6 posts tagged with "linear algebra"

From Camera to Clip Space​

Derivation of the Projection Matrices​

My latest article examines the transformation from camera space to clip space and ultimately to Normalized Device Coordinates (NDC). In the article, we derive the complete orthographic and perspective projection matrices, with a special emphasis on OpenGL conventions.

Model Matrix​

Rotation: World vs. Local Origin​

In my latest article, I introduce the mathematical foundation for rotation as linear transformations applied to model matrices in 3D computer graphics. The article focuses on the distinction between rotating an object around an external point and rotating it around its own local origin, and derives the necessary transformations from two conceptually different but mathematically equivalent perspectives: Active transformations and passive transformations.

Model Matrix​

Transformations: A Change-of-Coordinates Perspective​

My article examines model matrices through the lens of change-of-coordinates transformations.

Rather than treating model matrices as mere transformation tools, viewing them as change-of-coordinates matrices reveals why certain algebraic properties hold. Specifically, I demonstrate the mathematical equivalence between two common rotation approaches:

• Matrix-first: $\boldsymbol{R}\ \underset{W \leftarrow M}{P}\ \vec{v}$ (rotate the model matrix, then transform)
• Transform-first: $\boldsymbol{R}\ (\underset{W \leftarrow M}{P}\ \vec{v})$ (transform to world space, then rotate)

Change of Coordinates​

... and Applications to View Matrices​

My article introduces the mathematical derivation of the lookAt view matrix commonly used in computer graphics APIs like OpenGL.

As part of the graphics pipeline, the purpose of this matrix is to perform the view transformation, converting global world coordinates into a coordinate system defined by position and direction of an observer.

Rotations as a Special Case of Vector Transformations​

... with Applications to Game World Modeling​

Rotations, as a class of affine vector transformations, represent rigid body transformations that change the direction of a vector within a given plane.
Efficient and accurate handling of such transformations poses challenges not only for game and animation engines, but also for simulations and real-time systems, where the limitations of floating-point arithmetic and operation throughput can negatively impact the computational outcomes.

The Geometry of the Dot Product​

... and Applications to Visibility Modeling​

The Dot Product is a fundamental building block for vector operations in video games and simulations. A solid understanding is crucial for applications involving view-related coordinate transformations and even physical modeling within a game world. For many practical use cases, the dot product offers an elegant alternative to constructing explicit visibility cones or relying on computationally expensive raytracing algorithms.