WEBVTT

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Game Basics
Mathematics for 3D Games Part 2: Cross Product of Vectors

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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, we will take a look at an important operation on vectors

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which is the cross product

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We will explore the cross product operation

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and its properties

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Next, we will compare the differences

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between the cross product and the cross product operation

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Finally, we’ll look at various ways

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to apply the cross product

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The Cross Product Operation

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Let’s start by understanding the cross product operation

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The cross product operation

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can only be performed in three-dimensional space,

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meaning it applies exclusively

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to 3D vectors

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The result of the cross product

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unlike the dot product

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is a vector value

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The dot product yields a scalar value,

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but the cross product produces a vector

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The operation is performed in 3D space

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on 3D vectors

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and the result is also a 3D vector

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So, let’s explore how the cross product operation

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is performed

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Here, when performing the cross product of an arbitrary vector, u

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and another arbitrary vector v

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For the first component, x,

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you exclude x and use the y and z components

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instead and proceed

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The first step in calculating the cross product of u and v

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is to take uy, vz  in order.

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Then, you put a minus

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and reverse the product

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Thus, vy and uz is subtracted

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which is a key characteristic of the cross product operation

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Next, let’s look at the second component

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The second component is y

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so you exclude y and use the other two components

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We follow the cyclic order x,y,z,x,y,z, as previously mentioned

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Excluding y

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the sequence proceeds as z, x

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First, calculate uz and vx

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Then, subtract the reverse product

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for vz ux

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Finally, for the z-component

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the sequence proceeds as x,y

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First, calculate ux*vy

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Then, subtract the reverse product

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vx*uy

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This completes the vector derived from the cross product operation

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We should always examine

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the cross product and dot product of vectors

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by comparing them

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Although these two operations are contrasting

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you can think of the cross product as

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complementing the aspects that the dot product lacks

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Looking at this diagram

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when two arbitrary vectors are used

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the dot product involves

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multiplying corresponding components and then adding them together

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However, in the cross product

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for any x-component,

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it uses the components other than x

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For y, it uses components other than y

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so the operation relies only

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on non-corresponding components

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This way, you can understand that the cross product complements

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the properties of the dot product

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Now, let’s examine the properties

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of the dot product and the cross product operations

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Previously, we studied the various properties

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of the dot product operation

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Let’s review the two operations

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by comparing them side by side

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The dot product operation satisfies the commutative property

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but the cross product does not

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As you can see, the cross product involves subtraction

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so if you swap v and u in the operation

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the resulting value will differ from the previous calculation

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Thus, the commutative property does not hold for the cross product

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The important point here is that 
although the commutative property does not apply

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adding a negative sign to the vector,

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when reversing the order

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results in the same cross product value as the original

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This means a vector in the opposite direction is created

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So, while the commutative property does not apply

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you can think of it as

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following a specific rule

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Neither the dot product nor the cross product

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satisfies the associative property

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It’s good to note that

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neither operation adheres

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to the associative property

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Next, one important property is that

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over addition, the dot product satisfies

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the distributive property

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The same applies to the cross product

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The distributive property holds for 
addition in the cross product, as shown here

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Make sure to remember this property

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Let’s examine the key properties

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of the cross product

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If we consider vectors in the same or opposite directions

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this means

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vectors that are either aligned in the same or opposite directions

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This implies that the vectors are parallel

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This implies that the vectors are parallel vectors

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always results in a zero vector

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Let’s explore why this happens

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When you take the cross product

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of the same vector with itself,

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the result is a zero vector

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due to the negative sign introduced by the operation

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Similarly, the cross product of vectors in opposite directions

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also results in a zero vector

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The only difference is the sign—positive or negative

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Even if you multiply a vector by a scalar

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and take the cross product with another vector

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in the same direction,

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the result remains equivalent to scaling the zero vector

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Since scaling a zero vector still results in a zero vector

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parallel vectors always produce a zero vector,

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which is a fundamental property of the cross product

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Let’s explore

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how to apply this property

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When we have two vectors, u and v

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with these two,

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we can decompose u with respect to v

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into two components

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One component is parallel, and the other is perpendicular

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The parallel component will naturally produce 
a zero vector in the cross product

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The perpendicular component will result in a non-zero value

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Previously, we divided the vector into two components like this

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Since the distributive property holds for addition

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the cross product of u and v

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simplifies as follows

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The parallel component becomes a zero vector and cancels out

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Only the perpendicular component remains

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Thus, when dealing with

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such arbitrary vectors,

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if we consistently decompose the vector 
and exclude the parallel component,

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regardless of the vector’s magnitude

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the result is influenced by

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the angle between the vector and v

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This is because only the perpendicular component contributes to the result

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Ultimately, the smaller the angle,

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the smaller the perpendicular component becomes

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As a result, the cross product value decreases

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On the other hand, the larger the angle,

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the larger the perpendicular component becomes

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Thus, the cross product value increases

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This proportional relationship

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is related to the sine function of trigonometry

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In this context, the sine function corresponds to this part

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which represents the perpendicular component

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Thus, the magnitude of the cross product vector

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is proportional to the sine of the angle between the two vectors

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This is the conclusion we can draw

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In the case of the dot product we learned earlier

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it was proportional to the cosine function

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The cosine and sine functions,

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as well as the dot product and cross product, complement each other

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Remember this

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Now, let’s examine the cross product

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of two arbitrary vectors

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After calculating the cross product

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we’ll take the result and compute the dot product with either u or v

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According to the equation, the result of the dot product

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will repeat identical values, ultimately resulting in 0

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This is when we take the dot product with u

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and this is when we do it with v

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Since the dot product satisfies the commutative property

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it doesn’t matter which vector is placed first

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In any case, taking the dot product resulted in 0

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What does this mean?

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The property of the dot product being proportional to the cosine function

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is called orthogonality

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Thus, a dot product value of 0

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means that the two vectors are orthogonal

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Previously, we discussed that two arbitrary vectors

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with different slopes

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can form a plane

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When you take the cross product of these two vectors

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a vector orthogonal to both

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is produced

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This means that it generates a vector orthogonal to the plane

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This is how the cross product operation works

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The direction of orthogonality to the plane

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essentially indicates the shortest distance from the plane

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we can understand this in this way

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It can also be understood as the direction the plane is facing

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The vector produced by the cross product of two vectors

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is commonly called a normal vector

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The cross product operation is used to create this normal vector

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This has practical applications in computer graphics

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particularly for light calculations

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So it’s important to understand this

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When creating an orthogonal vector

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note that a plane has a front side and a back side

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There is a front side and a back side

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and whether it faces forward or backward

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is determined by orthogonality

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which means the dot product equation results in 0

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This is related to the order of the cross product we examined earlier

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If we take the cross product of u and v

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and get a direction,

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taking the cross product of v and u in reverse

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produces a vector in the opposite direction

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Thus, we need to decide

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which side is the true front

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This is determined by the right-hand rule

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The result of the cross product always follows the right-hand rule

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The direction of the plane depends

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on the priority given to the vectors

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In the case of the diagram shown here

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if you curl your right hand from u to v

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the direction your thumb points indicates

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the direction of the plane

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We’ve covered how to generate a normal vector

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Now let’s compare the dot product and cross product comprehensively

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using a diagram

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Let’s assume we have two vectors

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The dot product is proportional to the cosine function

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while the cross product is proportional to the sine function

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For the cross product, which is proportional to the sine function

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a positive value indicates

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the right side, as shown here

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A negative value indicates the left side

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On the other hand, the dot product

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is proportional to the cosine function

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so a positive value represents the front side

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Thus, the dot product is used to determine orthogonality

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The dot product helps identify orthogonality

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while the cross product determines parallelism

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The dot product is proportional to cosθ, and the cross product is proportional to sinθ

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The dot product is used to distinguish front and back

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while the cross product is used to distinguish left and right

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You can see that

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they complement each other quite well

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The dot product produces a scalar value

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while the cross product produces a vector value

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One key application of the dot product

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is creating projection vectors

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The cross product is used to create normal vectors

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Applications of the Cross Product

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Let’s explore how the cross product

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is practically applied in game development

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The first application we’ll look at is determining left and right

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The cross product can be used to determine whether

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something is on the left or the right

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When we set a viewing vector f

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and define the vector from the character to the object as v

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if we calculate the cross product

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from f to v

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the result of the cross product will point downward

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when using the right-hand rule, right?

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On the other hand, if the object

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is on the left side of the viewing vector, let’s call it v'

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the cross product will point upward, as you can see

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This means that relative to the character’s view

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an object on the left will result in an upward direction

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while an object on the right will point downward

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This is what you will be able to see

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By using this property, where upward means left

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and downward means right

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we can determine whether the object

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is on the left or right

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However, since the cross product always produces a vector

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it’s not possible to clearly distinguish

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between left and right

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based solely on the vector value

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00:14:21.119 --> 00:14:23.329
At this point, we introduce an "Up vector,"

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a vector pointing upward

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00:14:27.479 --> 00:14:29.847
Typically, this is the world’s y-axis

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00:14:29.847 --> 00:14:31.880
represented as (0, 1, 0)

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00:14:31.880 --> 00:14:36.360
We use the (0, 1, 0) vector

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00:14:36.360 --> 00:14:38.799
to perform an additional dot product calculation

280
00:14:38.799 --> 00:14:40.969
When taking the dot product

281
00:14:40.969 --> 00:14:42.604
 with the Up vector

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00:14:42.604 --> 00:14:45.289
if they are in the same direction, it indicates upward

283
00:14:45.289 --> 00:14:48.540
Thus, a positive result indicates left, and negative indicates right

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00:14:48.540 --> 00:14:50.100
you can understand it like this

285
00:14:50.100 --> 00:14:52.771
You can use the scalar triple product

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00:14:52.771 --> 00:14:55.679
to determine left and right like this

287
00:14:55.679 --> 00:14:58.185
The second application of the cross product

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00:14:58.185 --> 00:15:02.029
is generating a rotation matrix from vectors

289
00:15:02.029 --> 00:15:04.204
This is commonly used

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00:15:04.204 --> 00:15:06.389
for determining the orientation of an object’s viewing direction

291
00:15:06.389 --> 00:15:09.559
or a camera’s viewing direction

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00:15:09.559 --> 00:15:11.961
to derive a rotation matrix

293
00:15:11.961 --> 00:15:14.719
from the viewing direction

294
00:15:14.719 --> 00:15:16.679
Let’s go through an example

295
00:15:16.679 --> 00:15:18.699
Suppose there is a camera and an object

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00:15:18.699 --> 00:15:21.952
The direction from the camera to the object

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00:15:21.952 --> 00:15:25.840
can be found by subtracting the camera’s position from the object’s position

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00:15:25.840 --> 00:15:28.570
This vector essentially represents

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00:15:28.570 --> 00:15:31.280
the local z-axis

300
00:15:31.280 --> 00:15:34.573
Once we have the local z-axis

301
00:15:34.573 --> 00:15:38.520
we need to calculate the remaining local x-axis and y-axis

302
00:15:38.520 --> 00:15:42.980
The local x-axis can be derived using the world’s Up vector

303
00:15:42.980 --> 00:15:45.520
and the plane formed by

304
00:15:45.520 --> 00:15:48.362
the z-local and the Up vector

305
00:15:48.362 --> 00:15:55.520
The vector orthogonal to this plane becomes the x-local axis

306
00:15:55.520 --> 00:15:58.420
Thus, we calculate the x-local axis

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00:15:58.420 --> 00:16:01.320
which points beyond the camera monitor

308
00:16:01.320 --> 00:16:04.640
Using the cross product, this becomes the normal vector

309
00:16:04.640 --> 00:16:07.280
After calculating the x-local axis this way

310
00:16:07.280 --> 00:16:10.374
the y-local axis can be found

311
00:16:10.374 --> 00:16:14.280
by taking the cross product of z-local and x-local

312
00:16:14.280 --> 00:16:17.719
This completes the three axes

313
00:16:17.719 --> 00:16:23.536
However, it’s important to note that these vectors

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00:16:23.536 --> 00:16:26.799
may not all have a magnitude of 1

315
00:16:26.799 --> 00:16:30.559
For the z-local axis,

316
00:16:30.559 --> 00:16:32.359
we can normalize it

317
00:16:32.359 --> 00:16:36.359
Dividing the vector by its magnitude normalizes it

318
00:16:36.359 --> 00:16:40.389
Similarly, the x-vector generated from the cross product

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00:16:40.389 --> 00:16:42.519
may not have a magnitude of 1

320
00:16:42.519 --> 00:16:44.939
This is because it’s not guaranteed that

321
00:16:44.939 --> 00:16:47.490
the y-axis and x-axis are orthogonal

322
00:16:47.490 --> 00:16:51.220
Thus, the angle between them may vary,

323
00:16:51.220 --> 00:16:53.216
so we must normalize it

324
00:16:53.216 --> 00:16:56.440
after calculation

325
00:16:56.440 --> 00:16:59.179
Since the z and x axes

326
00:16:59.179 --> 00:17:01.640
are designed to be orthogonal

327
00:17:01.640 --> 00:17:04.618
the y-axis can be directly calculated

328
00:17:04.618 --> 00:17:07.239
using the cross product without additional normalization

329
00:17:07.239 --> 00:17:08.807
In summary, you can calculate

330
00:17:08.807 --> 00:17:12.000
the three local axes as follows

331
00:17:12.000 --> 00:17:14.666
These axes can then be used

332
00:17:14.666 --> 00:17:18.479
to construct a rotation matrix in 3D space

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00:17:18.479 --> 00:17:22.800
There are some considerations to keep in mind when creating such a rotation matrix

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00:17:22.800 --> 00:17:24.680
For instance, there are cases where the camera might flip upside down

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00:17:24.680 --> 00:17:27.599
Sometimes an object moves in an inverted manner

336
00:17:27.599 --> 00:17:29.839
Although these cases are rare

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00:17:29.839 --> 00:17:32.599
it’s good to account for them just in case

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00:17:32.599 --> 00:17:35.526
You can typically provide an option

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00:17:35.526 --> 00:17:37.920
to specify the direction of the Up vector

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00:17:37.920 --> 00:17:40.040
in such scenarios

341
00:17:40.040 --> 00:17:42.729
Keep in mind that such functionality

342
00:17:42.729 --> 00:17:45.239
might be necessary

343
00:17:45.239 --> 00:17:50.199
The second issue occurs when the camera’s viewing direction

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00:17:50.199 --> 00:17:53.239
is parallel to the specified Up vector

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00:17:53.239 --> 00:17:55.536
In this case, taking their cross product

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00:17:55.536 --> 00:17:57.199
results in a zero vector

347
00:17:57.199 --> 00:18:00.239
which means you cannot proceed further

348
00:18:00.239 --> 00:18:03.239
This becomes a significant obstacle

349
00:18:03.239 --> 00:18:06.119
In such cases,

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00:18:06.119 --> 00:18:12.237
having the Up vector

351
00:18:12.237 --> 00:18:15.439
and the local z-axis parallel is meaningless, right?

352
00:18:15.439 --> 00:18:20.610
So, you need to arbitrarily designate a vector

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00:18:20.610 --> 00:18:23.960
orthogonal to the viewing direction

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00:18:23.960 --> 00:18:27.669
You can specify an x-axis separately or arbitrarily assign it

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00:18:27.669 --> 00:18:31.040
In this case, the x-axis refers to the world’s x-axis

356
00:18:31.040 --> 00:18:35.576
You can assign either the world x-axis or the world z-axis,

357
00:18:35.576 --> 00:18:39.385
and use them to calculate

358
00:18:39.385 --> 00:18:40.970
the remaining axes

359
00:18:40.970 --> 00:18:42.932
Such exceptions

360
00:18:42.932 --> 00:18:45.537
must be considered during implementation

361
00:18:47.211 --> 00:18:49.399
The final application of the cross product is

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00:18:49.399 --> 00:18:52.515
back-face culling, which is a technique

363
00:18:52.515 --> 00:18:54.711
commonly used in rendering

364
00:18:54.711 --> 00:18:58.640
When dealing with a triangle

365
00:18:58.640 --> 00:19:02.075
there’s always a front face and a back face

366
00:19:02.075 --> 00:19:06.309
If it faces the back, there’s no need to render it

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00:19:06.309 --> 00:19:08.720
Because it’s not visible to the eye, typically

368
00:19:08.720 --> 00:19:10.514
in the case of opaque objects

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00:19:10.514 --> 00:19:13.000
the back face is not rendered as it’s not visible

370
00:19:13.000 --> 00:19:16.611
Using back-face culling

371
00:19:16.611 --> 00:19:18.380
to omit the back face

372
00:19:18.380 --> 00:19:21.726
can significantly reduce the number of triangles rendered

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00:19:21.726 --> 00:19:24.209
by nearly half

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00:19:24.209 --> 00:19:28.239
This leads to substantial performance improvements in game development

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00:19:28.239 --> 00:19:31.610
Let’s look at how this is implemented

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00:19:31.610 --> 00:19:35.439
Typically, when we define a triangle by specifying three vertices

377
00:19:35.439 --> 00:19:36.830
we need to determine

378
00:19:36.830 --> 00:19:40.360
which way the face of the triangle is oriented

379
00:19:40.360 --> 00:19:43.420
Since a triangle always has a front face and a back face

380
00:19:43.420 --> 00:19:48.299
it’s necessary to specify which side is the front face

381
00:19:48.299 --> 00:19:52.160
However, if we explicitly define the front face for every triangle

382
00:19:52.160 --> 00:19:54.582
we’d have to store

383
00:19:54.582 --> 00:19:57.359
additional directional data

384
00:19:57.359 --> 00:19:59.780
which would be a significant waste of storage

385
00:19:59.780 --> 00:20:01.750
This is where the cross product comes in

386
00:20:01.750 --> 00:20:08.359
We implicitly use the order of the vertices

387
00:20:08.359 --> 00:20:13.719
to define the direction of the triangle’s face using the normal vector

388
00:20:13.719 --> 00:20:16.199
generated by the cross product

389
00:20:16.199 --> 00:20:17.479
Let’s take an example

390
00:20:17.479 --> 00:20:20.642
Here for 0, 1, 2, the triangle is defined

391
00:20:20.642 --> 00:20:23.399
in a counter-clockwise order

392
00:20:23.399 --> 00:20:28.000
In this case, using the right-hand rule

393
00:20:28.000 --> 00:20:30.720
the result of the cross product points beyond the monitor

394
00:20:30.720 --> 00:20:33.199
This indicates the direction of the face

395
00:20:33.199 --> 00:20:35.048
Now, for a triangle defined

396
00:20:35.048 --> 00:20:36.569
in a clockwise order

397
00:20:36.569 --> 00:20:39.840
the result points toward the front, or toward me

398
00:20:39.840 --> 00:20:43.630
Thus, this side is implicitly defined

399
00:20:43.630 --> 00:20:45.390
as the direction of the face

400
00:20:45.390 --> 00:20:48.839
Using this method, there’s no need

401
00:20:48.839 --> 00:20:52.809
to explicitly store face direction data in the mesh

402
00:20:52.809 --> 00:20:56.199
By clearly distinguishing between the front and back faces

403
00:20:56.199 --> 00:20:59.099
we can implement the technique of back-face culling

404
00:20:59.099 --> 00:21:03.690
For example, if the camera has a viewing direction

405
00:21:03.690 --> 00:21:08.160
only the faces that face the viewing direction are rendered

406
00:21:08.160 --> 00:21:10.360
If the face direction

407
00:21:10.360 --> 00:21:13.525
is the same as the camera’s viewing direction

408
00:21:13.525 --> 00:21:15.260
it means the camera is looking at the back face

409
00:21:15.260 --> 00:21:17.730
In this case, rendering is unnecessary

410
00:21:17.730 --> 00:21:20.089
When the triangle is oriented clockwise

411
00:21:20.089 --> 00:21:21.821
and the camera looks in this direction

412
00:21:21.821 --> 00:21:24.438
since the camera and face are facing each other,

413
00:21:24.438 --> 00:21:26.410
it should be rendered

414
00:21:26.410 --> 00:21:27.909
So how do we determine this?

415
00:21:27.909 --> 00:21:30.622
By calculating the dot product of the camera’s viewing direction vector

416
00:21:30.622 --> 00:21:33.880
and the face’s direction vector,

417
00:21:33.880 --> 00:21:36.360
if their directions align, the result is positive

418
00:21:36.360 --> 00:21:39.000
If the result is negative, they face each other

419
00:21:39.000 --> 00:21:41.710
so it should be rendered

420
00:21:41.710 --> 00:21:44.079
This screen here shows

421
00:21:44.079 --> 00:21:47.159
both the front and back faces rendered on the left

422
00:21:47.159 --> 00:21:53.240
When we render rectangles or cubes in 3D,

423
00:21:53.240 --> 00:21:55.551
we typically render even the parts at the back

424
00:21:55.551 --> 00:21:57.640
This is how a wireframe is drawn

425
00:21:57.640 --> 00:21:59.559
However, using back-face culling

426
00:21:59.559 --> 00:22:03.159
all back faces that face away from the camera

427
00:22:03.159 --> 00:22:05.841
are omitted

428
00:22:05.841 --> 00:22:09.396
This gives the appearance of

429
00:22:09.396 --> 00:22:11.890
a solid, three-dimensional object

430
00:22:11.890 --> 00:22:16.919
In this way, we’ve covered the basic operations and properties of the cross product of vectors

431
00:22:16.919 --> 00:22:19.700
We also explored

432
00:22:19.700 --> 00:22:22.439
its various applications

433
00:22:22.439 --> 00:22:24.449
That concludes the summary of the cross product

434
00:22:24.449 --> 00:22:27.770
We’ll meet again next time with a different topic

435
00:22:27.770 --> 00:22:29.929
This concludes today’s lecture

436
00:22:29.929 --> 00:22:31.760
Thank you for your attention

437
00:22:31.760 --> 00:22:32.580
Thank you!

438
00:22:32.969 --> 00:22:35.463

Vector Cross Product Operations
Cross Product Operation
Performed only in 3D space, resulting in a vector
u=ux, uy, uz, v=vx, vy, vz

439
00:22:35.463 --> 00:22:37.909
u x v= uyvz-vyuz,uzvx-vzux,uxvy-vxuy

Properties of the outer product operation
Commutative law does not hold
Associative law does not hold
Distributive law holds for addition

440
00:22:37.909 --> 00:22:39.951
Operation of vector cross product
Main properties of cross product
When you cross product parallel vectors, you get a zero vector
The size of the cross product vector is proportional to the sine function of the angle between them
Creates a vector orthogonal to the plane

441
00:22:39.951 --> 00:22:41.641
Comparison of outer and inner products
The inner product is used for orthogonal discrimination, and the outer product is used for parallel discrimination
The inner product is proportional to cosθ, and the outer product is proportional to sinθ
The inner product is used for front and back discrimination, and the outer product is used for left and right discrimination

442
00:22:41.641 --> 00:22:42.928
Dot Product Applications: Projection vectors
Cross Product Applications: Normal vectors

443
00:22:42.928 --> 00:22:45.606
Application of vector outer product
Left-right determination
Determination using scalar triple product
f x v y > 0
f x v y <0
Generating rotation matrix from vector

444
00:22:45.606 --> 00:22:46.806
xlocal=y x zlocal/y x zlocal, ylocal=zlocal x xlocal, zlocal-v/v

445
00:22:46.806 --> 00:22:47.988
R=xx yx zx 0, xy yy zy 0, xz yz zz 0, 0, 0, 0, 1

446
00:22:47.988 --> 00:22:49.238
Application of Vector Cross Product
Backface Culling
Opaque objects usually have no back-facing graphics, so backface culling can drastically reduce triangle drawing

447
00:22:49.238 --> 00:22:50.517
The normal vector created externally using the vertex placement order is implicitly designated as the direction of the triangle
Only the faces facing the camera's view direction are drawn

448
00:22:50.517 --> 00:22:51.717
Dot Product of Camera View Vector and Triangle Normal:
Positive: Same direction, skip rendering

449
00:22:51.717 --> 00:22:52.938
Negative: Opposite direction, render