the translation vector.

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So, the x, y, and z axes are

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initially arranged as columns

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So, the x, y, and z axes are arranged as columns

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This transposed matrix represents the inverse rotation matrix

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used for the camera transform

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For the translation inverse matrix, it’s straightforward

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As previously mentioned, you simply negate the x, y,

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and z components of the translation vector

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That gives you the translation inverse matrix

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You first apply the translation inverse,

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followed by the rotation inverse

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By multiplying these two matrices,

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the translation component becomes

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part of the result as a dot product

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At this stage, we need to add something

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If we stop here, this is what we get

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This is what we get if we stop here

273
00:14:00.940 --> 00:14:03.890
To fully convert the matrix into the view space,

274
00:14:03.890 --> 00:14:07.419
the x-axis and z-axis values must be inverted

275
00:14:07.419 --> 00:14:09.969
By reversing the x and z components

276
00:14:09.969 --> 00:14:11.979
in the original transform information,

277
00:14:11.979 --> 00:14:16.299
both the basic x and z values,

278
00:14:16.299 --> 00:14:19.099
as well as the corresponding elements

279
00:14:19.099 --> 00:14:21.299
in the dot product, are negated

280
00:14:21.299 --> 00:14:24.499
This adjustment produces

281
00:14:24.499 --> 00:14:26.700
this view coordinate system

282
00:14:26.700 --> 00:14:32.179
That was on the foundational setup for 3D space

283
00:14:32.179 --> 00:14:35.429
Next, I will demonstrate

284
00:14:35.429 --> 00:14:36.661
how to apply the view matrix to render a 3D cube

285
00:14:36.661 --> 00:14:38.419
using the program I created

286
00:14:38.419 --> 00:14:41.979
First, the object will be positioned at the origin point (0, 0, 0)

287
00:14:41.979 --> 00:14:43.629
You can imagine

288
00:14:43.629 --> 00:14:46.079
placing a rectangular cube on the monitor

289
00:14:46.079 --> 00:14:50.059
that you are currently looking at as the object being discussed

290
00:14:50.059 --> 00:14:54.059
Next, the object will be rotated 45 degrees
around the y-axis

291
00:14:54.059 --> 00:14:56.539
and 60 degrees around the x-axis

292
00:14:56.539 --> 00:14:59.900
This means the object will first spin around the y-axis,

293
00:14:59.900 --> 00:15:03.639
like the rotation of a helicopter propeller,

294
00:15:03.639 --> 00:15:06.739
and then tilt by -60 degrees around the x-axis,

295
00:15:06.739 --> 00:15:09.074
something like this,

296
00:15:09.074 --> 00:15:10.780
as if for nodding

297
00:15:10.780 --> 00:15:14.448
These two rotations combined will result in a state

298
00:15:14.448 --> 00:15:17.299
where the object has been rotated in both directions

299
00:15:17.299 --> 00:15:19.299
After applying these rotations, the camera will

300
00:15:19.299 --> 00:15:21.419
be positioned at (0, 0, 500)

301
00:15:21.419 --> 00:15:25.607
This placement assumes the positive z-axis points outward

302
00:15:25.607 --> 00:15:27.340
from the monitor toward you,

303
00:15:27.340 --> 00:15:31.260
meaning the camera is essentially placed at your position

304
00:15:31.260 --> 00:15:34.910
Since the camera is facing the monitor,

305
00:15:34.910 --> 00:15:37.020
the view needs to be adjusted by a 180-degree rotation,

306
00:15:37.020 --> 00:15:39.320
in the y-axis,

307
00:15:39.320 --> 00:15:42.619
so that the camera can face the object

308
00:15:42.619 --> 00:15:45.369
This is the result

309
00:15:45.369 --> 00:15:48.059
of that scene we just created

310
00:15:48.059 --> 00:15:50.966
This concludes the demonstration

311
00:15:50.966 --> 00:15:54.059
of how to set up and configure a 3D scene

312
00:15:54.059 --> 00:15:57.459
The result displayed here shows

313
00:15:57.459 --> 00:16:00.709
the rectangular shape that appears slightly unnatural

314
00:16:00.709 --> 00:16:02.380
due to the lack of perspective applied

315
00:16:02.380 --> 00:16:04.580
In the upcoming section, we will explore

316
00:16:04.580 --> 00:16:08.180
methods to incorporate perspective into the scene

317
00:16:08.180 --> 00:16:11.059
and compare the resulting outputs

318
00:16:11.059 --> 00:16:15.020
This concludes the basic setups for the 3D space

319
00:16:15.215 --> 00:16:19.017
Euler Angles

320
00:16:20.158 --> 00:16:22.958
To understand Euler angles,

321
00:16:22.958 --> 00:16:26.658
we must first revisit how to handle rotations

322
00:16:26.658 --> 00:16:30.257
in 3D space, as discussed previously

323
00:16:30.257 --> 00:16:34.299
When constructing a rotation matrix,

324
00:16:34.299 --> 00:16:40.026
the standard basis vectors  e_1 ,  e_2 , and  e_3

325
00:16:40.026 --> 00:16:41.497
are transformed

326
00:16:41.497 --> 00:16:44.776
while preserving their original orthogonality,

327
00:16:44.776 --> 00:16:52.020
rotating the axes to result in this rotation matrix

328
00:16:52.020 --> 00:16:57.070
The transformed vectors, which are now defined

329
00:16:57.070 --> 00:17:00.570
as local X, local Y, and local Z,

330
00:17:00.570 --> 00:17:02.739
these three local vectors

331
00:17:02.739 --> 00:17:08.939
of x, y, and z

332
00:17:08.939 --> 00:17:10.904
are arranged as columns

333
00:17:10.904 --> 00:17:14.060
to create a matrix

334
00:17:14.060 --> 00:17:17.171
While constructing the rotation matrix is straightforward

335
00:17:17.171 --> 00:17:20.459
if the values of these transformed local axes are known,

336
00:17:20.459 --> 00:17:25.009
the challenge lies in determining

337
00:17:25.009 --> 00:17:27.900
the new local axis values after rotation

338
00:17:27.900 --> 00:17:31.800
If we want to rotate it by some angle,

339
00:17:31.800 --> 00:17:35.600
what is the value of each local axis

340
00:17:35.600 --> 00:17:39.410
that has been rotated? That is still a question

341
00:17:39.410 --> 00:17:41.560
For instance, if a game developer

342
00:17:41.560 --> 00:17:45.540
who is not familiar with math

343
00:17:45.540 --> 00:17:50.540
wants to rotate an object 30 degrees around the y-axis,

344
00:17:50.540 --> 00:17:54.540
with that command that resulted in the 30 degrees rotation,

345
00:17:54.540 --> 00:17:59.090
calculating the exact values of

346
00:17:59.090 --> 00:18:01.240
the local x, y, and z vectors after the rotation

347
00:18:01.240 --> 00:18:02.900
becomes a complex problem

348
00:18:02.900 --> 00:18:06.540
It could be a very complex problem, indeed

349
00:18:06.540 --> 00:18:09.340
The second issue arises

350
00:18:09.340 --> 00:18:12.859
when managing rotations using matrices

351
00:18:12.859 --> 00:18:14.659
As shown, a rotation matrix

352
00:18:14.659 --> 00:18:18.182
requires three 3D vectors,

353
00:18:18.182 --> 00:18:20.219
resulting in a total of nine data elements

354
00:18:20.219 --> 00:18:23.619
When issuing a simple command like

355
00:18:23.619 --> 00:18:25.869
rotate 30 degrees around the y-axis,

356
00:18:25.869 --> 00:18:30.140
determining how to correctly arrange and calculate

357
00:18:30.140 --> 00:18:33.890
these nine values can be cumbersome,

358
00:18:33.890 --> 00:18:36.819
especially in the developer's point of view

359
00:18:36.819 --> 00:18:40.119
From a developer’s perspective,

360
00:18:40.119 --> 00:18:41.893
a simpler method

361
00:18:41.893 --> 00:18:44.393
for representing rotations in 3D space

362
00:18:44.393 --> 00:18:46.380
is highly desirable

363
00:18:46.380 --> 00:18:49.699
his is where the Euler angle representation comes into play

364
00:18:49.699 --> 00:18:52.199
As shown on the screen,

365
00:18:52.199 --> 00:18:55.449
the Euler angle representation
expresses rotations in 3D space

366
00:18:55.449 --> 00:18:58.739
by specifying the amount of rotation

367
00:18:58.739 --> 00:19:03.900
around the local x, y, and z axes

368
00:19:03.900 --> 00:19:07.597
The newly defined basis axes

369
00:19:07.597 --> 00:19:10.297
e1, e2, and e3 correspond to

370
00:19:10.297 --> 00:19:13.780
the standard basis vectors (1, 0, 0), (0, 1, 0), and (0, 0, 1)

371
00:19:13.780 --> 00:19:16.380
So they do not need to be explicitly included

372
00:19:16.380 --> 00:19:22.680
Instead, only the rotation values for these axes

373
00:19:22.680 --> 00:19:24.080
can represent the 3D rotations

374
00:19:24.080 --> 00:19:26.060
as the following,

375
00:19:26.060 --> 00:19:29.900
as theta_x, theta_y, and theta_z

376
00:19:29.900 --> 00:19:33.914
 This approach significantly reduces complexity,

377
00:19:33.914 --> 00:19:35.514
as it replaces the need for nine matrix elements

378
00:19:35.514 --> 00:19:38.714
with just three values

379
00:19:38.714 --> 00:19:41.020
for representing rotations in 3D space

380
00:19:41.020 --> 00:19:45.520
In Unity, rotation values for each axis are managed

381
00:19:45.520 --> 00:19:48.020
using the Vector3 structure,

382
00:19:48.020 --> 00:19:50.020
while in Unreal Engine,

383
00:19:50.020 --> 00:19:53.780
rotations are handled through a Rotator structure

384
00:19:53.780 --> 00:19:56.291
A key distinction is that Unity

385
00:19:56.291 --> 00:19:58.307
uses x, y, and z values directly,

386
00:19:58.307 --> 00:20:02.120
whereas Unreal Engine employs Yaw, Pitch, and Roll

387
00:20:02.120 --> 00:20:06.440
to define rotations,

388
00:20:06.440 --> 00:20:08.699
which are modeled after airplane movements

389
00:20:08.699 --> 00:20:11.349
Let's talk more about

390
00:20:11.349 --> 00:20:13.080
the Yaw, Roll, and Pitch rotations

391
00:20:13.080 --> 00:20:15.720
Yaw, Pitch, and Roll are specific types of rotations

392
00:20:15.720 --> 00:20:19.973
based on how an object moves relative to its axes,

393
00:20:19.973 --> 00:20:21.923
often visualized

394
00:20:21.923 --> 00:20:23.920
using an airplane as you can see

395
00:20:23.920 --> 00:20:27.480
This is the rotation in an airplane

396
00:20:27.480 --> 00:20:30.530
We do not determine this

397
00:20:30.530 --> 00:20:32.320
as x, y, and z axis

398
00:20:32.320 --> 00:20:36.360
We instead determine the rotation based on the shape

399
00:20:36.360 --> 00:20:37.880
Let me explicate

400
00:20:37.880 --> 00:20:40.360
The Yaw rotation

401
00:20:40.360 --> 00:20:42.359
is rotating around the Up vector,

402
00:20:42.359 --> 00:20:44.263
akin to

403
00:20:44.263 --> 00:20:46.599
a helicopter propeller's rotation

404
00:20:46.599 --> 00:20:48.400
that we all know of

405
00:20:48.400 --> 00:20:51.819
That shape of rotation

406
00:20:51.819 --> 00:20:53.456
occurring in a 3D space

407
00:20:53.456 --> 00:20:55.599
is what we call a Yaw rotation

408
00:20:55.599 --> 00:20:58.472
As for the Pitch rotation,

409
00:20:58.472 --> 00:21:00.807
it rotates around the left-to-right axis,

410
00:21:00.807 --> 00:21:02.161
resembling

411
00:21:02.161 --> 00:21:04.879
a nodding motion like so

412
00:21:04.879 --> 00:21:08.457
The Roll rotation happens around the forward axis,

413
00:21:08.457 --> 00:21:10.107
where the object spins

414
00:21:10.107 --> 00:21:11.768
like a barrel roll

415
00:21:11.768 --> 00:21:15.239
These three types of rotation
define how objects move in 3D space

416
00:21:15.239 --> 00:21:18.160
If the rotation occurs around the Up vector,
it is referred to as Yaw

417
00:21:18.160 --> 00:21:21.460
If the rotation occurs around the left-right axis

418
00:21:21.460 --> 00:21:23.680
with a nodding motion, it is Pitch

419
00:21:23.680 --> 00:21:25.718
Finally, if the rotation happens around the forward axis,

420
00:21:25.718 --> 00:21:27.919
it is Roll

421
00:21:27.919 --> 00:21:32.040
Regardless of which axes (x, y, z) are used,

422
00:21:32.040 --> 00:21:35.360
the shapes of the rotations remain consistent

423
00:21:35.360 --> 00:21:42.320
This consistency allows for easier exchange of rotation data

424
00:21:42.320 --> 00:21:44.360
between different programs
that use varying 3D coordinate systems

425
00:21:44.360 --> 00:21:46.134
By standardizing rotations

426
00:21:46.134 --> 00:21:48.080
with Yaw, Pitch, and Roll
based on these rotational shapes,

427
00:21:48.080 --> 00:21:51.880
it becomes much clearer

428
00:21:51.880 --> 00:21:54.599
and more precise to share rotational information

429
00:21:54.599 --> 00:21:56.699
across different

430
00:21:56.699 --> 00:21:59.879
systems or applications

431
00:21:59.879 --> 00:22:05.056
For us, Yaw, Roll and Pitch are the following

432
00:22:05.056 --> 00:22:08.156
Our coordinate system is the right-hand one

433
00:22:08.156 --> 00:22:10.112
The left-hand one does not matter here

434
00:22:10.112 --> 00:22:12.395
So for each of x, y, and x axis,

435
00:22:12.395 --> 00:22:14.800
we can assign Yaw, Roll, and Pitch

436
00:22:14.800 --> 00:22:16.550
In the context of a Y-up coordinate system,

437
00:22:16.550 --> 00:22:18.679
Yaw corresponds to rotation around the y-axis

438
00:22:18.679 --> 00:22:20.639
or the vertical axis

439
00:22:20.639 --> 00:22:24.449
Now the x axis is the left-right, while the z is the forward-backward

440
00:22:24.449 --> 00:22:27.599
Pitch corresponds to rotation around the x-axis

441
00:22:27.599 --> 00:22:30.600
Roll corresponds to rotation around the z-axis

442
00:22:30.600 --> 00:22:35.350
That's how each of them is assigned

443
00:22:35.350 --> 00:22:37.240
in our program

444
00:22:37.240 --> 00:22:39.052
The Yaw, Pitch, and Roll convention

445
00:22:39.052 --> 00:22:42.093
originally came from aeronautics,

446
00:22:42.093 --> 00:22:45.079
adopted in mechanical engineering

447
00:22:45.079 --> 00:22:48.119
over time with the original methods

448
00:22:48.119 --> 00:22:50.819
and later became a standard in computer graphics,

449
00:22:50.819 --> 00:22:53.904
where it is still widely used today

450
00:22:53.904 --> 00:22:55.534
As previously mentioned,

451
00:22:55.534 --> 00:22:58.279
different programs may use axes differently,

452
00:22:58.279 --> 00:23:00.629
but representing rotations based on their shapes

453
00:23:00.629 --> 00:23:02.839
ensures that the rotation information

454
00:23:02.839 --> 00:23:05.919
can be clearly exchanged between systems

455
00:23:05.919 --> 00:23:09.080
That's the major advantage

456
00:23:09.080 --> 00:23:13.759
Ultimately, Euler angles

457
00:23:13.759 --> 00:23:17.309
need to be converted

458
00:23:17.309 --> 00:23:19.191
into rotation matrices, why?

459
00:23:19.191 --> 00:23:22.279
To create a TRS (Translation, Rotation, Scaling) matrix,

460
00:23:22.279 --> 00:23:25.960
a rotation matrix is necessary

461
00:23:25.960 --> 00:23:28.759
Even though Euler angles are user-friendly,

462
00:23:28.759 --> 00:23:31.309
for operations involving a large number of points

463
00:23:31.309 --> 00:23:33.360
like tens of thousands or millions,

464
00:23:33.360 --> 00:23:34.800
matrices are essential for efficient transformations

465
00:23:34.800 --> 00:23:37.960
Therefore, based on the information of the Euler angle,

466
00:23:37.960 --> 00:23:40.000
creating rotation matrix is necessary

467
00:23:40.000 --> 00:23:42.320
Let's dive deeper into it

468
00:23:42.320 --> 00:23:44.479
First, Euler angles

469
00:23:44.479 --> 00:23:46.979
do not represent

470
00:23:46.979 --> 00:23:48.639
a single instantaneous rotation

471
00:23:48.639 --> 00:23:52.720
It's not an instantaneous thing

472
00:23:52.720 --> 00:23:54.870
Instead, they describe a sequence

473
00:23:54.870 --> 00:23:57.600
of three rotations, each around a local axis

474
00:23:57.600 --> 00:23:59.759
First, the object is rotated around the x-axis

475
00:23:59.759 --> 00:24:04.199
After this rotation, the local y- and z-axes
shift to new positions

476
00:24:04.199 --> 00:24:07.149
Now based on its new local axes,

477
00:24:07.149 --> 00:24:08.600
the object is rotated around the y-axis

478
00:24:08.600 --> 00:24:13.399
Finally, the object is rotated around the z-axis,
again using its updated local axes

479
00:24:13.399 --> 00:24:17.000
This sequence of rotations, broken into three steps,

480
00:24:17.000 --> 00:24:20.279
intuitively represents 3D rotations

481
00:24:20.279 --> 00:24:24.640
I say intuitive for users,

482
00:24:24.640 --> 00:24:27.040
but in terms of theory, it involves

483
00:24:27.040 --> 00:24:30.407
a layered transformation process

484
00:24:30.407 --> 00:24:31.839
That's how it works, essentially

485
00:24:31.839 --> 00:24:37.600
So the order of these opretaions is critical

486
00:24:37.600 --> 00:24:42.160
x, y, and z, or Yaw, Roll, and Pitch

487
00:24:42.160 --> 00:24:44.799
These three rotations, Yaw, Pitch, and Roll,

488
00:24:44.799 --> 00:24:48.399
are applied one by one

489
00:24:48.399 --> 00:24:51.239
Depending on the chosen order,

490
00:24:51.239 --> 00:24:54.239
six possible cases of applying the rotations can occur,

491
00:24:54.239 --> 00:24:57.079
of which we need to choose one

492
00:24:57.079 --> 00:25:00.689
To determine the order of applying

493
00:25:00.689 --> 00:25:02.359
rotations to the local axes,

494
00:25:02.359 --> 00:25:04.920
a common sequence

495
00:25:04.920 --> 00:25:08.200
is Roll, Pitch,

496
00:25:08.200 --> 00:25:12.239
and then Yaw, in that order

497
00:25:12.239 --> 00:25:15.239
This method is widely adopted in game engines

498
00:25:15.239 --> 00:25:18.399
such as Unreal Engine and Unity,

499
00:25:18.399 --> 00:25:20.359
making it the standard approach

500
00:25:20.359 --> 00:25:22.279
for handling Euler angles

501
00:25:22.279 --> 00:25:25.000
If you are wondering

502
00:25:25.000 --> 00:25:27.119
how to remember this order,

503
00:25:27.119 --> 00:25:30.320
think of the process

504
00:25:30.320 --> 00:25:32.000
as similar to controlling an airplane

505
00:25:32.000 --> 00:25:33.160
First, the airplane stabilizes

506
00:25:33.160 --> 00:25:35.440
its horizontal orientation to be able to fly

507
00:25:35.440 --> 00:25:39.760
by rolling around its forward axis

508
00:25:39.760 --> 00:25:43.279
Next, to fly upward,

509
00:25:43.279 --> 00:25:44.959
the airplane nods

510
00:25:44.959 --> 00:25:46.519
upward or downward

511
00:25:46.519 --> 00:25:49.640
by pitching around its left-right axis

512
00:25:49.640 --> 00:25:51.399
Finally, to steer toward the destination,

513
00:25:51.399 --> 00:25:53.600
the airplane turns left or right

514
00:25:53.600 --> 00:25:55.600
by yawing around its vertical axis

515
00:25:55.600 --> 00:25:58.959
This sequence,stabilize the roll, adjust the pitch,

516
00:25:58.959 --> 00:26:00.640
and steer with yaw,

517
00:26:00.640 --> 00:26:02.799
matches the order

518
00:26:02.799 --> 00:26:07.320
Roll, Pitch, and Yaw,

519
00:26:07.320 --> 00:26:11.760
making it intuitive for applications

520
00:26:11.760 --> 00:26:15.359
Once the order is set,

521
00:26:15.359 --> 00:26:17.259
the next step involves

522
00:26:17.259 --> 00:26:19.679
calculating the rotation matrix

523
00:26:19.679 --> 00:26:23.279
based on this sequence

524
00:26:23.279 --> 00:26:25.200
Each axis's rotation matrix

525
00:26:25.200 --> 00:26:27.799
will be derived like so

526
00:26:27.799 --> 00:26:30.040
The rotation matrix for the x-axis

527
00:26:30.040 --> 00:26:33.559
means x is the basis

528
00:26:33.559 --> 00:26:35.859
that represents a rotation

529
00:26:35.859 --> 00:26:37.519
around the x-axis,

530
00:26:37.519 --> 00:26:41.040
where the x-axis itself remains unchanged

531
00:26:41.040 --> 00:26:44.920
In this case, only the y- and z-axes undergo rotation,

532
00:26:44.920 --> 00:26:48.279
without the x axis rotating,

533
00:26:48.279 --> 00:26:50.640
effectively describing a planar rotation within the yz-plane

534
00:26:50.640 --> 00:26:52.760
For the x, y, and z axis,

535
00:26:52.760 --> 00:26:56.239
Since I'm a right-handed person, like this,

536
00:26:56.239 --> 00:27:01.040
let's think about a yz plane based on this x axis

537
00:27:01.040 --> 00:27:07.079
In a right-handed coordinate system,

538
00:27:07.079 --> 00:27:13.079
y axis is Up, x axis is Left-Right, and z axis, Forward-Backward

539
00:27:13.079 --> 00:27:16.239
This right-handed positioning

540
00:27:16.239 --> 00:27:18.489
is the 3D coordinate

541
00:27:18.489 --> 00:27:20.320
that my program uses

542
00:27:20.320 --> 00:27:23.160
For a rotation about the x-axis,

543
00:27:23.160 --> 00:27:26.480
for this rotation,

544
00:27:26.480 --> 00:27:29.839
the yz-plane rotates,

545
00:27:29.839 --> 00:27:32.839
with the direction of rotation

546
00:27:32.839 --> 00:27:36.539
 being from the y-axis

547
00:27:36.539 --> 00:27:38.239
toward the z-axis

548
00:27:38.239 --> 00:27:43.079
When looking at the rotation matrix for the y- and z-axes,

549
00:27:43.079 --> 00:27:48.244
it includes cosine and negative sine components,

550
00:27:48.244 --> 00:27:50.178
which form the basis of the rotation matrix

551
00:27:50.178 --> 00:27:53.600
Thus, for the x-axis rotation matrix,
it can be represented in this way

552
00:27:53.600 --> 00:27:56.119
Moving on to the rotation matrix for the y-axis,

553
00:27:56.119 --> 00:27:59.399
the y-axis remains unchanged

554
00:27:59.399 --> 00:28:00.999
What’s critical here

555
00:28:00.999 --> 00:28:04.920
is whether this y axis rotation

556
00:28:04.920 --> 00:28:08.600
goes from the x-axis to the z-axis or vice versa

557
00:28:08.600 --> 00:28:13.239
It's determining the sequence of rotation within the plane

558
00:28:13.239 --> 00:28:17.239
When the rotation is based on the y-axis,
it starts from the z-axis

559
00:28:17.239 --> 00:28:20.559
and moves to the x-axis, forming the zx-plane rotation

560
00:28:20.559 --> 00:28:24.362
If you list the sequence x, y, z repeatedly,

561
00:28:24.362 --> 00:28:26.440
like so,

562
00:28:26.440 --> 00:28:29.720
and you rotate about the x-axis,
the remaining plane is yz

563
00:28:29.720 --> 00:28:32.679
Then, for the y-axis,

564
00:28:32.679 --> 00:28:35.200
the sequence becomes zx,

565
00:28:35.200 --> 00:28:37.679
and for the z-axis, it becomes xy

566
00:28:37.679 --> 00:28:42.280
This is how the order of rotations is determined
when designing the  matrices

567
00:28:42.280 --> 00:28:44.799
Because it moves from zx,

568
00:28:44.799 --> 00:28:48.919
it effectively flips the xz-plane,

569
00:28:48.919 --> 00:28:51.238
creating a transposed matrix

570
00:28:51.238 --> 00:28:52.280
It's a flip

571
00:28:52.280 --> 00:28:56.380
This inversion results in the rotation matrix for the y-axis

572
00:28:56.380 --> 00:28:58.416
having a negative sign

573
00:28:58.416 --> 00:29:00.280
in the lower-left corner

574
00:29:00.280 --> 00:29:03.200
It’s easy to get confused,

575
00:29:03.200 --> 00:29:05.479
so it’s important to understand this clearly

576
00:29:05.479 --> 00:29:08.359
For the z-axis rotation, it starts with the xy-plane,

577
00:29:08.359 --> 00:29:11.659
and its rotation

578
00:29:11.659 --> 00:29:15.067
behaves consistently as expected

579
00:29:15.067 --> 00:29:17.840
In this context, the y-axis corresponds to Yaw,

580
00:29:17.840 --> 00:29:21.389
the x-axis corresponds to Pitch,
and the z-axis corresponds to Roll

581
00:29:21.389 --> 00:29:25.317
Since we apply the rotations
in the order Roll, Pitch, and Yaw,

582
00:29:25.317 --> 00:29:28.617
Yaw is applied last,

583
00:29:28.617 --> 00:29:30.039
which places its rotation matrix at the far left

584
00:29:30.039 --> 00:29:31.880
R_y by

585
00:29:31.880 --> 00:29:34.039
Roll, being the first applied, goes on the far right,

586
00:29:34.039 --> 00:29:38.439
making the order of matrix multiplication

587
00:29:38.439 --> 00:29:41.735
R_y×R_x×R_z

588
00:29:41.735 --> 00:29:44.835
If Yaw, Pitch, and Roll are represented

589
00:29:44.835 --> 00:29:46.960
as α, β, and γ, respectively,

590
00:29:46.960 --> 00:29:50.119
each rotation matrix can be computed

591
00:29:50.119 --> 00:29:54.080
and multiplied to yield the final result

592
00:29:54.080 --> 00:29:57.000
Although this composite rotation matrix is lengthy,

593
00:29:57.000 --> 00:30:00.520
it represents the 3D rotation
derived from Euler angles effectively

594
00:30:00.520 --> 00:30:02.919
So, this is what it is,

595
00:30:02.919 --> 00:30:05.469
the 3D rotation matrix

596
00:30:05.469 --> 00:30:10.080
derived from Euler angles

597
00:30:10.080 --> 00:30:12.530
The calculated rotation matrix

598
00:30:12.530 --> 00:30:15.479
is identical to the rotation matrix for 3D space

599
00:30:15.479 --> 00:30:18.440
 The column vectors here

600
00:30:18.440 --> 00:30:23.080
are the rotated results

601
00:30:23.080 --> 00:30:27.880
of the local x, y, and z axes

602
00:30:27.880 --> 00:30:29.000
That's what it is

603
00:30:29.000 --> 00:30:32.550
If you take this part

604
00:30:32.550 --> 00:30:35.099
and extract it as a vector, it becomes the local x-axis

605
00:30:35.099 --> 00:30:37.000
If you take this part, it becomes the local y-axis

606
00:30:37.000 --> 00:30:40.559
Lastly, taking this part gives you the local z-axis

607
00:30:40.559 --> 00:30:44.599
In game design,

608
00:30:44.599 --> 00:30:47.799
this allows us to specify

609
00:30:47.799 --> 00:30:50.719
a desired 3D space like this

610
00:30:50.719 --> 00:30:53.520
Now, while this resolves two problems,

611
00:30:53.520 --> 00:30:56.000
Euler angles still have their own limitations

612
00:30:56.000 --> 00:30:58.239
Let’s take a look at this

613
00:30:58.239 --> 00:31:01.400
It might seem like a perfect solution,

614
00:31:01.400 --> 00:31:05.159
but there is a fundamental limitation to Euler angles

615
00:31:05.159 --> 00:31:06.799
As I explained earlier,

616
00:31:06.799 --> 00:31:10.719
if they look straight ahead and apply a Roll rotation,

617
00:31:10.719 --> 00:31:14.239
the face rotates, right?

618
00:31:14.239 --> 00:31:16.467
The face, or your nose as the center,

619
00:31:16.467 --> 00:31:18.760
turns sideways like this

620
00:31:18.760 --> 00:31:21.060
Then, if I apply a Pitch rotation

621
00:31:21.060 --> 00:31:22.239
about theneck,

622
00:31:22.239 --> 00:31:24.599
then it goes this way

623
00:31:24.599 --> 00:31:27.719
After this Pitch rotation,

624
00:31:27.719 --> 00:31:32.019
when I proceed to the Roll, Pitch, and Yaw rotations,

625
00:31:32.019 --> 00:31:33.640
and continue,

626
00:31:33.640 --> 00:31:35.840
what happens with the Yaw rotation?

627
00:31:35.840 --> 00:31:38.559
The face rotates around its center again,

628
00:31:38.559 --> 00:31:40.009
around the face

629
00:31:40.009 --> 00:31:41.880
It's rotating around the center of the face

630
00:31:41.880 --> 00:31:45.479
This is the replica,

631
00:31:45.479 --> 00:31:48.239
the exact same result

632
00:31:48.239 --> 00:31:51.139
as the initial Roll rotation

633
00:31:51.139 --> 00:31:52.440
as we've seen here

634
00:31:52.440 --> 00:31:56.200
Essentially, the rotations from two axes

635
00:31:56.200 --> 00:31:59.200
end up being used for rotating the face,

636
00:31:59.200 --> 00:32:02.559
This means that a rotation that should involve three axes

637
00:32:02.559 --> 00:32:06.000
becomes

638
00:32:06.000 --> 00:32:10.520
unnaturally constrained to two axes,

639
00:32:10.520 --> 00:32:13.239
resulting in an inability to properly represent the rotation

640
00:32:13.239 --> 00:32:16.960
This phenomenon is called gimbal lock

641
00:32:16.960 --> 00:32:19.660
To demonstrate this,

642
00:32:19.660 --> 00:32:21.440
let’s look at a simple example

643
00:32:21.440 --> 00:32:23.559
In this demonstration, as the object rotates freely,

644
00:32:23.559 --> 00:32:26.840
there’s a specific condition

645
00:32:26.840 --> 00:32:31.280
Here, despite using different keys for control,

646
00:32:31.280 --> 00:32:32.960
the same rotation is applied

647
00:32:32.960 --> 00:32:35.400
You can see that both the Yaw and Roll rotations
are continuously changing,

648
00:32:35.400 --> 00:32:38.640
but in the end, they only affect the face's rotation

649
00:32:38.640 --> 00:32:40.080
Even when pressing

650
00:32:40.080 --> 00:32:42.919
separate keys for Yaw or Roll,

651
00:32:42.919 --> 00:32:44.960
both seem to control the same motion

652
00:32:44.960 --> 00:32:48.200
As a result,

653
00:32:48.200 --> 00:32:49.650
the freedom to express rotation

654
00:32:49.650 --> 00:32:51.679
becomes severely limited in this situation

655
00:32:51.679 --> 00:32:54.760
This phenomenon can occur occasionally,

656
00:32:54.760 --> 00:32:57.679
but detecting the exact moment it happens

657
00:32:57.679 --> 00:32:59.000
is very challenging

658
00:32:59.000 --> 00:33:04.559
To prevent this issue,

659
00:33:04.559 --> 00:33:07.400
developers need to write defensive code,

660
00:33:07.400 --> 00:33:11.080
making it difficult

661
00:33:11.080 --> 00:33:13.039
to safely implement rotation

662
00:33:13.039 --> 00:33:16.339
Euler angles

663
00:33:16.339 --> 00:33:17.559
work fine when the object is static,

664
00:33:17.559 --> 00:33:19.559
or when the rotation doesn’t change

665
00:33:19.559 --> 00:33:22.919
It’s effective for representing fixed rotations

666
00:33:22.919 --> 00:33:27.080
However, when the rotation
dynamically changes and shifts freely,

667
00:33:27.080 --> 00:33:30.559
unexpected situations like gimbal lock can occur,

668
00:33:30.559 --> 00:33:32.640
making it very tricky to handle

669
00:33:32.640 --> 00:33:37.159
Thus, while setting fixed angles

670
00:33:37.159 --> 00:33:38.274
for static objects

671
00:33:38.274 --> 00:33:41.080
in engines is fine, like this

672
00:33:41.080 --> 00:33:44.400
since we’re assigning values without movement

673
00:33:44.400 --> 00:33:46.333
Euler angles pose critical problems

674
00:33:46.333 --> 00:33:49.320
when it comes to objects

675
00:33:49.320 --> 00:33:51.559
that need to rotate freely

676
00:33:51.559 --> 00:33:53.320
That's one

677
00:33:53.320 --> 00:33:57.559
The second issue is interpolation of rotations

678
00:33:57.559 --> 00:34:01.400
When we rotate an object using Euler angles along one axis,

679
00:34:01.400 --> 00:34:04.960
such as a Yaw rotation by 15 degrees

680
00:34:04.960 --> 00:34:08.159
followed by another Yaw rotation by 30 degrees,

681
00:34:08.159 --> 00:34:13.679
the result is intuitively a Yaw rotation of 45 degrees

682
00:34:13.679 --> 00:34:16.719
In terms of rotation, this means

683
00:34:16.719 --> 00:34:20.280
a 15-degree rotation plus a 30-degree rotation
equals a 45-degree rotation

684
00:34:20.280 --> 00:34:24.119
If we represent this as R_2 and R_1,

685
00:34:24.119 --> 00:34:26.569
combining a 15-degree and 30-degree rotation

686
00:34:26.569 --> 00:34:29.239
should logically result

687
00:34:29.239 --> 00:34:30.919
in a 45-degree rotation

688
00:34:30.919 --> 00:34:33.419
Mathematically,

689
00:34:33.419 --> 00:34:34.599
this holds true because,

690
00:34:34.599 --> 00:34:40.159
for Euler angles, only one axis rotates

691
00:34:40.159 --> 00:34:42.119
while the others remain stationary

692
00:34:42.119 --> 00:34:44.719
Since the other axes contribute
nothing to the rotation, or 0 degrees,

693
00:34:44.719 --> 00:34:47.479
so that's a Roll rotation

694
00:34:47.479 --> 00:34:50.479
For Pitch rotation,

695
00:34:50.479 --> 00:34:53.520
It's treated as an identity matrix, or I

696
00:34:53.520 --> 00:34:58.159
So the related things here use I

697
00:34:58.159 --> 00:35:00.315
Ultimately, these two matrices

698
00:35:00.315 --> 00:35:02.359
result in the same conclusion

699
00:35:02.359 --> 00:35:04.759
By adding two angles,

700
00:35:04.759 --> 00:35:06.798
we can create a new angle

701
00:35:06.798 --> 00:35:09.880
This demonstrates that interpolation is valid

702
00:35:09.880 --> 00:35:13.679
If the sum of these two angles results in the same outcome,

703
00:35:13.679 --> 00:35:15.840
for example,

704
00:35:15.840 --> 00:35:19.760
if we interpolate halfway,

705
00:35:19.760 --> 00:35:22.039
we only need to include half of each angle

706
00:35:22.039 --> 00:35:27.599
This allows us to determine the angle at any proportion

707
00:35:27.599 --> 00:35:30.840
between the start and end,

708
00:35:30.840 --> 00:35:34.159
making partial rotation possible

709
00:35:34.159 --> 00:35:39.200
However, if more than two rotations

710
00:35:39.200 --> 00:35:42.919
are involved and angles are applied simultaneously,

711
00:35:42.919 --> 00:35:44.919
issues arise at this point

712
00:35:44.919 --> 00:35:48.869
In such cases,

713
00:35:48.869 --> 00:35:50.599
the previously valid equations no longer hold

714
00:35:50.599 --> 00:35:54.599
For example, if you apply
both a roll rotation and a yaw rotation,

715
00:35:54.599 --> 00:36:01.520
the 45-degree rotation represented as R_yaw3×R_roll3

716
00:36:01.520 --> 00:36:06.280
differs from the outcome of applying each rotation

717
00:36:06.280 --> 00:36:07.520
separately, by 30, right?

718
00:36:07.520 --> 00:36:11.579
By 15 degrees or 30 degrees

719
00:36:11.579 --> 00:36:15.119
This is Ru3×Rl3

720
00:36:15.119 --> 00:36:23.559
While this is R_yaw2, R_roll2, R_yaw1, R_roll1

721
00:36:23.559 --> 00:36:25.463
For the angles to align,

722
00:36:25.463 --> 00:36:28.320
rotations must follow the same axis and sequence

723
00:36:28.320 --> 00:36:31.479
However, since roll and yaw

724
00:36:31.479 --> 00:36:33.039
alternate across different axes,

725
00:36:33.039 --> 00:36:37.290
their summed rotations

726
00:36:37.290 --> 00:36:40.390
do not guarantee consistent results

727
00:36:40.390 --> 00:36:41.479
for their rotations

728
00:36:41.479 --> 00:36:45.239
Consequently, the results differ

729
00:36:45.239 --> 00:36:49.679
This inconsistency makes interpolating Euler angles
with more than two rotations

730
00:36:49.679 --> 00:36:52.200
challenging and impractical

731
00:36:52.200 --> 00:36:57.250
Since these rotations cannot be expressed
as a simple sum of angles,

732
00:36:57.250 --> 00:36:58.239
the process becomes more complex

733
00:36:58.239 --> 00:37:00.939
For instance,

734
00:37:00.939 --> 00:37:02.520
in animations,

735
00:37:02.520 --> 00:37:05.560
let's say that we're going for some

736
00:37:05.560 --> 00:37:07.640
animation work in a timeframe

737
00:37:07.640 --> 00:37:11.240
if you simply define a starting and ending angle

738
00:37:11.240 --> 00:37:13.560
to create a smooth transition

739
00:37:13.560 --> 00:37:15.680
with keyframes in between

740
00:37:15.680 --> 00:37:18.359
this approach fails with Euler angles

741
00:37:18.359 --> 00:37:22.560
Instead, you need to use methods like axis-angle rotation,
where a specific axis is defined,

742
00:37:22.560 --> 00:37:27.599
and rotation occurs around it

743
00:37:27.599 --> 00:37:30.920
to interpolate between the start and end rotations

744
00:37:30.920 --> 00:37:35.538
Common techniques for this
include Rodrigues’ rotation formula or quaternions

745
00:37:35.538 --> 00:37:38.388
However, in this session,

746
00:37:38.388 --> 00:37:42.359
we will focus solely on Euler angles

747
00:37:42.359 --> 00:37:45.719
We explored how Euler angles are used

748
00:37:45.719 --> 00:37:48.920
to define rotations in 3D space and their limitations

749
00:37:48.920 --> 00:37:51.079
This concludes today’s lecture

750
00:37:51.079 --> 00:37:53.000
Thank you for your hard work

751
00:37:53.000 --> 00:37:53.973
Thank you

752
00:37:54.489 --> 00:37:55.679
Design of 3D Space
3D space is divided into left and right-handed coordinate systems,
defined by three axes
Here, we use the right-handed Y-up

753
00:37:55.679 --> 00:37:56.813
3D Transform Setup
The rotation mechanism:
M=T·R·S=
[x_x·s_x y_x·s_y z_x·s_z t_x]

754
00:37:56.813 --> 00:37:57.870
[x_y·s_x y_y·s_y z_y·s_z t_y]
[x_z·s_x y_z·s_y z_z·s_z t_z]
[ 0 0 0 1 ]

755
00:37:57.870 --> 00:37:58.977
3D Camera System
To view the world from the camera's perspective,
the x-axis points to the right

756
00:37:58.977 --> 00:37:59.996
Rotating the y-axis 180 degrees aligns the screen's axes with a Cartesian system
Flipp x and y axes to create the view coordinate

757
00:37:59.996 --> 00:38:01.043
View Matrix Construction
V=
[-x_x -x_y -x_x x·y]
[y_x y_y y_z -y·t]
[-z_x -z_y -z_z z·t]
[ 0 0 0 1 ]

758
00:38:01.043 --> 00:38:02.743
Euler angle representation
Rotation matrix: local vectors in columns
Rotations are expressed with three angles

759
00:38:02.743 --> 00:38:04.443
Unity Engine uses x, y, z in a Vector3 structure,
while Unreal Engine uses Yaw, Roll, Pitch in a Rotator structure

760
00:38:04.443 --> 00:38:06.143
Yaw, Roll, Pitch Rotation
Yaw: Rotation around the up vector, resembling a propeller

761
00:38:06.143 --> 00:38:07.834
Roll: Rotation about the forward axis
Pitch: Rotation about the side axis, like nodding

762
00:38:07.834 --> 00:38:08.786
Euler Angle Application Order
Roll→Pitch→Yaw
Rotation Matrix for Euler Angles
R=
[cos_α·cos_γ+sin_α·sin_β·sin_γ·cos_β·sin_γ

763
00:38:08.786 --> 00:38:09.758
-cos_a·sin_γ+sin_α·sin_β·cos_γ·cos_β·cos_γ sin_α·cos_β-sin_β]

764
00:38:09.758 --> 00:38:10.691
[-sin_α·cos_γ+cos_α·sin_β·sin_γ sin_α·sin_γ+cos_α·sin_β·cos_γ cos_α·cos_β]
Local Vectors with Euler Angles

765
00:38:10.691 --> 00:38:11.669
x_local=(cos_α·cos_γ+sin_α·sin_β·sin_γ, cos_β·sin_γ, -sin_α·cos_γ+cos_α·sin_β·sin_γ)

766
00:38:11.669 --> 00:38:12.597
y_local=(-cos_a·sin_γ+sin_α·sin_β·cos_γ, cos_β·cos_γ, -sin_α·sin_γ+cos_α·sin_β·cos_γ)

767
00:38:12.597 --> 00:38:13.465
z_local=(sin_α·cos_β, -sin_β, cos_α·cos_β)
Issues with Euler Angles

768
00:38:13.465 --> 00:38:14.325
Gimbal Lock: Two axes align under certain conditions
Interpolation of Rotation: Euler angles make interpolation difficult with many axes