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Model Matrix: Rotation - World vs. Local Origin

· One min read

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

· One min read

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

Change of Coordinates and Applications to View Matrices

· One min read

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

· One min read

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

· One min read

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.

A Brief Overview on Recent Cyberattacks

· 5 min read

In the first months of 2025, several high-impact cyberattacks made headlines - targeting everything from crypto exchanges and streaming platforms to core infrastructure components. This post takes a brief look at two selected incidents: How they worked, what security objectives were compromised, and what they reveal about the evolving threat landscape.

Contracted Events for Micro Frontend Communication

· 2 min read

Contracted Events for Micro Frontend Communication​

A non-formal approach to non-breaking events for displaying information in distributed Frontend Fragments​

Schema Versioning and Consumer-driven contracts play critical roles in the world of distributed systems, where data structures often evolve across releases. These concepts can be applied to Micro Frontends, where messages need to be exchanged across system boundaries: Release-agnostic systems, which can dynamically enter and exit the system of concern, must be able to properly respond to event messages: Updated message structures must not introduce breaking changes.

In a frontend, provide contracts for data structures consumed by conceptually related modules and process the data according to the individual requirements of the consuming context.

[DE] Betrachtung des Widerspruchsbeweis des speziellen Halteproblems nach Vossen und Witt

· 6 min read

Oft wird der Nachweis, dass das Halteproblem nicht entscheidbar ist, in der Fachliteratur (Schöning [📖Sch08, 119 f.], Asteroth und Baier [📖BA02, 106 f.], Sipser [📖Sip12, 216 f.]) mithilfe einer Turingmaschine und einem Widerspruchsbeweis gezeigt, in etwa:

Angenommen,

K′={w∈Σ∗∣w=⟨T⟩,T stoppt bei Eingabe w}K' = \{w \in \Sigma^* | w = \langle T \rangle, \text{$T$ stoppt bei Eingabe $w$}\}

ist entscheidbar.

Sei T′T' ist die Turingmaschine, die K′K' entscheidet.
Sei w=⟨T⟩w = \langle T \rangle die Codierung1 einer Turingmaschine, die wie folgt arbeitet:

  • TT simuliert das Verhalten von T′T' bei der Eingabe ww:
    • TT stoppt, wenn T′T' die Eingabe ww verwirft (w∉K′w \notin K')
    • TT stoppt nicht, wenn T′T' die Eingabe ww akzeptiert (w∈K′w \in K')

Damit folgt: TT stoppt bei Eingabe ww ⇔\Leftrightarrow T′T' verwirft ww ⇔\Leftrightarrow w∉K′w \notin K' ⇔\Leftrightarrow TT stoppt nicht

Widerspruch!

Footnotes​

  1. Anstatt Gödelnummer wird im Folgenden auch einfach der Begriff Codierung verwendet, wobei streng genommen τ(T)\tau(T) die Codierung einer normierten Turingmaschine ist, bevor ihr eine Gödelnummer zugewiesen wird ↩