WEBVTT

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Game Basics
Mathematics for 3D Games Part 3: Perspective Projection Transformation and Depth Values

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Hello, everyone?

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This is Lee Deuk-woo from Game Mathematics

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In this session,

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we’ll discuss perspective projection

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We’ll explain what perspective projection transformation is

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and how to implement it

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We’ll also explore the concepts

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of clip space,

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homogeneous coordinates,

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and vanishing points, which are used in the perspective projection process

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Next, we’ll look into depth values

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I’ll explain what depth values are

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and why they are necessary

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To handle depth values, we need to create

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a specific geometric shape

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which is called a frustum

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For the frustum

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we will examine how it is constructed

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We’ll also explore

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how to design the newly defined space

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using the frustum

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This is what we will explore today

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Finally, we’ll discuss

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the final perspective projection matrix

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that incorporates the frustum and calculates depth values

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Perspective Projection Transformation

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First, let’s discuss the perspective projection transformation

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This is referred to in English as

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the Perspective Projection Transformation

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That is what it is called

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It’s related to

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 the perspective techniques often mentioned in art

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You can think of it as a method

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to implement perspective drawing techniques

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This perspective technique in art

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was devised during the Renaissance period

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and has evolved into a technique

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that is now used in nearly all

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three-dimensional drawings

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In the past, to achieve this,

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a single focal point would be set

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and strings were extended

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to project the image onto a surface

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in a highly precise, detailed process

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This was done manually

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Even today, when drawing

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it’s common to use vanishing points as a reference

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and apply perspective techniques using lines

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is common

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Instead of applying these techniques

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step by step manually,

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we can use matrices

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to efficiently represent objects with perspective

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all at once

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This is achieved through

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the perspective projection transformation

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I explained earlier

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The principle behind perspective projection transformation is as follows

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We start with a real-world space where the three axes

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are orthogonal to each other

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The space we perceive with our eyes

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views this orthogonal space

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slightly differently

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In the real world

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objects are arranged orthogonally

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but the world we perceive

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begins from a single point, like a camera

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and expands outward

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in a specified field of view

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So you can think of the spaces

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as being configured differently

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You can understand it like this

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In this expanding space

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defined by the field of view

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we designate a plane

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to project the objects in the space

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onto this plane

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The farther this plane is from the camera

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the larger it becomes

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and the closer it is, the smaller it appears

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We need to specify a single plane

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that changes depending on the distance

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This specified plane is called the projection plane

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The shortest distance from the camera to this plane

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is called the focal length

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With these two concepts in mind,

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we can now project objects onto that plane

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to create a final image

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with perspective

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If we look at this from the side

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it can be visualized as follows

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First, there’s the field of view

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This is predetermined in the camera

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The field of view is defined

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Next, we specify the projection plane onto which objects will be projected

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The shortest distance between the camera and the projection plane

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is called the focal length, correct?

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Let’s denote this as d

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From this basic setup

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we need to determine

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how to implement this

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A commonly used method

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is to define the projected area

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known as NDC

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Typically, the final game screen

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is displayed on a monitor

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Depending on the monitor specifications

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resolutions can range from 1024x768

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to 1920x1080 or even 4K

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There are various types of resolutions, right?

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These different resolutions use varying pixel densities

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to form the screen

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It’s impractical to handle all these resolutions individually

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So, in programming,

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the resolution is handled separately

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while the projected space is designed

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to be as simple as possible

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using a unit scale of 1

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for the projection plane

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This is the fundamental principle of NDC

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Similarly, in textures

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no matter how large the texture image is

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we use UV coordinates from (0, 0)

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to (1, 1)

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Regardless of the screen resolution

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we define the projection plane

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as a 1 unit-sized plane

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This is the NDC coordinate system

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However, when specifying a size of 1,

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it doesn’t mean the entire size is 1

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As shown here

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only half the size is defined as 1

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This makes calculations easier

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If the half-size of the projection plane

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is defined as 1 within the field of view

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we’ll be working with only half the field of view

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For this half field of view, we can calculate

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the focal length and form a right triangle

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with one side equal to 1

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When viewed from the front, the resulting plane

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will take this form

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Since half the size is defined as 1

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the plane extends 1 unit up and down

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and 1 unit left and right from (0, 0)

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This creates a plane with a total size of 2 units

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The coordinates of the points

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projected onto this plane are referred to

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as NDC coordinates

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The mathematics behind NDC

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as explained earlier,

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starts with the half field of view, the focal length

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and a right triangle

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with one side equal to 1

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The focal length depends on the field of view

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so how do we calculate it

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What happens as the field of view decreases?

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As the field of view decreases

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the size corresponding to 1 moves farther back

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This is because the size of 1 remains constant

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Only at this distance

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can the size of 1 be maintained as the field of view narrows

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This demonstrates an inverse proportionality

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The focal length is related to tan θ/2

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the tangent of half the field of view

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Since it’s the height divided by the base

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it equals 1/focal length

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To calculate d,

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take the reciprocal of tan θ/2

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where θ is half the field of view

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This is how you can do it

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Let’s visualize how

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a point in view space

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is projected onto the projection plane

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using this triangle

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There are various points in this space

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Some points are in the foreground

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while others are in the background

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All points are projected

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toward the camera point onto the projection plane

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By drawing lines toward the camera

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we find the NDC coordinates

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where they intersect the projection plane

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This allows us to calculate the points in the NDC coordinate system

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Using these points, we construct triangles

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to render the scene

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This enables us to

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create a 3D space with perspective

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Our goal is to create

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a matrix that performs

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this transformation

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So, how do we create this matrix?

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We use the similarity ratio of two triangles

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Here,

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although we don’t know the angle

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we aim to obtain the projected coordinates

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We already know the focal length

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We know the focal length

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and in the arbitrary view space

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if a point exists in the view space

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where the three axes are orthogonal

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we can determine

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the point’s distance and height

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relative to the camera

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This information is derived from the coordinates in view space

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If we know the focal length,

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we can establish a proportional relationship

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between the focal length

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the distance from the camera

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the y-value, and the NDC value we want to calculate

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Previously, we mentioned that the view space

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always faces the back of the camera, right?

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Thus, all object coordinates in the view space

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in front of the camera

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always have negative values

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When setting up the proportional relationship

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the z-value, which is negative

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needs to be multiplied by -1

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to convert it into

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a positive value

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Once converted to positive

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the two proportional equations

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allow us to calculate the y-value

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for the Pndc coordinate

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For the x-axis,

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since it shares the same field of view

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as the y-axis,

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the principle is the same

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In essence, there’s no difference between the y and x axis

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So, applying the same calculations

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to the x-axis,

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we can obtain

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the projected coordinate values in this form

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This results in an NDC coordinate system

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where the size

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is a square extending 2 units

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horizontally and vertically

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So how do we project this onto

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an actual monitor?

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We expand it proportionally to the monitor's resolution

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and calculate accordingly

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For instance, if the monitor has

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a resolution of 800x600

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you scale the NDC area

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by 400 horizontally and 300 vertically

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extending it like this will help make it

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to fit it perfectly onto the screen

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However, there’s an issue

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when expanding the NDC space

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to fit the screen resolution

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As mentioned earlier,

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the NDC space is a uniform square

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But most monitor resolutions

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have different horizontal and vertical aspect ratios

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This creates a problem

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For example, let’s say we have

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a perfect circle in NDC space

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as shown here

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If we expand it

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 horizontally and vertically based on the screen resolution

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the differing aspect ratio

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causes the circle to distort into an ellipse

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as shown on the screen

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This can be confusing

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You might wonder why your perfect sphere

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that you made

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looks distorted

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To prevent this issue

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you need to address

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the differing aspect ratios

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The solution is

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to apply the aspect ratio

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 to the object in NDC space

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distorting it first, and then scaling it up

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By doing this, the distortion cancels out

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during scaling, and you end up

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with a perfect sphere

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What is the reference point for the aspect ratio?

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Does it use the horizontal

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or vertical dimension as a reference?

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It can vary depending on the choice

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In most cases, the vertical dimension is used as the reference

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So here, we’re using

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the vertical as the reference

285
00:13:19.765 --> 00:13:21.403
and the a value will be greater than 1

286
00:13:21.403 --> 00:13:22.883
This is because

287
00:13:22.883 --> 00:13:26.306
the horizontal length is usually greater than the vertical length

288
00:13:26.306 --> 00:13:29.226
To apply this,

289
00:13:29.226 --> 00:13:33.580
we account for the reciprocal of the aspect ratio in the calculations

290
00:13:33.580 --> 00:13:35.448
We don’t adjust

291
00:13:35.448 --> 00:13:37.195
the y-component specifically

292
00:13:37.195 --> 00:13:39.644
because we’re only distorting the x-component

293
00:13:39.644 --> 00:13:42.876
Thus, we apply the reciprocal of the aspect ratio

294
00:13:42.876 --> 00:13:44.768
only to the x-component

295
00:13:44.768 --> 00:13:50.043
The final NDC coordinates

296
00:13:50.043 --> 00:13:53.251
will have the same focal length (d) and depth values

297
00:13:53.251 --> 00:13:55.203
Everything remains unchanged,

298
00:13:55.203 --> 00:13:58.086
except that the aspect ratio adjustment

299
00:13:58.086 --> 00:14:00.063
is applied inversely to the x-component

300
00:14:00.063 --> 00:14:02.308
Using this calculation method

301
00:14:02.308 --> 00:14:05.304
we can determine the NDC coordinates

302
00:14:05.304 --> 00:14:07.652
The matrix reflecting these adjustments

303
00:14:07.652 --> 00:14:10.508
can be calculated as follows

304
00:14:10.508 --> 00:14:14.637
When we obtain the x, y, z values in the view coordinate system

305
00:14:14.637 --> 00:14:18.071
the matrix for deriving the Pndc values

306
00:14:18.071 --> 00:14:20.517
can be structured like this

307
00:14:20.517 --> 00:14:24.118
However, there is an issue when examining this matrix closely

308
00:14:24.118 --> 00:14:26.431
The problem lies in the fact

309
00:14:26.431 --> 00:14:29.197
that this matrix

310
00:14:29.197 --> 00:14:31.842
uses the vz value

311
00:14:31.842 --> 00:14:34.054
It relies on the z value

312
00:14:34.054 --> 00:14:36.863
of individual points

313
00:14:36.863 --> 00:14:38.843
Why is this a problem?

314
00:14:38.843 --> 00:14:41.281
In general, when using matrices

315
00:14:41.281 --> 00:14:45.282
we don’t create a unique matrix for each point

316
00:14:45.282 --> 00:14:47.068
Instead, we create a fixed matrix

317
00:14:47.068 --> 00:14:49.758
that can be applied to tens, hundreds, or millions of points at once

318
00:14:49.758 --> 00:14:53.188
Matrices are designed to be uniform

319
00:14:53.188 --> 00:14:56.094
and applied simultaneously to many points

320
00:14:56.094 --> 00:14:59.569
If the z-coordinate of a point

321
00:14:59.569 --> 00:15:01.832
is directly integrated into the matrix design

322
00:15:01.832 --> 00:15:03.485
it means we would need

323
00:15:03.485 --> 00:15:05.372
to create a new matrix for each point

324
00:15:05.372 --> 00:15:07.546
being transformed

325
00:15:07.546 --> 00:15:11.353
This undermines the purpose of using a fixed, reusable matrix

326
00:15:11.353 --> 00:15:13.431
For instance, if there are 100,000 points

327
00:15:13.431 --> 00:15:15.965
100,000 matrices would need to be generated

328
00:15:15.965 --> 00:15:18.683
This approach is clearly impractical

329
00:15:18.683 --> 00:15:22.023
Fortunately, there are solutions to this issue

330
00:15:22.023 --> 00:15:25.094
The solution lies in what we call clip space

331
00:15:25.094 --> 00:15:28.175
which I’ll now explain

332
00:15:28.175 --> 00:15:30.926
Clip space is used to create a general-purpose matrix

333
00:15:30.926 --> 00:15:34.756
that works independently of specific points

334
00:15:34.756 --> 00:15:36.686
It serves as an intermediate space

335
00:15:36.686 --> 00:15:39.191
This is what we call a Clip space

336
00:15:39.191 --> 00:15:41.713
Here, we’ve summarized

337
00:15:41.713 --> 00:15:45.735
the concept of clip space as follows

338
00:15:45.735 --> 00:15:49.833
It’s essentially

339
00:15:49.833 --> 00:15:54.595
a 3D space transformed from the view space

340
00:15:54.595 --> 00:15:57.809
It’s an intermediate space

341
00:15:57.809 --> 00:16:00.419
between the view space and the final 3D NDC space

342
00:16:00.419 --> 00:16:02.440
Think of it as a transitional step

343
00:16:02.440 --> 00:16:04.198
This space retains all three dimensions

344
00:16:04.198 --> 00:16:06.244
as part of the intermediate stage

345
00:16:06.244 --> 00:16:08.224
Unlike NDC space,

346
00:16:08.224 --> 00:16:11.700
which only requires 2D since it’s a plane

347
00:16:11.700 --> 00:16:14.106
clip space is structured in 3D

348
00:16:14.106 --> 00:16:16.576
But why is it structured as 3D?

349
00:16:16.576 --> 00:16:20.233
In this equation,

350
00:16:20.233 --> 00:16:24.056
all coordinates are divided by -vz

351
00:16:24.056 --> 00:16:29.746
So, before dividing, the idea is to use this value

352
00:16:29.746 --> 00:16:31.437
to construct the coordinates

353
00:16:31.437 --> 00:16:35.433
After constructing the coordinates, dividing by -vz

354
00:16:35.433 --> 00:16:37.866
yields the basic NDC coordinates

355
00:16:37.866 --> 00:16:42.014
we intended to obtain

356
00:16:42.014 --> 00:16:44.723
beforehand

357
00:16:44.723 --> 00:16:45.999
This means applying the matrix

358
00:16:45.999 --> 00:16:48.612
doesn’t directly yield NDC coordinates

359
00:16:48.612 --> 00:16:50.571
but instead produces the clip space coordinates

360
00:16:50.571 --> 00:16:53.202
denoted as Pclip

361
00:16:53.202 --> 00:16:57.227
which becomes the primary goal of the matrix

362
00:16:57.227 --> 00:16:59.253
So, how can we set this up?

363
00:16:59.253 --> 00:17:02.626
To get -vz

364
00:17:02.626 --> 00:17:05.920
simply place -1 in the last element of the matrix

365
00:17:05.920 --> 00:17:08.445
This allows us to design such a matrix

366
00:17:08.445 --> 00:17:11.096
As you can see, the P-matrix

367
00:17:11.096 --> 00:17:14.453
used for perspective projection

368
00:17:14.453 --> 00:17:16.392
doesn’t contain point-specific information

369
00:17:16.392 --> 00:17:18.361
Instead, it only depends on the camera’s focal length

370
00:17:18.361 --> 00:17:21.489
field of view,

371
00:17:21.489 --> 00:17:25.259
and horizontal-to-vertical aspect ratio

372
00:17:25.259 --> 00:17:27.127
With these parameters

373
00:17:27.127 --> 00:17:28.399
we can construct the perspective projection matrix

374
00:17:28.399 --> 00:17:31.652
Thus, even if there are 100,000 points

375
00:17:31.652 --> 00:17:34.325
a single precomputed matrix

376
00:17:34.325 --> 00:17:37.771
based on the camera settings,

377
00:17:37.771 --> 00:17:41.151
can be applied universally to all points

378
00:17:41.151 --> 00:17:44.356
Once the coordinates are generated

379
00:17:44.356 --> 00:17:47.510
manually dividing by the last element

380
00:17:47.510 --> 00:17:49.492
yields the desired

381
00:17:49.492 --> 00:17:50.838
NDC coordinates

382
00:17:50.838 --> 00:17:52.351
This approach satisfies

383
00:17:52.351 --> 00:17:55.396
both objectives

384
00:17:55.396 --> 00:17:58.087
The concept of clip space

385
00:17:58.087 --> 00:18:01.795
is closely related to what is called

386
00:18:01.795 --> 00:18:05.773
homogeneous coordinates

387
00:18:05.773 --> 00:18:09.160
In 3D computer graphics

388
00:18:09.160 --> 00:18:11.494
you can think of it as involving two types of spaces

389
00:18:11.494 --> 00:18:14.002
First, there’s the world space

390
00:18:14.002 --> 00:18:16.056
which is used for placement and overall structure

391
00:18:16.056 --> 00:18:20.552
where x, y, z are orthogonal

392
00:18:20.552 --> 00:18:23.122
Next, the world space

393
00:18:23.122 --> 00:18:24.716
is transformed into view space

394
00:18:24.716 --> 00:18:27.845
where it is adjusted to match the camera’s perspective

395
00:18:27.845 --> 00:18:30.446
This transformation mimics the rules

396
00:18:30.446 --> 00:18:32.614
by which humans perceive the world

397
00:18:32.614 --> 00:18:34.460
through their eyes

398
00:18:34.460 --> 00:18:35.580
It follows those rules

399
00:18:35.580 --> 00:18:38.192
in the same procedure

400
00:18:38.192 --> 00:18:41.343
This process occurs

401
00:18:41.343 --> 00:18:43.886
in what we previously described as clip space

402
00:18:43.886 --> 00:18:47.282
Mathematically, this is referred to as projective space

403
00:18:47.282 --> 00:18:50.388
It’s called "projective space"

404
00:18:50.388 --> 00:18:52.890
Let me explain the differences

405
00:18:52.890 --> 00:18:54.654
between these two spaces

406
00:18:54.654 --> 00:18:57.989
In Euclidean space

407
00:18:57.989 --> 00:18:59.894
parallel lines never intersect

408
00:18:59.894 --> 00:19:02.454
This is something we are all familiar with

409
00:19:02.454 --> 00:19:10.519
However, let’s consider an assumption

410
00:19:10.519 --> 00:19:13.343
regarding these parallel lines

411
00:19:13.343 --> 00:19:16.312
This pertains to results within a 2D space

412
00:19:16.312 --> 00:19:19.408
Let’s assume that these 2D results

413
00:19:19.408 --> 00:19:22.690
are part of a system

414
00:19:22.690 --> 00:19:24.956
constructed within a 3D space

415
00:19:24.956 --> 00:19:26.376
Let's make such an assumption

416
00:19:26.376 --> 00:19:29.091
The current coordinates

417
00:19:29.091 --> 00:19:32.836
we see are just a partial projection

418
00:19:32.836 --> 00:19:35.529
of a 3D space

419
00:19:35.529 --> 00:19:40.167
Unseen elements from the 3D space

420
00:19:40.167 --> 00:19:42.479
influence this projection

421
00:19:42.479 --> 00:19:44.585
So, how do they affect it?

422
00:19:44.585 --> 00:19:46.869
The value divided by the last dimension

423
00:19:46.869 --> 00:19:50.400
in this case represented

424
00:19:50.400 --> 00:19:54.221
by ax + b

425
00:19:54.221 --> 00:19:57.204
is the result of dividing

426
00:19:57.204 --> 00:20:00.442
by the new dimension, z

427
00:20:00.442 --> 00:20:01.495
Let’s analyze it this way

428
00:20:01.495 --> 00:20:08.411
So, when we divide the 3D space

429
00:20:08.411 --> 00:20:11.210
represented by x', y' z',

430
00:20:11.210 --> 00:20:15.167
by z', it results in x, y, 1

431
00:20:15.167 --> 00:20:18.380
This shows that

432
00:20:18.380 --> 00:20:21.434
the plane is constructed using x and y

433
00:20:21.434 --> 00:20:23.887
In this mechanism

434
00:20:23.887 --> 00:20:25.907
the essence of x is

435
00:20:25.907 --> 00:20:30.494
that it’s derived by dividing x' by z'

436
00:20:30.494 --> 00:20:32.361
Similarly, the essence of y

437
00:20:32.361 --> 00:20:35.557
is that it’s derived by dividing y' by z'

438
00:20:35.557 --> 00:20:39.814
If we substitute this into the linear equation

439
00:20:39.814 --> 00:20:42.741
y = ax + b discussed earlier,

440
00:20:42.741 --> 00:20:45.414
we can express it in terms

441
00:20:45.414 --> 00:20:47.533
of the underlying spatial rules

442
00:20:47.533 --> 00:20:50.954
Multiplying both sides by z'

443
00:20:50.954 --> 00:20:54.266
yields the equation y' = ax' + bz'

444
00:20:54.266 --> 00:20:56.780
This is the equation that is created

445
00:20:56.780 --> 00:20:58.334
Looking at these equations,

446
00:20:58.334 --> 00:21:01.243
since x', y' and z' all share

447
00:21:01.243 --> 00:21:03.642
the same linear degree

448
00:21:03.642 --> 00:21:06.684
this is called a homogeneous equation

449
00:21:06.684 --> 00:21:09.611
A space constructed based on this theory

450
00:21:09.611 --> 00:21:12.194
is referred to as homogeneous space

451
00:21:12.194 --> 00:21:15.374
This is equivalent to the previously mentioned

452
00:21:15.374 --> 00:21:19.147
projective space

453
00:21:19.147 --> 00:21:23.495
Thus, dividing by the last element

454
00:21:23.495 --> 00:21:25.350
is analogous to the process

455
00:21:25.350 --> 00:21:27.383
of converting from clip space

456
00:21:27.383 --> 00:21:29.615
to NDC space

457
00:21:29.615 --> 00:21:31.726
In other words, this process is the same

458
00:21:31.726 --> 00:21:35.913
as the conversion from clip space to NDC space

459
00:21:35.913 --> 00:21:39.150
Therefore, clip space can be thought of as

460
00:21:39.150 --> 00:21:41.728
a space created relative to NDC space,

461
00:21:41.728 --> 00:21:43.554
where the concept is

462
00:21:43.554 --> 00:21:47.900
that it exists in one additional dimension

463
00:21:47.900 --> 00:21:49.783
beyond what NDC space can perceive

464
00:21:49.783 --> 00:21:52.598
You can understand it as a concept like this

465
00:21:52.598 --> 00:21:55.833
Let’s examine how

466
00:21:55.833 --> 00:21:57.553
clip space operates

467
00:21:57.553 --> 00:22:01.957
One key example is the principle of vanishing points, which we discussed earlier

468
00:22:01.957 --> 00:22:03.701
In Euclidean space,

469
00:22:03.701 --> 00:22:07.442
parallel lines never intersect, right?

470
00:22:07.442 --> 00:22:10.936
But let’s consider this in projective space

471
00:22:10.936 --> 00:22:13.274
In projective space,

472
00:22:13.274 --> 00:22:17.934
if we think about it this way,

473
00:22:17.934 --> 00:22:21.981
as x and y increase

474
00:22:21.981 --> 00:22:24.068
z consistently influences

475
00:22:24.068 --> 00:22:26.709
the space

476
00:22:26.709 --> 00:22:28.669
If we change the z value

477
00:22:28.669 --> 00:22:31.869
along a line in projective space

478
00:22:31.869 --> 00:22:33.803
in Euclidean space

479
00:22:33.803 --> 00:22:36.338
where x, y, and z are orthogonal

480
00:22:36.338 --> 00:22:38.968
z does not influence x and y at all

481
00:22:38.968 --> 00:22:41.969
Thus, z simply moves independently,

482
00:22:41.969 --> 00:22:43.362
unaffected by x and y

483
00:22:43.362 --> 00:22:46.555
However, in projective space

484
00:22:46.555 --> 00:22:49.859
z affects x and y

485
00:22:49.859 --> 00:22:52.094
This interaction means that

486
00:22:52.094 --> 00:22:53.733
as z increases

487
00:22:53.733 --> 00:22:56.070
 and y values also increase

488
00:22:56.070 --> 00:22:59.102
They have such characteristics

489
00:22:59.102 --> 00:23:00.833
What happens if z decreases

490
00:23:00.833 --> 00:23:02.352
and approaches 0?

491
00:23:02.352 --> 00:23:04.407
The x and y values will also ultimately

492
00:23:04.407 --> 00:23:05.809
appear as 0

493
00:23:05.809 --> 00:23:08.287
For lines with different slopes

494
00:23:08.287 --> 00:23:10.680
when z = 0

495
00:23:10.680 --> 00:23:12.410
no matter what values they have

496
00:23:12.410 --> 00:23:13.863
they will converge

497
00:23:13.863 --> 00:23:15.726
to 0

498
00:23:15.726 --> 00:23:22.321
So, values in projective space

499
00:23:22.321 --> 00:23:24.891
with the same slope

500
00:23:24.891 --> 00:23:26.688
can be thought of as belonging

501
00:23:26.688 --> 00:23:29.497
to the same category

502
00:23:29.497 --> 00:23:32.768
The points along this line

503
00:23:32.768 --> 00:23:35.586
are considered of the same type

504
00:23:35.586 --> 00:23:37.147
However, even if

505
00:23:37.147 --> 00:23:39.309
 the points have

506
00:23:39.309 --> 00:23:41.350
different slopes

507
00:23:41.350 --> 00:23:43.155
they all ultimately converge at the origin

508
00:23:43.155 --> 00:23:44.842
Similarly, lines extending in the opposite direction

509
00:23:44.842 --> 00:23:47.642
converge at a point

510
00:23:47.642 --> 00:23:49.677
at infinity

511
00:23:49.677 --> 00:23:50.830
This is the fundamental principle

512
00:23:50.830 --> 00:23:53.857
of projective space

513
00:23:53.857 --> 00:23:57.578
This may seem like a difficult concept

514
00:23:57.578 --> 00:23:59.980
but to summarize

515
00:23:59.980 --> 00:24:01.415
the last element always

516
00:24:01.415 --> 00:24:05.354
influences x and y

517
00:24:05.354 --> 00:24:08.080
There are sets of points

518
00:24:08.080 --> 00:24:09.873
sharing the same ratio,

519
00:24:09.873 --> 00:24:12.529
and these points form a line

520
00:24:12.529 --> 00:24:14.580
The points forming the line

521
00:24:14.580 --> 00:24:17.467
are described as having the same properties

522
00:24:17.467 --> 00:24:19.980
There are infinitely many points

523
00:24:19.980 --> 00:24:21.592
with the same properties

524
00:24:21.592 --> 00:24:23.568
Among them, we use only

525
00:24:23.568 --> 00:24:25.586
one specific point

526
00:24:25.586 --> 00:24:27.016
Specifically, the point

527
00:24:27.016 --> 00:24:30.181
where z = 1

528
00:24:30.181 --> 00:24:33.288
This is because it represents

529
00:24:33.288 --> 00:24:38.272
the actual displayable area on the monitor

530
00:24:38.272 --> 00:24:40.430
Thus, the points in the same category

531
00:24:40.430 --> 00:24:42.863
converge here

532
00:24:42.863 --> 00:24:45.344
By dividing by the last element,

533
00:24:45.344 --> 00:24:47.870
the z-value becomes 1

534
00:24:47.870 --> 00:24:51.311
allowing us to extract just one point

535
00:24:51.311 --> 00:24:54.470
from the many

536
00:24:54.470 --> 00:24:56.917
sharing the same properties

537
00:24:56.917 --> 00:24:59.108
This rule ensures that

538
00:24:59.108 --> 00:25:02.751
when we render points on the screen

539
00:25:02.751 --> 00:25:04.842
what happens if

540
00:25:04.842 --> 00:25:06.696
the vz value is set to infinity?

541
00:25:06.696 --> 00:25:09.089
If vz is infinite,

542
00:25:09.089 --> 00:25:12.143
it is technically -vz

543
00:25:12.143 --> 00:25:14.411
because it is the opposite direction of the camera

544
00:25:14.411 --> 00:25:16.833
I mean it aligns with the camera’s viewing direction

545
00:25:16.833 --> 00:25:18.153
If we set it to infinity

546
00:25:18.153 --> 00:25:20.246
along the camera’s direction

547
00:25:20.246 --> 00:25:22.361
the value approaches 0

548
00:25:22.361 --> 00:25:23.756
as it increases toward infinity

549
00:25:23.756 --> 00:25:26.226
Thus, points far away from the camera converge

550
00:25:26.226 --> 00:25:30.253
near the center of the NDC space at 0

551
00:25:30.253 --> 00:25:32.736
All objects located infinitely

552
00:25:32.736 --> 00:25:35.315
far from the camera

553
00:25:35.315 --> 00:25:37.827
converge near 0 in the NDC space

554
00:25:37.827 --> 00:25:39.397
We can understand it like this

555
00:25:39.397 --> 00:25:42.266
Consequently, these points

556
00:25:42.266 --> 00:25:44.854
converge to the NDC

557
00:25:44.854 --> 00:25:48.099
coordinate (0,0)

558
00:25:48.099 --> 00:25:50.853
This is the same principle

559
00:25:50.853 --> 00:25:53.166
as the vanishing point

560
00:25:53.166 --> 00:25:55.802
often discussed in art

561
00:25:55.802 --> 00:26:01.393
Taking this vanishing point principle into account,

562
00:26:01.393 --> 00:26:03.588
let’s design the system accordingly

563
00:26:03.588 --> 00:26:05.483
The last dimension

564
00:26:05.483 --> 00:26:07.508
doesn’t hold much significance in this context

565
00:26:07.508 --> 00:26:10.790
You only need to replace the z-value with -z

566
00:26:10.790 --> 00:26:15.664
A 4x4 matrix must be used

567
00:26:15.664 --> 00:26:17.974
to integrate it

568
00:26:17.974 --> 00:26:19.886
into the rendering pipeline

569
00:26:19.886 --> 00:26:22.593
This allows us to construct the perspective projection matrix

570
00:26:22.593 --> 00:26:25.175
 as follows

571
00:26:25.175 --> 00:26:27.989
This concludes our discussion on the matrices

572
00:26:27.989 --> 00:26:30.894
for perspective projection transformation

573
00:26:30.894 --> 00:26:35.222
Depth Values

574
00:26:35.222 --> 00:26:38.920
First, let’s discuss depth values

575
00:26:38.920 --> 00:26:40.958
Why are depth values necessary?

576
00:26:40.958 --> 00:26:42.626
Depth values represent

577
00:26:42.626 --> 00:26:46.173
the numerical data

578
00:26:46.173 --> 00:26:48.740
that indicate how far an object is

579
00:26:48.740 --> 00:26:51.209
from the camera

580
00:26:51.209 --> 00:26:53.077
If we were to render multiple objects

581
00:26:53.077 --> 00:26:54.710
in 3D space

582
00:26:54.710 --> 00:26:57.664
without depth values

583
00:26:57.664 --> 00:26:59.976
as shown in the image

584
00:26:59.976 --> 00:27:01.510
we wouldn’t be able to determine

585
00:27:01.510 --> 00:27:02.945
which objects are in front

586
00:27:02.945 --> 00:27:04.056
and which are behind

587
00:27:04.056 --> 00:27:07.302
The large character face here

588
00:27:07.302 --> 00:27:09.972
is actually positioned in the foreground

589
00:27:09.972 --> 00:27:13.255
but because it was drawn first

590
00:27:13.255 --> 00:27:15.939
 it gets covered by faces drawn afterward

591
00:27:15.939 --> 00:27:17.753
As a result, it appears to be

592
00:27:17.753 --> 00:27:20.291
in the background when viewed

593
00:27:20.291 --> 00:27:22.308
So, how do we solve this issue?

594
00:27:22.308 --> 00:27:23.334
What we have to do is

595
00:27:23.334 --> 00:27:26.000
we need to calculate depth values

596
00:27:26.000 --> 00:27:28.893
and use them

597
00:27:28.893 --> 00:27:30.681
to ensure that objects in the foreground

598
00:27:30.681 --> 00:27:33.607
aren’t overwritten by those in the background

599
00:27:33.607 --> 00:27:36.057
This must be calculated and applied

600
00:27:36.057 --> 00:27:38.143
at the pixel level

601
00:27:38.143 --> 00:27:40.618
This is referred to as a depth buffer

602
00:27:40.618 --> 00:27:43.604
We’ll explore how to implement a depth buffer

603
00:27:43.604 --> 00:27:47.392
in more detail next time

604
00:27:47.392 --> 00:27:50.316
but today, we’ll focus on

605
00:27:50.316 --> 00:27:51.995
how to calculate the depth values

606
00:27:51.995 --> 00:27:53.486
used in it

607
00:27:53.486 --> 00:27:55.622
That is what we will discuss today

608
00:27:57.910 --> 00:28:00.839
To address depth values,

609
00:28:00.839 --> 00:28:03.337
we ultimately need to expand

610
00:28:03.337 --> 00:28:06.117
the existing 2D NDC space

611
00:28:06.117 --> 00:28:08.274
by adding

612
00:28:08.274 --> 00:28:09.319
an additional dimension

613
00:28:09.319 --> 00:28:11.691
We extend the NDC space

614
00:28:11.691 --> 00:28:13.939
into 3D as follows,

615
00:28:13.939 --> 00:28:15.914
so that the depth values are defined

616
00:28:15.914 --> 00:28:17.911
within this range

617
00:28:17.911 --> 00:28:19.902
We must specify the depth range

618
00:28:19.902 --> 00:28:23.360
from one boundary to another

619
00:28:23.360 --> 00:28:27.207
This is wher

620
00:28:27.207 --> 00:28:28.836
the frustum comes into play

621
00:28:28.836 --> 00:28:32.606
A frustum is formed when

622
00:28:32.606 --> 00:28:34.373
we define a field of view from the camera

623
00:28:34.373 --> 00:28:37.041
It extends outward from the camera

624
00:28:37.041 --> 00:28:39.055
to infinity

625
00:28:39.055 --> 00:28:42.356
But there’s no defined start or end point

626
00:28:42.356 --> 00:28:46.780
The starting point is essentially the camera’s position,

627
00:28:46.780 --> 00:28:48.449
but it can also extend backward

628
00:28:48.449 --> 00:28:50.148
In projective space

629
00:28:50.148 --> 00:28:51.916
if the camera is at the center

630
00:28:51.916 --> 00:28:54.273
it projects forward

631
00:28:54.273 --> 00:28:56.655
but can also extend backward

632
00:28:56.655 --> 00:28:59.129
Effectively, it creates a space

633
00:28:59.129 --> 00:29:00.422
that extends infinitely

634
00:29:00.422 --> 00:29:02.936
in both directions

635
00:29:02.936 --> 00:29:05.932
Instead of calculating depth values

636
00:29:05.932 --> 00:29:08.376
across an infinite range

637
00:29:08.376 --> 00:29:11.720
we define a finite range, similar to NDC space

638
00:29:11.720 --> 00:29:14.212
We specify a fixed range

639
00:29:14.212 --> 00:29:17.395
for rendering

640
00:29:17.395 --> 00:29:20.632
Then, we define one end as the minimum value

641
00:29:20.632 --> 00:29:23.834
and the other as the maximum value

642
00:29:23.834 --> 00:29:26.969
This effectively slices a finite section

643
00:29:26.969 --> 00:29:28.654
out of the infinite projective space

644
00:29:28.654 --> 00:29:32.414
Only the objects within this range are rendered

645
00:29:32.414 --> 00:29:35.147
The resulting 3D shape

646
00:29:35.147 --> 00:29:39.495
is called a frustum

647
00:29:39.495 --> 00:29:43.958
When sliced at two points,

648
00:29:43.958 --> 00:29:47.819
it forms a truncated pyramid-like shape

649
00:29:47.819 --> 00:29:50.507
where the point is cut

650
00:29:50.507 --> 00:29:52.454
as you can see

651
00:29:52.454 --> 00:29:54.910
The first slice,

652
00:29:54.910 --> 00:29:57.994
closer to the camera,

653
00:29:57.994 --> 00:30:00.449
is called the near plane

654
00:30:00.449 --> 00:30:03.828
Since we cannot draw the end of infinity

655
00:30:03.828 --> 00:30:07.796
the farthest slice within the drawable range

656
00:30:07.796 --> 00:30:11.769
is called the far plane

657
00:30:11.769 --> 00:30:15.409
The depth value for the near plane

658
00:30:15.409 --> 00:30:18.387
is typically set to -1

659
00:30:18.387 --> 00:30:20.726
and the depth value for the far plane

660
00:30:20.726 --> 00:30:22.388
is set to 1

661
00:30:22.388 --> 00:30:25.683
Objects within the frustum

662
00:30:25.683 --> 00:30:29.031
will have depth values between -1 and 1

663
00:30:29.031 --> 00:30:32.267
This prevents the issue we saw earlier,

664
00:30:32.267 --> 00:30:36.435
where objects in the foreground

665
00:30:36.435 --> 00:30:37.617
appeared in the background

666
00:30:37.617 --> 00:30:41.198
We can prevent this

667
00:30:41.198 --> 00:30:43.657
Let’s look at how this is applied

668
00:30:43.657 --> 00:30:46.015
If we extend this

669
00:30:46.015 --> 00:30:49.358
to NDC space,

670
00:30:49.358 --> 00:30:51.574
the coordinates will look like this

671
00:30:51.574 --> 00:30:54.672
The values closest to the camera

672
00:30:54.672 --> 00:30:57.598
have a z value of -1

673
00:30:57.598 --> 00:30:58.909
and the farthest values

674
00:30:58.909 --> 00:31:00.970
have a z value of 1

675
00:31:00.970 --> 00:31:03.762
This is how you can think about

676
00:31:03.762 --> 00:31:06.467
the 3D NDC coordinate system

677
00:31:06.467 --> 00:31:08.439
One notable aspect here is that

678
00:31:08.439 --> 00:31:11.709
since x ranges from

679
00:31:11.709 --> 00:31:13.891
-1 to 1

680
00:31:13.891 --> 00:31:16.037
and y ranges from -1 ro 1

681
00:31:16.037 --> 00:31:18.558
you might expect z to follow the same range

682
00:31:18.558 --> 00:31:22.336
That would make the three axes align perfectly

683
00:31:22.336 --> 00:31:24.176
However, from a practical perspective,

684
00:31:24.176 --> 00:31:26.954
when calculating depth values

685
00:31:26.954 --> 00:31:31.691
they are often represented as images

686
00:31:31.691 --> 00:31:35.536
Images, after all, represent color

687
00:31:35.536 --> 00:31:37.921
and color typically

688
00:31:37.921 --> 00:31:39.809
does not exist in minus

689
00:31:39.809 --> 00:31:41.759
it ranges from 0 to 1

690
00:31:41.759 --> 00:31:44.525
0 corresponds to black,

691
00:31:44.525 --> 00:31:46.420
and 1 corresponds to white

692
00:31:46.420 --> 00:31:49.572
To make depth values useful when expressed as colors

693
00:31:49.572 --> 00:31:51.640
between 0 and 1

694
00:31:51.640 --> 00:31:53.519
some systems avoid negative values

695
00:31:53.519 --> 00:31:55.064
nd start from 0

696
00:31:55.064 --> 00:31:57.023
In graphics libraries like DirectX

697
00:31:57.023 --> 00:31:59.654
the depth value

698
00:31:59.654 --> 00:32:02.872
for the near plane

699
00:32:02.872 --> 00:32:04.579
is sometimes set to 0

700
00:32:04.579 --> 00:32:05.934
This is something

701
00:32:05.934 --> 00:32:08.940
you should keep in mind

702
00:32:08.940 --> 00:32:11.262
If we summarize

703
00:32:11.262 --> 00:32:14.291
this coordinate system

704
00:32:14.291 --> 00:32:19.361
in NDC space,

705
00:32:19.361 --> 00:32:22.544
the farther an object is from the camera

706
00:32:22.544 --> 00:32:23.694
what happens?

707
00:32:23.694 --> 00:32:25.199
Its size appears to increase

708
00:32:25.199 --> 00:32:28.323
But as we’ve seen before,

709
00:32:28.323 --> 00:32:29.684
in view space,

710
00:32:29.684 --> 00:32:34.601
which uses a right-handed coordinate system

711
00:32:34.601 --> 00:32:36.833
the x-axis typically points left

712
00:32:36.833 --> 00:32:39.926
and the y-axis points up

713
00:32:39.926 --> 00:32:43.225
this is typical

714
00:32:43.225 --> 00:32:46.440
Even if the camera faces a certain direction

715
00:32:46.440 --> 00:32:47.978
the z-axis isn't aligned

716
00:32:47.978 --> 00:32:49.969
with the camera’s view direction

717
00:32:49.969 --> 00:32:53.709
Instead, priority is given to how the screen is structured

718
00:32:53.709 --> 00:32:58.197
Thus, the z-axis in the view space, structured using a right-handed coordinate system

719
00:32:58.197 --> 00:33:01.443
points opposite to the viewing direction

720
00:33:01.443 --> 00:33:04.046
because it uses a right-handed coordinate system

721
00:33:04.046 --> 00:33:06.891
However,

722
00:33:06.891 --> 00:33:08.329
the objects we render

723
00:33:08.329 --> 00:33:09.974
aren’t behind the camera

724
00:33:09.974 --> 00:33:12.961
but rather in front of it

725
00:33:12.961 --> 00:33:15.666
When transformed

726
00:33:15.666 --> 00:33:20.170
into view space

727
00:33:20.170 --> 00:33:22.536
all points of objects in front of the camera

728
00:33:22.536 --> 00:33:24.615
will always have negative z-values

729
00:33:24.615 --> 00:33:27.104
we can make this conclusion

730
00:33:27.104 --> 00:33:30.532
This is due to the use of a right-handed coordinate system

731
00:33:30.532 --> 00:33:33.939
However, in NDC space,

732
00:33:33.939 --> 00:33:37.633
for objects in front of the camera

733
00:33:37.633 --> 00:33:39.973
the z-value or depth value

734
00:33:39.973 --> 00:33:43.988
increases as the object moves farther away

735
00:33:43.988 --> 00:33:45.082
This means that

736
00:33:45.082 --> 00:33:48.039
NDC space

737
00:33:48.039 --> 00:33:49.878
uses a left-handed coordinate system

738
00:33:49.878 --> 00:33:51.397
It’s not a right-handed system

739
00:33:51.397 --> 00:33:54.024
but a left-handed one

740
00:33:54.024 --> 00:33:58.028
Let’s briefly review

741
00:33:58.028 --> 00:34:00.669
what we’ve learned so far

742
00:34:00.669 --> 00:34:03.335
In our example,

743
00:34:03.335 --> 00:34:05.274
we started with world space

744
00:34:05.274 --> 00:34:07.035
World space uses

745
00:34:10.033 --> 00:34:12.381
a right-handed coordinate system

746
00:34:12.381 --> 00:34:15.862
for ease of object placement and viewing

747
00:34:15.862 --> 00:34:17.511
I previously mentioned that

748
00:34:17.511 --> 00:34:18.748
engines like Unity and Unreal

749
00:34:18.748 --> 00:34:21.168
use left-handed coordinate systems

750
00:34:21.168 --> 00:34:23.545
Thus, we constructed the world

751
00:34:23.545 --> 00:34:25.477
using a right-handed coordinate system

752
00:34:25.477 --> 00:34:27.059
and the camera’s view space

753
00:34:27.059 --> 00:34:29.049
also uses

754
00:34:29.049 --> 00:34:31.561
a right-handed system

755
00:34:31.561 --> 00:34:32.719
This is something

756
00:34:32.719 --> 00:34:34.391
important to remember

757
00:34:34.391 --> 00:34:37.484
However, during the conversion

758
00:34:37.484 --> 00:34:42.066
from view space to NDC space

759
00:34:42.066 --> 00:34:45.661
the system transitions to a left-handed coordinate system

760
00:34:45.661 --> 00:34:48.564
This goes back to when we discussed

761
00:34:48.564 --> 00:34:50.837
perspective projection matrices

762
00:34:50.837 --> 00:34:56.011
We always multiplied the z-value by -1

763
00:34:56.011 --> 00:35:00.758
to construct the z-value in clip space

764
00:35:00.758 --> 00:35:03.509
Since we constructed the z-value in clip space,

765
00:35:03.509 --> 00:35:06.469
this step indicates a transition from a right-handed coordinate system

766
00:35:06.469 --> 00:35:09.220
to a left-handed one

767
00:35:09.220 --> 00:35:11.250
When we transitioned clip space

768
00:35:11.250 --> 00:35:13.108
we summarized it as follows

769
00:35:13.108 --> 00:35:15.164
In local space

770
00:35:15.164 --> 00:35:16.886
the coordinate system is

771
00:35:16.886 --> 00:35:18.489
typically determined by the modeling program

772
00:35:18.489 --> 00:35:20.373
Most modeling programs use

773
00:35:20.373 --> 00:35:22.658
a right-handed coordinate system

774
00:35:22.658 --> 00:35:25.814
In world space,

775
00:35:25.814 --> 00:35:27.626
the coordinate system depends on the game engine

776
00:35:27.626 --> 00:35:29.910
Unity and Unreal Engine use left-handed systems,

777
00:35:29.910 --> 00:35:32.772
but in this course

778
00:35:32.772 --> 00:35:35.733
we’re using a right-handed coordinate system

779
00:35:35.733 --> 00:35:38.060
View space also uses a right-handed system,

780
00:35:38.060 --> 00:35:39.827
even in game engines

781
00:35:39.827 --> 00:35:42.793
However, when transitioning to clip space,

782
00:35:42.793 --> 00:35:46.283
the z-value is multiplied by -1

783
00:35:46.283 --> 00:35:48.939
switching to a left-handed system

784
00:35:48.939 --> 00:35:51.524
Simply dividing by the last element

785
00:35:51.524 --> 00:35:54.808
results in NDC space also adopting a left-handed system

786
00:35:54.808 --> 00:35:57.342
It’s useful to understand

787
00:35:57.342 --> 00:35:59.079
these coordinate system characteristics

788
00:35:59.079 --> 00:36:02.188
so I provided this summary

789
00:36:02.188 --> 00:36:04.937
Now, let’s look at

790
00:36:04.937 --> 00:36:07.577
the final perspective projection matrix

791
00:36:07.577 --> 00:36:08.700
that incorporates

792
00:36:08.700 --> 00:36:11.253
depth values

793
00:36:11.253 --> 00:36:14.139
Previously, we wasted a row

794
00:36:14.139 --> 00:36:19.353
in the 4x4 matrix

795
00:36:19.353 --> 00:36:22.524
just to make it conform to the format

796
00:36:22.524 --> 00:36:24.894
But now, we’ll assign

797
00:36:24.894 --> 00:36:26.933
meaningful values

798
00:36:26.933 --> 00:36:28.725
to calculate depth

799
00:36:28.725 --> 00:36:31.029
For the third row,

800
00:36:31.029 --> 00:36:32.844
let’s assume there are four components

801
00:36:32.844 --> 00:36:36.335
i, j, k, l as unknowns

802
00:36:36.335 --> 00:36:38.126
We will assume this

803
00:36:38.126 --> 00:36:41.054
If we determine these four values,

804
00:36:41.054 --> 00:36:44.138
we can calculate

805
00:36:44.138 --> 00:36:46.319
the depth as well

806
00:36:46.319 --> 00:36:49.438
We don’t know the exact depth values

807
00:36:49.438 --> 00:36:53.288
resulting from multiplying i, j, k, l

808
00:36:53.288 --> 00:36:57.365
but depth

809
00:36:57.365 --> 00:36:59.766
is independent of x and y

810
00:36:59.766 --> 00:37:02.692
Depth is unrelated to x and y values

811
00:37:02.692 --> 00:37:04.968
In other words, depth depends

812
00:37:04.968 --> 00:37:08.180
on the z-values in view space

813
00:37:08.180 --> 00:37:11.054
Therefore, the first and second components,

814
00:37:11.054 --> 00:37:15.400
which are related to a

815
00:37:15.400 --> 00:37:17.389
should effectively be 0

816
00:37:17.389 --> 00:37:18.812
This is because they don’t influence depth

817
00:37:18.812 --> 00:37:22.265
So, we’ve already resolved two of the four components

818
00:37:22.265 --> 00:37:24.026
Next, we need to find k

819
00:37:24.026 --> 00:37:26.149
 and l

820
00:37:26.149 --> 00:37:28.278
To determine these values,

821
00:37:28.278 --> 00:37:34.874
we use the characteristics of the near plane and the far plane

822
00:37:34.874 --> 00:37:37.357
and derive

823
00:37:37.357 --> 00:37:40.165
two sample data points

824
00:37:40.165 --> 00:37:43.150
The simplest sample

825
00:37:43.150 --> 00:37:47.900
is a point on the near plane

826
00:37:47.900 --> 00:37:50.220
where both x and y

827
00:37:50.220 --> 00:37:52.748
are 0

828
00:37:52.748 --> 00:37:55.528
In view space,

829
00:37:55.528 --> 00:37:57.036
how is the camera’s view direction

830
00:37:57.036 --> 00:37:58.513
defined?

831
00:37:58.513 --> 00:37:59.726
It becomes a negative value, right?

832
00:37:59.726 --> 00:38:03.357
The near plane

833
00:38:03.357 --> 00:38:04.749
has a value of n

834
00:38:04.749 --> 00:38:06.921
based on the specified size

835
00:38:06.921 --> 00:38:11.255
In view space, the point is (0,0,-n)

836
00:38:11.255 --> 00:38:13.929
Similarly, the far plane point

837
00:38:13.929 --> 00:38:15.530
in view space

838
00:38:15.530 --> 00:38:18.576
is 0, 0, -f

839
00:38:18.576 --> 00:38:20.536
However, once transformed

840
00:38:20.536 --> 00:38:24.629
into NDC space,

841
00:38:24.629 --> 00:38:27.663
the depth values range from -1 to 1

842
00:38:27.663 --> 00:38:29.593
Thus, the near plane point becomes 0, 0, -1

843
00:38:29.593 --> 00:38:32.396
and the far plane pint becomes 0,0,1

844
00:38:32.396 --> 00:38:34.966
Using this basic data,

845
00:38:34.966 --> 00:38:38.485
let’s derive the intermediate values

846
00:38:38.485 --> 00:38:41.256
for k and l

847
00:38:41.256 --> 00:38:44.836
The point in view space

848
00:38:44.836 --> 00:38:46.275
is 0,0,-n, 1

849
00:38:46.275 --> 00:38:50.163
When this point is multiplied

850
00:38:50.163 --> 00:38:52.005
by a matrix containing two unresolved unknowns,

851
00:38:52.005 --> 00:38:55.888
what is the result?

852
00:38:55.888 --> 00:39:00.289
Ultimately, we need to examine

853
00:39:00.289 --> 00:39:02.503
what this value turns out to be

854
00:39:02.503 --> 00:39:06.330
At this point, these aren’t NDC values

855
00:39:06.330 --> 00:39:08.242
but clip space value

856
00:39:08.242 --> 00:39:09.878
When the matrix is applied,

857
00:39:09.878 --> 00:39:12.925
the result will be in clip space

858
00:39:12.925 --> 00:39:15.306
Here, if we take -n

859
00:39:15.306 --> 00:39:17.578
the fourth component will definitely be n

860
00:39:17.578 --> 00:39:19.785
Similarly, in this case

861
00:39:19.785 --> 00:39:21.808
the fourth component will definitely be f

862
00:39:21.808 --> 00:39:24.430
The key is determining the third component

863
00:39:24.430 --> 00:39:26.701
The value divided by the last element

864
00:39:26.701 --> 00:39:28.034
must equal -1

865
00:39:28.034 --> 00:39:30.983
so here it will be -n

866
00:39:30.983 --> 00:39:32.276
On the other hand,

867
00:39:32.276 --> 00:39:35.089
since the NDC value here is 0,0,1

868
00:39:35.089 --> 00:39:37.786
dividing by f to get 1

869
00:39:37.786 --> 00:39:39.125
means this must be f

870
00:39:39.125 --> 00:39:41.124
Thus, the respective clip coordinates

871
00:39:41.124 --> 00:39:44.802
0, 0,-n,n here

872
00:39:44.802 --> 00:39:49.255
and 0,0,f,f there

873
00:39:49.255 --> 00:39:54.129
Using this, let’s proceed with the calculations

874
00:39:54.129 --> 00:40:00.193
0,0,-n,1 should transform into 0,0,-n,n

875
00:40:00.193 --> 00:40:06.048
and 0,0,-f,1 should correspont to

876
00:40:06.048 --> 00:40:09.123
0,0,f,f

877
00:40:09.123 --> 00:40:10.852
When multiplied directly

878
00:40:10.852 --> 00:40:16.859
what is -kn + l

879
00:40:16.859 --> 00:40:19.954
the result for the third element is -n

880
00:40:19.954 --> 00:40:22.556
-kf + l

881
00:40:22.556 --> 00:40:24.095
must equal f

882
00:40:24.095 --> 00:40:26.484
However, n and fare

883
00:40:26.484 --> 00:40:28.760
predefined by the user

884
00:40:28.760 --> 00:40:31.546
as the near and far planes of the camera

885
00:40:31.546 --> 00:40:34.139
Since they represent the near and far plane values

886
00:40:34.139 --> 00:40:36.714
of the frustum,

887
00:40:36.714 --> 00:40:38.120
they are treated as constants

888
00:40:38.120 --> 00:40:39.293
The unknowns here are

889
00:40:39.293 --> 00:40:41.267
k and l

890
00:40:41.267 --> 00:40:44.019
Eliminating variables

891
00:40:44.019 --> 00:40:49.805
k is caluclated as n+f/n-f

892
00:40:49.805 --> 00:40:54.411
and l is calculated as 2nf/n-f

893
00:40:54.411 --> 00:40:56.321
Substituting these values

894
00:40:56.321 --> 00:40:59.116
yields the final

895
00:40:59.116 --> 00:41:02.200
perspective projection matrix

896
00:41:02.200 --> 00:41:04.288
This completes the calculation

897
00:41:04.288 --> 00:41:06.614
of the final perspective projection matrix

898
00:41:06.614 --> 00:41:10.412
we’ve been developing

899
00:41:10.412 --> 00:41:12.545
That’s it for today’s lecture

900
00:41:12.545 --> 00:41:14.864
Thank you for attending

901
00:41:14.864 --> 00:41:15.744
Thank you

902
00:41:16.424 --> 00:41:17.716
Summary
perspective projection transformation is implemented by drawing lines based on vanishing points
Projection Plane: mapped based on distance

903
00:41:17.716 --> 00:41:19.036
Focal Length:shortest distance from camera-projection plane
method to normalize screen size while considering resolution
NDC
regardless of monitor spe

904
00:41:19.036 --> 00:41:20.215
Defines projection plane size as 1 for easy calculation
focal length d=1/tan(θ/2)
Objects may appear distorted

905
00:41:20.215 --> 00:41:21.337
due to different monitor aspect ratios
this is corrected in NDC space
Final NDC coordinates Pndc = -vz/d a/vx, vy

906
00:41:21.337 --> 00:41:22.692
Clip Space
Intermediary space used to create general-purpose matrices
Perspective projection matrices constructed with camera focal length&aspect rate

907
00:41:22.692 --> 00:41:23.862
pv = a/d 0 0, 0 d 0, 0 0 -1 vx vy vz=a/d vx, d vy,-vz
Vanishing Point Principle: In projective space, all lines converge to origin as z-values decreas

908
00:41:23.862 --> 00:41:25.080
Clip Space Coordinates represent lines where points within same category are gathered
Dimension-Aligned Perspective Projection matrix

909
00:41:25.080 --> 00:41:26.367
p= a/d 0 0 0, 0 d 0 0, 0 0 -1 0, 0 0 0 1

910
00:41:26.367 --> 00:41:27.779
Necessity of Depth Values
First object drawn may appear in background
Ensure that objects behind dodo not overwrite those in front through pixel value

911
00:41:27.779 --> 00:41:29.168
Frustum
A 3D geometric region with a truncated pyramid shape, created by defining a range for z-values within the camera's usable space

912
00:41:29.168 --> 00:41:30.337
Coordinate Systems
When transformed to NDC space, right-handed coordinate systems are converted into left
Local Space: Most modeling programs use righ

913
00:41:30.337 --> 00:41:31.347
World Space: Uses the coordinate system defined by the game engine.
View Space: Uses right-handed
Clip Space: Uses left-handed 
NDC: Uses left-handed

914
00:41:31.347 --> 00:41:36.268
Final Projection Matrix
A final projection matrix that calculates depth values while accounting for frustum
p= a/d 0 0 0, 0 d 0 0, 0 0 n-f/n+f n-f/nf,