Emotion Vectors for Affective Computing

• Emotion Vectors (EVs) are continuous, multidimensional representations that mathematically encode emotional states using a geometric framework.
• The Coordinate Heart System employs eight anchor points within a unit disk to ensure complete coverage, enabling precise interpolation and mixing of emotions.
• The model integrates vector operations, conflict resolution, and a stability parameter to provide robust, real-time quantification of emotional blending and resilience.

Emotion Vectors (EVs) are continuous, multidimensional mathematical representations designed to encode emotional states for use in artificial intelligence, computational linguistics, affective computing, and multimodal emotion analysis. By providing a geometrically grounded structure and supporting well-defined vector operations, EVs enable systems to represent emotion in a way that supports interpolation, blending, intensity scaling, and principled conflict resolution. The Coordinate Heart System (CHS) exemplifies an advanced geometric framework for emotion representation, offering complete and gapless coverage of the emotion space and a robust mechanism for mapping natural language input to emotional states via coordinate mixing and vector arithmetic (Al-Desi, 19 Jul 2025).

1. Geometric Foundations: Eight-Core Emotion Coordinates

The Coordinate Heart System proposes a unit disk model in which seven primary emotions are positioned on the unit circle ($S^1$), each defined by a unique angular coordinate $\theta_i \in [0, 360^\circ)$ and associated with a vector representation $E_i = (\cos \theta_i, \sin \theta_i)$. The model positions "Love" at the origin, serving as a neutral or baseline state:

• Guilt ($0^\circ$): $(1.0,\, 0.0)$
• Anger ($90^\circ$): $(0.0,\, 1.0)$
• Pride ($180^\circ$): $(-1.0,\, 0.0)$
• Joy ($270^\circ$): $(0.0,\,-1.0)$
• Fear ($300^\circ$): $(0.5,\,-0.866)$
• Sadness ($330^\circ$): $(0.866,\,-0.5)$
• Disgust ($240^\circ$): $(-0.5,\,-0.866)$
• Love: $(0.0,\,0.0)$

This structure ensures all pure and composite emotional states can be mapped within the closed unit disk $D = \{(x,\,y): x^2 + y^2 \leq 1\}$, generalizing classic emotion-circle or circumplex models to achieve full geometric coverage. The inclusion of the origin enables representation of affective neutrality or compositional emotion states approaching "zero" intensity (Al-Desi, 19 Jul 2025).

2. Mathematical Guarantees: Convex Hull Coverage and Systematic Completeness

The model rigorously demonstrates (convex-hull argument) that five equally spaced emotion coordinates are insufficient; five points on $S^1$ leave $72^\circ$ angular gaps such that intermediate emotional directions cannot be reached by convex combinations of the base vectors. Adding the origin as an eighth "anchor" and using seven peripherally placed emotions shrink maximum gaps to $360^\circ/7 \approx 51^\circ$. Further, the radial anchor (Love) combined with the peripherally distributed coordinates eliminates all representational blind spots. The formal proof shows that the convex hull of the eight bases exactly fills the unit disk, guaranteeing that any blended or intermediate emotional state can be expressed as a convex combination of these anchors (Al-Desi, 19 Jul 2025).

3. Vectorial Operations: Mixing, Distance, and Conflict

a) Emotion Mixing (Linear Interpolation and Weighted Summation)

Given vectors $E_A = (x_A, y_A)$, $E_B = (x_B, y_B)$ and intensities $I_A, I_B \geq 0$, mixed emotion states are computed as: $$t = \frac{I_B}{I_A + I_B}, \quad v = (x_A + t(x_B - x_A),\, y_A + t(y_B - y_A))$$ For $n$ emotions, the system recursively cascades pairwise interpolations or computes a weighted sum.

b) Distance Metrics

The model provides both Euclidean and angular distances: $$d_\text{euclid}(P, Q) = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}, \quad d_\text{angular}(P, Q) = \arccos\!\left( \frac{x_1x_2 + y_1y_2}{\|P\|\|Q\|} \right)$$ where $\|P\| = \sqrt{x_1^2 + y_1^2}$.

c) Conflict Resolution

For directly opposed emotion pairs (e.g., joy/anger, guilt/pride), conflict is quantified as: $$I_{\min} = \min(I_j, I_k), \quad I_j^{\rm res} = \max(0, I_j - I_{\min}), \quad I_k^{\rm res} = \max(0, I_k - I_{\min}), \quad C_{\rm penalty} += I_{\min}$$ This mechanism annihilates opposing intensities where possible and accumulates penalty scores referencing psychological conflict load.

4. Stability Modeling: The $S$ Parameter

A scalar stability parameter $S \in [0,1]$, interpreted as “psychological resilience,” is defined as: $$S = 1.0 - E_\mathrm{drain} - C_\mathrm{drain} - X_\mathrm{drain}$$ where:

• $$E_\mathrm{drain} = \max(0, \sum_{i=1}^8 I_i^\text{res} - I_{capacity})$$ quantifies affective overload,
• $C_\mathrm{drain}$ accumulates penalties from conflict resolution,
• $X_\mathrm{drain}$ captures contextually induced drains (exogenous, non-emotional loads).

Example (single intense emotion, $I_1 = 1.0$, capacity $=0.5$): $E_{\rm drain} = 0.5$, $C,X = 0$, so $S = 0.5$.

Example (conflict: joy $0.7$ + sadness $0.7$): $I_\mathrm{res} = 0$, $C = 0.7$, hence $S = 0.3$.

The model thus formalizes not only the geometry of the emotion state but also a global homeostatic parameter $S$ integrating total intensity, conflict, and context, offering a quantitative basis for psychological state assessment (Al-Desi, 19 Jul 2025).

5. Natural Language Mapping and Temporal Smoothing

The CHS pipeline for language-to-emotion mapping is as follows:

1. LLM Analysis: A LLM processes input text to estimate raw emotional scores $\{I_i^{\rm raw}\}$ and contextual drain $X_{\rm raw}$.
2. Conflict Resolution: Raw scores are adjusted ($I_i^{\rm res}$; $C_{\rm drain}$ updated).
3. Normalization/Mixing: Intensities are normalized and the final coordinates $(x, y)$ are computed as a weighted sum over $E_i$.
4. Stability Computation: $E_{\rm drain}$, $X_{\rm drain}$, and $S$ are computed.

For dynamic tracking across time, the CHS introduces hybrid temporal smoothing. Given two states, $E_1, S_1$ (previous) and $E_2, S_2$ (new), smoothing weight $w = (S_1+\epsilon)/(S_1+S_2+2\epsilon)$ ($\epsilon \ll 1$) yields

$$x_s = w x_1 + (1-w) x_2,\quad y_s = w y_1 + (1-w) y_2,\quad I_{s,i} = w I_{1,i} + (1-w) I_{2,i},\quad S_s = w S_1 + (1-w) S_2$$

This enables real-time, stability-weighted interpolation of affective state, supporting fine-grained temporal emotion modeling.

6. Worked Examples and Computational Implications

Example 1: Emotion Mixing

For interpolation between joy $(0, -1)$ and sadness $(0.866, -0.5)$ with $I_\text{joy}=0.6$, $I_\text{sad}=0.4$: $$t = 0.4/(0.6+0.4) = 0.4;\quad v_x = 0 + 0.4 \times 0.866 = 0.3464;\quad v_y = -1 + 0.4\times(0.5) = -0.8;$$ Result: $(0.3464, -0.8)$.

Example 2: Stability-Weighted Smoothing

Previous state: $(0.3, -0.7), S_1 = 0.8$; new raw state: $(0.1, -0.6), S_2 = 0.4$; $\epsilon = 10^{-3}$: $$w = 0.801/(0.801+0.401) \approx 0.6667;\quad x_s = 0.2333;\; y_s = -0.6667;\; S_s = 0.6667$$

This framework enables robust modeling of psychologically conflicted and dynamically shifting states, outperforming categorical models particularly in the context of AI emotion recognition, conflict identification, and the quantification of emotion blends (Al-Desi, 19 Jul 2025).

7. Significance and Systemic Integration

The emotion vector formalism as instantiated in the Coordinate Heart System synthesizes geometric completeness, vectorial algebra, stability-aware affect regulation, and advanced natural language interpretability. Unlike traditional categorical or lexicon-based emotion models, the CHS/EV methodology guarantees full coverage of the emotion disk, provides explicit computational routines for blending, conflict, and load, and integrates stability as a formally defined scalar resilient to contextual and emotional drain. The approach is both mathematically rigorous and practically amenable to LLM-driven, real-time AI systems, opening new avenues in interpretable affective modeling, emotion-driven reasoning, and the operationalization of psychological well-being constructs within artificial agents (Al-Desi, 19 Jul 2025).