From Computer Vision to Semantics: RDP-Based Simplification of Embedding Trajectories

• Many algorithms find new applications far beyond the domain they were originally designed for, and the reason behind this is a strong structural logic. If a method relies not on a specific data type but on the geometric organization of the data, it naturally continues to function when transferred to a different domain. 

• The expansion of diffusion models from visual generation to text and other modalities, or the way the Transformer architecture became a foundational building block in tasks far beyond translation, are good examples of this phenomenon. 

• The Ramer–Douglas–Peucker (RDP) algorithm fits precisely into this category. In this article, starting from RDP’s classical role in computer vision, we examine—step by step, intuitively, and with visual support—why it works equally well on high-dimensional textual embedding sequences.

• The goal here is not to “reinvent” RDP for NLP, but to make the geometric nature of embedding sequences visible. 

• Throughout the article, we first review the fundamental structure of RDP, then demonstrate its behavior using 2D and 3D examples, and finally move into embedding space to show, through an experimental example, how RDP operates over semantic flow.

The Core Idea of RDP: The Structural Essence of a Curve

RDP’s effectiveness rests on a surprisingly simple idea: to make a complex curve meaningful to the human eye by removing unnecessary fluctuations while preserving only the characteristic “turning points” of the structure. When we unpack the algorithm’s steps, this simplicity becomes even more evident.

RDP Algorithm Mechanics

• The first and last points of a curve are taken as reference.

• A straight line is drawn between these two points.

• The distance of all intermediate points to this line is computed.

• Points with large deviations indicate genuine structural turns in the curve.

• If the deviation exceeds a given threshold, that point is preserved and the curve is split into two segments.

• The same process is applied recursively to each sub-segment.

• Through this recursive process, small oscillations—essentially noise—are removed, while the critical structures where the curve changes direction are preserved.

Figure 1: RPD Example (Source: Makris, A., Silva, C. L. D., Bogorny, V., Alvares, L. O., Macedo, J. A., & Tserpes, K. (2021). Evaluating the effect of compressing algorithms for trajectory similarity and classification problems. GeoInformatica, 25(4), 679-711.)

RDP’s characteristic behavior has enabled its use across a wide range of domains: from simplifying GPS trajectories to filtering robotic arm motions, from analyzing surface contours to compressing time series. In all these applications, the need to “discard unnecessary variation while preserving the structural essence” appears in the same way.

This point becomes crucial when transitioning into embedding space. Embedding sequences, much like curves, also contain “small oscillations” and “major turning points”; the only difference is that the space in which these structures live is far higher-dimensional.

RDP’s Behavior on a 2D Curve

To understand what RDP considers important, we begin with its most classical use case: a 2D example. The signal carries an underlying trend while also containing small oscillations, micro-noise, and occasional sharp jumps. To the human eye, this signal includes fluctuations that are difficult to immediately distinguish, yet structurally it contains certain “turning regions.”

When RDP is applied, the result becomes far more interpretable. Minor oscillations are removed, the visual clutter of the signal is reduced, and a simplified structure remains that preserves only the major directional changes. As the threshold value increases, an even more simplified version of the curve emerges—demonstrating that RDP’s degree of simplification is adjustable.

This stage makes it clear that RDP not only simplifies the curve but also produces a concise summary of the curve’s structural “character.”

Figure 2: RDP Processing in 2D Step-by-step

Increasing the Dimensionality: Why Does RDP Behave the Same in 3D?

The transition from 2D to 3D is important for illustrating RDP’s dimension-agnostic nature. Even if the embedding space has 768 dimensions, the problem is still geometrically the same—just a higher-dimensional version of it.

In the 3D example, we construct an extension of the previous signal: a path oscillating along the x, y, and z axes. Although the signal appears much more complex, the result obtained after applying RDP is strikingly familiar. Again, small variations are removed and only the prominent turns are preserved. RDP’s behavior does not degrade in any way with increasing dimensionality.

Figure 3: RDP Processing in 3D Step-by-step

A more fundamental question naturally arises here: 

Why does the algorithm behave identically in 2D and 3D?

The answer is actually very simple: the only mathematical operation RDP performs is computing the distance of a point to a line. This distance is well-defined in 2D, in 3D, and even in 1,000 dimensions. RDP is not tied to any special geometry; it relies solely on the notion of linear distance. This property is precisely what makes the transition into embedding space possible

The Geometric Nature of Embedding Space: A High-Dimensional Curve

Once the 3D example is established, we can turn to the main point: an embedding sequence is essentially a high-dimensional curve. Each word’s embedding vector consists of hundreds of dimensions, and when words are fed into the model in sequence, the resulting embedding stream forms a path composed of consecutive points—just like a trajectory in 2D or 3D.

The properties of this sequence are quite clear:
Words within the same theme occupy nearby regions in embedding space, causing the path to flow in small bends. When the theme or topic shifts, the embedding direction changes sharply. These directional changes are the geometric counterpart of what we refer to as “semantic transitions.”

Because proximity and distance in embedding space are tied to semantics, it is not surprising that an embedding sequence follows a fully geometric trajectory. In fact, modern embedding models are trained precisely to preserve this structure: placing similar meanings close together and pushing dissimilar meanings apart. For this reason, an embedding sequence naturally carries the geometric trace of a “semantic journey.”

Projecting the Embedding Path into 3D: Making It Visually Interpretable

Since high-dimensional embedding vectors cannot be directly visualized, we use dimensionality-reduction techniques such as UMAP to obtain a three-dimensional projection of the path. The goal of this projection is to make the structure of the embedding trajectory visually interpretable. In the 3D projection, the groups typically appear as clusters, and changes in direction become clearly visible when transitioning between them. This occurs because the words used in the experiment span distinct thematic categories such as math, music, food, technology, animals, and emotions, which naturally leads to cluster formation in the visualization. At this point, for example, the fact that the word “apple” lies physically closer to technology-related terms rather than to other fruits becomes a textbook illustration of how semantic similarity manifests as geometric proximity.

Figure 6: RDP Simplified Path in 3D

Throughout the experiment, we consistently observed that RDP behaves the same way across different data types. From 2D to 3D and all the way to high-dimensional embedding space, the algorithm’s behavior remains completely intact. This serves as a strong confirmation of RDP’s dimension-agnostic nature.

When examining embedding sequences, it becomes clear that semantic transitions indeed manifest as geometric changes in direction. Because RDP preserves only these directional changes, the embedding path becomes the “pure skeleton of semantic flow.” This skeleton provides a fast and intuitive way to analyze where meaning shifts occur in a text—especially in long sequences.

Our Mind

One of the most striking findings of this work was that using RDP in an NLP context is, in fact, entirely natural. The embedding sequence already possessed a geometric structure from the start; the only unnoticed aspect was that this structure was invisible. RDP makes this hidden geometric flow visible. The fact that semantic transitions emerge as “direction changes” is not only intuitive but also fully aligned with how embedding models operate. In this sense, RDP acts as a structural tool for semantic analysis.

KEY TAKEAWAYS

• Embedding sequences are, in essence, high-dimensional curves.

• RDP’s dimension-agnostic nature allows it to operate in embedding space exactly as it does in lower dimensions.

• Minor semantic variations are discarded, while major thematic transitions are preserved.

• The resulting simplified path represents the structural skeleton of semantic flow.

• The UMAP projection is purely for visualization; RDP itself is applied in the full high-dimensional space.

• The method is coarse but effective as a tool for producing a semantic summary.

CONCLUSION

This article has provided a step-by-step explanation of why RDP behaves consistently across data sequences of different dimensionalities and has demonstrated how semantic flow in embedding space can be effectively simplified. RDP is a lightweight and intuitive method with potential applications far beyond its classical role in computer vision. It can be regarded as a practical, interpretable, and powerful tool for revealing the underlying structure of embedding sequences.