Why is Vector Calculus important for Computer Graphics?

• spatial relationships, transformations
• PDEs
• Vector valued data everywhere

Index

1. Geometric interpretation of Vectors

Euclidean Norm

• Notion of “size” of a vector
• In Geometric calculations, the most important and often used norm is the Euclidean Norm
• Euclidean norm : Any notion of length preserved by rotations/translations/reflections of space
  • This is not true for all norms, hence we use Euclidean Norm in geometric calculations
• In orthonormal coordinates : *only in orthonormal coordinates does the Euclidean norm encode geometric length

$$ |\bold{u}| = \sqrt{u_1^2+\dots+u_n^2} $$

<aside> 🤔 고등학교 때 벡터의 크기를 배울 땐, 벡터를 항상 orthonormal basis로 encode된 벡터들을 다뤘기 때문에, Euclidean norm을 사용하여 벡터의 크기를 geometric length로 배웠다. 그러나 다양한 Norm도 존재하고, 다양한 basis도 존재하기 때문에 이전에 생각하던 방식으로 생각하면 안된다.

</aside>

Euclidean Inner Product / Dot Product

• Notion of “alignment” between vectors
• For Geometric Calculations, we want inner product to capture something about geometry
• Euclidean inner product for n-dim vectors

$$ \braket{\bold{u},\bold{v}}=|\bold{u}||\bold{v}|\cos{\theta} $$

• In orthonormal Cartesian coordinates it can be represented as dot product