Add Simple and Advanced Geometry Dataset Generators

[Schmeitzke]

Description
We would like to introduce two new dataset generators for geometry:

1. Simple Geometry Dataset for basic angle-finding tasks (e.g., interior angles of polygons).
2. Advanced Geometry Dataset for more complex problems (e.g., orthocenter, incircle radius, angle chasing using coordinate geometry).

Proposed Solution

1. Simple Geometry

   • Task Scope: Generate regular polygon or polygon-like problems (e.g., “Find the missing interior angle in a convex polygon”).
   • Implementation:
     • Configuration class (SimpleGeometryConfig) with parameters for number of sides, angle ranges, and dataset size.
     • Dataset class (SimpleGeometryDataset) that uses a known formula (sum of interior angles = 180°(n−2)) to verify correctness.
   • Output: A text prompt (the polygon angles) and a numeric answer (the missing angle).

2. Advanced Geometry

   • Task Scope: Coordinate-based geometry with Sympy for verification (e.g., orthocenter, incircle radius, angle measures).
   • Implementation:
     • Configuration class (AdvancedGeometryConfig) specifying coordinate ranges, possible tasks, and dataset size.
     • Dataset class (AdvancedGeometryDataset) that:
       • Randomly generates non-degenerate triangles.
       • Selects a task type (orthocenter, incircle, angle measure, etc.).
       • Uses Sympy geometry methods to compute exact/approximate solutions.
   • Output: A text prompt (e.g., triangle coordinates and requested property) and a numeric or symbolic solution.

Tasks

• Create configuration and dataset classes for both simple and advanced geometry.
• Implement unit tests for deterministic generation, correct structure, and verification logic.
• Ensure code aligns with existing repository conventions (e.g., ProceduralDataset base class, register_dataset).
• Document configuration parameters, usage examples, and test coverage.

Additional Context

• The simple dataset builds on existing arithmetic or logic dataset patterns by focusing on basic geometry.
• The advanced dataset extends geometric reasoning significantly, providing diverse and verifiable tasks for more robust model training.
• Both datasets will aid in testing a model’s ability to handle numeric computations, symbolic reasoning, and multi-step geometry tasks.

[andreaskoepf]

Great proposal, please go ahead, assigned the task to you.

[andreaskoepf]

@Schmeitzke Thanks for the PR! We are now missing a good score_answer() implementation for these datasets, e.g. to generate a score between 0-1 depending on how far the answer-value is from the right value. Also the question should include a short hint about the desired precision for the answer, e.g. rounded to how many decimal digts. Please let me know if you are in adding the score_answer methods.