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Vectors describe positions or directions within a coordinate space.
Matrices describe how to transform vectors from one space to another.

[ Xaxis.x  Yaxis.x  Zaxis.x  Tx ]
[ Xaxis.y  Yaxis.y  Zaxis.y  Ty ]
[ Xaxis.z  Yaxis.z  Zaxis.z  Tz ]
[   0        0        0      1 ]

A 4×4 transformation matrix is made up of:

Action Result
Set w = 1 Vector becomes a position → gets full matrix transform (T * R * S)
Set w = 0 Vector becomes a direction → skips translation
Divide by w Converts clip space → NDC (perspective divide)
Store something in w Use w as a packed value (for instancing, motion, etc.)

YES — asking what is a basis means you’re ready to understand all of 3D spacebuilding and matrix logic. Let’s break it down clearly, intuitively, and practically for tech art 👇


What Is a Basis?

A basis is a set of vectors that define a coordinate space.
It tells you:

It’s like building a mini coordinate system — your own local grid.


✅ Example: Standard Basis in 3D

The default basis in world space is:

X = (1, 0, 0)
Y = (0, 1, 0)
Z = (0, 0, 1)

This is called an orthonormal basis:

This is what Unity uses for world space, object space (by default), and more.


🧩 Why Is a Basis Useful?

A basis lets you build your own space inside another space.

You can:


🧱 Basis = The Building Blocks of a Matrix

In a transformation matrix, the 3 main columns are your basis vectors:

[ Right  Up  Forward  Position ]
[   X     Y     Z        T     ]

So:

✅ A matrix is just a basis with a location.


🧪 Real Tech Art Examples

Use Case Basis Involved? What It Does
Tangent space normal maps TBN basis Converts normals from UV space to world space
Custom pivot control Custom basis Reorients local X/Y/Z axes
Surface alignment shader Basis from world normals Builds a local space aligned to geometry
Procedural animation Create and apply basis Bones, aim constraints, etc.
Billboarding Rebuild basis using camera Rotate object to face view direction


✅ A basis is a set of vectors that define a space's axes.
A matrix = basis + position.
You use it to move, rotate, or remap things between spaces.

Let me know if you’d like a diagram or Unity version of any of these!


Revision #5
Created 23 March 2025 21:54:12 by victor
Updated 31 March 2025 00:40:59 by victor