FormalAnalyticGeo: A Neural-Symbolic Based Framework for Multimodal Analytic Geometry Problem Generation

# FormalAnalyticGeo: A Neural-Symbolic Based Framework for Multimodal Analytic Geometry Problem Generation
Source: [https://arxiv.org/html/2607.12982](https://arxiv.org/html/2607.12982)
###### Abstract\.

Math reasoning has achieved significant progress with the rapid advancement of Multimodal Large Language Models \(MLLMs\), however analytic geometry remains largely underexplored, primarily due to the scarcity of annotated samples\.

Existing diagram generation approaches struggle with analytic geometry: template methods cannot handle constraint\-driven layouts, and generative models lack the geometric precision to render annotated conic curves correctly\.

We present FormalAnalyticGeo, a scalable framework for fully automatic generation of multimodal analytic geometry problems\. Leveraging the rigor of formal languages, we design the framework around CDL \(Condition Description Language\), a formal intermediate representation that bridges free\-form problem text with precise diagram rendering via a Signed Distance Field \(SDF\) engine\. The framework employs four specialized LLM components in sequence: a Generator that produces diverse analytic geometry problems, a Formalizer that converts each problem into CDL for SDF\-based rendering, a Measurer that extracts ground\-truth answers through vision\-based measurement on the rendered diagrams, and a Quality Verifier that checks outputs at three stages\. Structured feedback from the Quality Verifier drives automatic retry, forming a closed loop that eliminates any need for human annotation\.

Applying FormalAnalyticGeo at scale yields AnalyticGeo7K, a dataset of over 7K verified multimodal problems, each with aligned text, diagram, formal annotation, and ground truth\.Experiments show that the generated problems achieve a median ground\-truth relative error of 0\.70%, with 82\.3% of answers falling within 5% of the exact symbolic solution\. Our framework and dataset will be publicly released\.

††copyright:none\{NoHyper\}††footnotetext:Xi’an Jiaotong\-Liverpool University \*Equal contribution\.†Corresponding author: qiufeng\.wang@xjtlu\.edu\.cn

## 1\.Introduction

Multimodal Large Language Models \(MLLMs\) have exhibited impressive performance in mathematical reasoning\. However, most existing research on multimodal mathematical reasoning primarily focuses on elementary plane geometry and solid geometry, leaving analytic geometry as a largely underexplored domain, while the capacity to jointly comprehend visual diagrams and complex algebraic structures continues to pose a considerable challenge, it demands an integrated understanding of equations, coordinate systems, geometric figures including conic sections, and the intricate spatial relationships between them\.

Despite its importance in both education and standardized examinations, the multimodal understanding of analytic geometry by MLLMs remains largely underexplored, in part because existing geometry benchmarks address plane geometry exclusively and the only large\-scale conic\-section resource \(Conic10K\(Wuet al\.,[2023](https://arxiv.org/html/2607.12982#bib.bib1)\)\) is purely text\-based\. Automated data generation is therefore essential, yet building a reliable generation framework for analytic geometry poses methodological challenges that existing approaches have not addressed\.

![Refer to caption](https://arxiv.org/html/2607.12982v1/Comparison.png)Figure 1\.Diagram generation for the same analytic geometry problem\. \(a\) ChatGPT: the circle does not use the focal distance as its diameter, andPPlies on the circle but not on the hyperbola\. \(b\) Nano Banana: geometrically meaningless output\. \(c\) Gemini 3 Pro: distorted curves and inaccurate point placement\. \(d\) Ours: geometrically exact rendering with known coordinate\-to\-pixel mapping, enabling direct visual measurement of ground\-truth answers\.We identify three such challenges\.\(i\) Representation gap\.Generating multimodal problems requires a formal representation that captures both algebraic relationships and geometric constructions needed to render a diagram\. Existing geometry formalisms \(Inter\-GPS\(Luet al\.,[2021](https://arxiv.org/html/2607.12982#bib.bib2)\), FormalGeo\(Zhanget al\.,[2024d](https://arxiv.org/html/2607.12982#bib.bib3)\)\) target plane geometry and cannot express coordinate systems or conic curves; Conic10K’s\(Wuet al\.,[2023](https://arxiv.org/html/2607.12982#bib.bib1)\)Assertional Logic encodes algebraic relations but lacks the construction primitives required for rendering\.\(ii\) Rendering gap\.Analytic geometry diagrams involve algebraically defined curves embedded in a coordinate system\. Existing tools such as GeoGPT4V\(Caiet al\.,[2024](https://arxiv.org/html/2607.12982#bib.bib4)\)are restricted to Euclidean constructions; MAVIS\(Zhanget al\.,[2024c](https://arxiv.org/html/2607.12982#bib.bib13)\)spans analytic geometry but lacks formal text–diagram verification\. Traditional plotting libraries require all positions to be pre\-computed, offering no mechanism for constraint\-driven layout\.\(iii\) Verification gap\.Existing generation approaches are single\-pass, accepting outputs without cross\-stage verification; errors at any stage propagate undetected into the final dataset\.

We propose FormalAnalyticGeo, a scalable framework that addresses all three challenges through formal language\. To close the*representation gap*, we design CDL \(Condition Description Language\), a formal intermediate representation that captures both algebraic relations and geometric constructions, enabling automatic consistency checking\. To close the*rendering gap*, we develop an SDF\-based rendering engine that compiles CDL programs into differentiable Signed Distance Fields, unifying constraint solving and diagram rendering so that under\-determined positions are resolved via gradient descent\. To close the*verification gap*, a Quality Verifier checks each stage’s output at three gate positions and returns structured feedback for automatic retry, forming a closed loop that eliminates any need for human annotation\. The framework comprises four stages: a Generator produces diverse problems, a Formalizer translates each into CDL, the SDF engine renders a diagram, and a Measurer extracts ground\-truth answers through vision\-based measurement on the rendered diagram\. Applying FormalAnalyticGeo at scale yields over 7K verified problems, each comprising a natural language question, a standardized diagram, a CDL annotation, and a visually extracted ground\-truth answer, with a median relative error of 0\.70%\.

Our main contributions are as follows:

- •A closed\-loop generation framework: FormalAnalyticGeo decomposes multimodal analytic geometry data generation into specialized stages connected by a formal language, with feedback\-driven quality gates eliminating the need for human annotation\.
- •A formal language for analytic geometry: CDL bridges natural language problem descriptions and geometric rendering, enabling automatic syntactic validation and completeness checking\.
- •An SDF\-based rendering engine: SDF Engine compiles CDL into differentiable Signed Distance Fields, unifying constraint solving and diagram rendering for exact analytic curves\.
- •A large\-scale multimodal dataset: AnalyticGeo7k as a product of the framework, it covers over 7K verified problems with ablation studies validating each component\.
- •Comprehensive Empirical Analysis: We conduct comprehensive experiments on 8 state\-of\-the\-art models and providing actionable insights for advancing multimodal spatial and mathematical reasoning\.

## 2\.Related Work

We review related work in three areas: multimodal mathematical reasoning benchmarks, geometry data generation, and multi\-agent systems and tool use\.

### 2\.1\.Multimodal Math Reasoning Benchmarks

Recent benchmarks have driven rapid progress in multimodal mathematical reasoning\. MathVista\(Luet al\.,[2024](https://arxiv.org/html/2607.12982#bib.bib18)\)consolidates 28 existing datasets into a 6,141\-problem suite and shows GPT\-4V\(OpenAI,[2024a](https://arxiv.org/html/2607.12982#bib.bib20)\)achieves only 49\.9% versus 60\.3% for humans\. MathVerse\(Zhanget al\.,[2024b](https://arxiv.org/html/2607.12982#bib.bib17)\)further reveals that some MLLMs perform*better*when diagrams are withheld, indicating that models routinely ignore visual content\. MATH\-Vision\(Wanget al\.,[2024](https://arxiv.org/html/2607.12982#bib.bib16)\)extends this to 3,040 competition\-level problems across 16 subjects including analytic geometry, with GPT\-4V\(OpenAI,[2024a](https://arxiv.org/html/2607.12982#bib.bib20)\)scoring only 23\.98%\. DynaMath\(Zouet al\.,[2024](https://arxiv.org/html/2607.12982#bib.bib19)\)constructs 5,010 variants from 501 seed problems to evaluate VLM robustness under systematic visual perturbations\.

For plane geometry, GeoQA\(Chenet al\.,[2021](https://arxiv.org/html/2607.12982#bib.bib5)\), UniGeo\(Chenet al\.,[2022](https://arxiv.org/html/2607.12982#bib.bib29)\), Geometry3K\(Luet al\.,[2021](https://arxiv.org/html/2607.12982#bib.bib2)\), and PGPS9K\(Zhanget al\.,[2023](https://arxiv.org/html/2607.12982#bib.bib14)\)together supply over 30,000 problems focused on Euclidean constructions \(triangles, circles, and quadrilaterals\), accompanied by symbolic annotations and program solutions\.

Analytic geometry remains largely uncovered\. GeoEval\(Zhanget al\.,[2024a](https://arxiv.org/html/2607.12982#bib.bib15)\)explicitly includes analytic geometry but contributes fewer than 1% of its corpus\.000GeoEval contains 28 analytic geometry problems out of 5,050 total\.Conic10K\(Wuet al\.,[2023](https://arxiv.org/html/2607.12982#bib.bib1)\)is the only large\-scale conic\-section dataset \(10,861 problems\), with formal AL annotations and multi\-step reasoning chains, but it is purely text\-based: it provides neither rendered diagrams nor a multimodal evaluation protocol\. FormalAnalyticGeo directly addresses this gap by providing an annotation\-free generation framework that pairs each problem with a standardized, SDF\-rendered diagram\.

### 2\.2\.Geometry Data Generation

Automated geometry data generation has taken several forms\. AlphaGeometry\(Trinhet al\.,[2024](https://arxiv.org/html/2607.12982#bib.bib12)\)synthesizes 100 million proof traces by combining a neural language model with a symbolic deduction engine \(DDAR\), achieving gold\-medalist performance on IMO plane\-geometry problems\. R\-CoT\(Lingeret al\.,[2025](https://arxiv.org/html/2607.12982#bib.bib21)\)proposes a two\-stage forward\-then\-reverse pipeline that generates diagram descriptions before deriving problems from them\. These approaches are confined to Euclidean proof tasks and produce no multimodal pairs suitable for algebraic or metric reasoning\.

More recent work targets multimodal geometry data directly\. GeoGPT4V\(Caiet al\.,[2024](https://arxiv.org/html/2607.12982#bib.bib4)\)uses GPT\-4\(OpenAI,[2024a](https://arxiv.org/html/2607.12982#bib.bib20)\)and GPT\-4V\(OpenAI,[2024a](https://arxiv.org/html/2607.12982#bib.bib20)\)with Wolfram Alpha to generate 4\.9K plane\-geometry problems with diagram images\. G\-LLaVA\(Gaoet al\.,[2025](https://arxiv.org/html/2607.12982#bib.bib11)\)converts formal logic representations to Geo170K\(Gaoet al\.,[2025](https://arxiv.org/html/2607.12982#bib.bib11)\)multimodal pairs, while TrustGeoGen\(Fuet al\.,[2026](https://arxiv.org/html/2607.12982#bib.bib22)\)introduces formal\-verification guarantees for a Euclidean geometry data engine\. NeSyGeo\(Wuet al\.,[2025](https://arxiv.org/html/2607.12982#bib.bib10)\)proposes a neuro\-symbolic framework that defines an entity\-attribute\-relation DSL and combines symbolic construction with LLM\-based question generation to synthesize 100K plane\-geometry caption and reasoning samples\. AutoGeo\(Huanget al\.,[2025](https://arxiv.org/html/2607.12982#bib.bib9)\)and the concurrent Socratic\-Geo\(Jiaoet al\.,[2026](https://arxiv.org/html/2607.12982#bib.bib8)\)further demonstrate multi\-agent pipelines for plane\-geometry diagram synthesis\. Despite their advances, all of these systems target Euclidean or plane geometry exclusively\.

Geometry formal languages and formalization methods underpin rendering and reasoning pipelines\. Inter\-GPS\(Luet al\.,[2021](https://arxiv.org/html/2607.12982#bib.bib2)\)defines 91 geometric predicates for plane\-figure logic forms; FormalGeo\(Zhanget al\.,[2024d](https://arxiv.org/html/2607.12982#bib.bib3)\)extends this to 88 predicates and 196 theorems\. AutoGPS\(Pinget al\.,[2026](https://arxiv.org/html/2607.12982#bib.bib26)\)introduces a Multimodal Problem Formalizer that extracts formal representations from geometry diagrams and text, feeding a Deductive Symbolic Reasoner for interpretable, step\-by\-step problem solving\. Conic10K’s\(Wuet al\.,[2023](https://arxiv.org/html/2607.12982#bib.bib1)\)AL formalizes conic\-section problems with 90 operators, but AL encodes only algebraic relations and lacks the geometric construction information required to render diagrams \(e\.g\., explicit point positions, intersections, tangent lines\)\. Our CDL extends AL with construction primitives that drive SDF\-based rendering\. To the best of our knowledge, no prior work has addressed analytic geometry data generation for multimodal benchmarks\.

### 2\.3\.LLM Agents and Tool Use

LLM\-based agents, characterized by iterative perception, reasoning, and action, have become a dominant paradigm for complex task automation\. The ReAct\(Yaoet al\.,[2023](https://arxiv.org/html/2607.12982#bib.bib7)\)framework structures agent behavior into interleaved reasoning traces and tool\-invocation actions, grounding language model outputs in external observations\. Multi\-agent platforms such as AutoGen\(Wuet al\.,[2024](https://arxiv.org/html/2607.12982#bib.bib23)\)and MetaGPT\(Honget al\.,[2024](https://arxiv.org/html/2607.12982#bib.bib24)\)extend this to collaborative pipelines where specialized agents with distinct roles coordinate to solve tasks beyond the capacity of a single model\.

Tool\-augmented reasoning is particularly critical for tasks requiring grounded computation\. Toolformer\(Schicket al\.,[2023](https://arxiv.org/html/2607.12982#bib.bib28)\)demonstrates that LLMs can autonomously learn to invoke external APIs \(calculators, search engines\) at appropriate locations\. In multi\-agent data synthesis, MATRIX\(Tanget al\.,[2025](https://arxiv.org/html/2607.12982#bib.bib27)\)and AgenticMath\(Liuet al\.,[2026](https://arxiv.org/html/2607.12982#bib.bib25)\)show that specialized generator–verifier pipelines produce higher\-quality training data than single\-model generation\.

FormalAnalyticGeo adopts this multi\-component paradigm with a domain\-specific quality layer\. Unlike general\-purpose frameworks, our framework embeds a dedicated Quality Verifier equipped with geometry\-specific tools including CDL syntax validation, symbolic solvability checking, and cross\-component answer verification, applied at three gate positions\. This design provides closed\-loop feedback that drives upstream retries until each generated problem meets measurability and correctness thresholds, yielding annotation\-free ground truth for multimodal analytic geometry\.

![Refer to caption](https://arxiv.org/html/2607.12982v1/Framework.001.jpeg)Figure 2\.The framework comprises three specialized generative components and one dedicated verification component\. Each generative component operates in a task\-specific ReAct loop and accepts structured feedback from the Quality Verifier for retry; the Quality Verifier enforces data quality at three gate positions between stages\.

## 3\.FormalAnalyticGeo

FormalAnalyticGeo is a scalable framework that automatically generates multimodal analytic geometry problems with formally verified text–diagram consistency\. Unlike prior data\-generation approaches that rely on template\-based synthesis or single\-model bootstrapping, FormalAnalyticGeo decomposes the task into specialized stages connected by a formal intermediate language and enforced by quality gates with feedback\-driven retry\. We first give an overview of the framework, then describe Condition Description Language, a formal language that bridges natural language and diagram rendering, and the SDF\-based rendering engine\. Finally, we detail the LLM components and quality gates that drive the framework end\-to\-end\.

### 3\.1\.Framework Overview

FormalAnalyticGeo is organized as a sequential system in which four specialized LLM components and one deterministic rendering engine collaborate to produce multimodal analytic geometry problems\. Each problem comprises four aligned artifacts: a natural language problem text, a formal CDL annotation, a standardized geometric diagram, and a ground\-truth answer\. The framework proceeds as follows: the Generator produces a problem together with its conic equation, the Formalizer translates the problem into a CDL program \(Section[3\.2](https://arxiv.org/html/2607.12982#S3.SS2)\), the SDF engine renders a diagram from the CDL \(Section[3\.3](https://arxiv.org/html/2607.12982#S3.SS3)\), and the Measurer extracts the ground\-truth answer by measuring the rendered diagram\.

A key design principle is*closed\-loop quality control*\. A dedicated Quality Verifier checks each component’s output at three gate positions using task\-specific verification tools\. When a check fails, the Quality Verifier returns structured feedback that pinpoints the error and suggests a correction; the upstream component then retries with this feedback as additional context\. This feedback\-driven retry mechanism is critical, as unverified outputs can actively degrade downstream data quality\. Problems that exhaust all retries are discarded, ensuring that only verified data enters the final dataset\.

All inter\-agent communication passes through structured formats \(CDL programs, measurement instructions, verification verdicts\), and the entire framework runs without human intervention\.

### 3\.2\.Condition Description Language

The framework requires a formal intermediate representation that is both human\-readable and machine\-renderable\. Existing formal languages for geometry, such as Assertional Logic\(Wuet al\.,[2023](https://arxiv.org/html/2607.12982#bib.bib1)\), were designed for semantic parsing rather than diagram synthesis and lack a direct mapping to visual primitives\. We propose CDL \(Condition Description Language\), a declarative formal language in which every statement maps to either a visual element or a differentiable constraint in the rendering engine\. Renderability is the primary design goal: every valid CDL program can be deterministically compiled into an SDF scene\.

Statement Categories\.A CDL program is a sequence of semicolon\-separated statements organized into seven categories\.*Declarations*and*Equations*define geometric objects and their algebraic equations, each creating an SDF primitive with determined parameters\.*Coordinates*fix point positions as non\-learnable constants\.*Derived objects*such asFocus,Intersection, andDirectrixcompute new entities from existing ones via conic\-section formulas at parse time\.*Predicates*and*Metric constraints*assert geometric relationships \(e\.g\.,PointOnCurve,IsTangent\) and numeric measurements \(e\.g\.,Distance,Slope\), each mapping to a differentiable loss function for the SDF optimizer \(Section 3\.3\)\.*Inequality constraints*restrict parameter ranges for well\-posedness\.

Machine Verifiability\.CDL’s formal structure enables two levels of automatic quality control: syntactic validation by the CDL parser, which detects undeclared names and parsing failures with immediate feedback for the Formalizer’s self\-correction, and completeness checking by the Quality Verifier, which verifies that all geometric objects in the problem text are encoded in the CDL program \(Section 3\.4\)\.

Scope and Limitations\.The current CDL specification covers five primitive types \(line, circle, ellipse, parabola, hyperbola\), 12 derived\-object operators, and metric and predicate constraints sufficient for the five goal types in our dataset\. Region\-based rendering \(e\.g\., shading the area inside a curve and above a line\) is not yet supported, as it requires signed\-region SDF composition beyond point\-level constraints; we treat this as future work\.

### 3\.3\.SDF Engine

We now describe the rendering engine that compiles CDL programs into geometric diagrams\. Traditional plotting libraries such as Matplotlib and TikZ can render curves at given coordinates but treat constraint solving as an external responsibility: when a CDL program specifies that “pointPPlies on ellipseCCclosest to focusFF,” the user must resolvePP’s coordinates separately before plotting\. We instead adopt Signed Distance Fields \(SDFs\) as both the geometric representation and the rendering primitive, unifying constraint solving and rendering in a single differentiable framework\. Because SDFs are differentiable, under\-determined point positions can be resolved automatically via gradient descent over geometric constraints; because rendering reduces to thresholding on a pixel grid, the resulting coordinate\-to\-pixel mapping is exact – a property the Measurer later exploits for visual ground\-truth extraction; and because all geometric elements share the same SDF representation, conic curves, coordinate axes, directrix lines, and asymptotes are rendered through one unified mechanism without element\-specific drawing logic\.

Formally, an SDF is a scalar functionF:ℝ2→ℝF:\\mathbb\{R\}^\{2\}\\to\\mathbb\{R\}that assigns to every point𝐱\\mathbf\{x\}the signed distance to the nearest boundary of a geometric shape:

\(1\)F​\(𝐱\)=\{−d​\(𝐱,∂Ω\)if​𝐱∈Ω,d​\(𝐱,∂Ω\)if​𝐱∉Ω,F\(\\mathbf\{x\}\)=\\begin\{cases\}\-d\(\\mathbf\{x\},\\partial\\Omega\)&\\text\{if \}\\mathbf\{x\}\\in\\Omega,\\\\ \\phantom\{\-\}d\(\\mathbf\{x\},\\partial\\Omega\)&\\text\{if \}\\mathbf\{x\}\\notin\\Omega,\\end\{cases\}whered​\(𝐱,∂Ω\)=inf𝐲∈∂Ω‖𝐱−𝐲‖d\(\\mathbf\{x\},\\partial\\Omega\)=\\inf\_\{\\mathbf\{y\}\\in\\partial\\Omega\}\\\|\\mathbf\{x\}\-\\mathbf\{y\}\\\|\. The zero\-level set\{𝐱:F​\(𝐱\)=0\}\\\{\\mathbf\{x\}:F\(\\mathbf\{x\}\)=0\\\}recovers the shape boundary exactly\.

SDF Primitives\.We implement SDF primitives for the five geometric element types in analytic geometry: line, circle, ellipse, parabola, and hyperbola\. Lines and circles admit standard closed\-form SDFs\. For ellipses and parabolas, we adopt analytical cubic solvers that reduce the closest\-point problem to a depressed cubic solvable via Cardano’s formula\. For hyperbolas, where no closed\-form closest\-point solution exists, we use the implicit approximationSDFhyp​\(𝐱\)≈\|f​\(𝐱\)\|/‖∇f​\(𝐱\)‖\\text\{SDF\}\_\{\\text\{hyp\}\}\(\\mathbf\{x\}\)\\approx\{\|f\(\\mathbf\{x\}\)\|\}/\{\\\|\\nabla f\(\\mathbf\{x\}\)\\\|\}, wheref​\(𝐱\)=x2/a2−y2/b2−1f\(\\mathbf\{x\}\)=x^\{2\}/a^\{2\}\-y^\{2\}/b^\{2\}\-1is the implicit equation and∇f\\nabla fits gradient; this first\-order approximation provides sub\-pixel accuracy near the curve and converges to the exact SDF on the zero\-level set\. All primitives are implemented as PyTorch modules with batch dimensions for GPU\-accelerated parallel optimization\.

Constraints and Optimization\.Geometric constraints from the CDL program are compiled into differentiable loss functions over the set of geometric elementsE=\{e1,e2,…,eN\}E=\\\{e\_\{1\},e\_\{2\},\\ldots,e\_\{N\}\\\}\. Each constraintcic\_\{i\}maps a configuration to a non\-negative scalar \(zero when satisfied\), and the optimizer seeks the configurationE∗E^\{\*\}that minimizes the total loss:

\(2\)L​\(E\)=∑i=1mci​\(E\)\+λ​∑j<k\[max⁡\(0,τ−‖ej−ek‖\)\]2,L\(E\)=\\sum\_\{i=1\}^\{m\}c\_\{i\}\(E\)\+\\lambda\\sum\_\{j<k\}\\bigl\[\\max\\bigl\(0,\\;\\tau\-\\\|e\_\{j\}\-e\_\{k\}\\\|\\bigr\)\\bigr\]^\{2\},where the second term is a crowd regularization penalty that prevents elements from collapsing into degenerate configurations by penalizing pairs closer than a thresholdτ\\tau\. We optimizeLLwith AdamW and cosine\-annealing learning rate scheduling over a batch ofBBparallel initializations, selecting the sample with the lowest loss\. For fully determined problems where all coordinates are resolved symbolically from the CDL, optimization is skipped entirely\.

RenderingThe boundary of each shape is extracted by sampling the SDF on anN×NN\\times Npixel grid and thresholding at the zero\-level set\. All geometric elements – conic curves, coordinate axes, directrix lines, and asymptotes – are rendered through this unified SDF mechanism, ensuring pixel\-level geometric consistency across all element types\. The viewport metadata \(coordinate bounds, pixel size\) is recorded alongside each image as a sidecar file, providing exact coordinate\-to\-pixel mappings that the Measurer later uses for visual measurement\.

### 3\.4\.Generator

Given target objects \(line, circle, ellipse, hyperbola, or parabola\), the Generator produces a natural language problem together with their polynomial equations\. It operates in a ReAct loop with four domain\-specific tools: a seed retrieval tool that samples real\-world raw text from the Conic10K\(Wuet al\.,[2023](https://arxiv.org/html/2607.12982#bib.bib1)\)dataset, a property enumeration tool that lists the geometric properties derivable from a given equation via SymPy, and two validation tools \(rule\-based and symbolic\) that verify the generated problem’s self\-consistency\. The Generator supports two generation paths: a*forward*path in which it freely composes a problem, and a*reverse*path in which it grounds the problem in symbolically verified properties from the seed library\. The reverse path is essential because LLMs often generate problems with incorrect geometric relationships when composing freely; anchoring the generation in SymPy\-verified properties significantly reduces such errors\. Each problem also carries an explicit information partition: letItI\_\{t\}denote the conditions stated in the problem text andIvI\_\{v\}those that must be read from the diagram\. The constraintIv∩It=∅I\_\{v\}\\cap I\_\{t\}=\\emptysetensures that solving the problem requires both modalities\.

### 3\.5\.Formalizer

The Formalizer translates the natural language problem into a CDL program at temperature 0\. A key design is*grammar prompting*: the CDL specification provided to the Formalizer is dynamically tailored to the detected curve type, pruning irrelevant operators \(e\.g\., removingAsymptotefor ellipses\) and supplying curve\-specific few\-shot examples\. This reduces attention dilution over the full operator set and prevents the Formalizer from hallucinating unsupported constructs\. The output CDL is immediately parsed by the CDL parser; any syntactic error triggers an automatic retry with the parser’s error message as feedback, before the result reaches the Quality Verifier\.

### 3\.6\.Measurer ReAct Loop

The Measurer extracts ground\-truth answers by measuring the rendered diagram\. A critical challenge is*information leakage*: if the Measurer receives both the diagram and the curve equations, the underlying LLM bypasses visual measurement and solves the problem analytically, producing answers that are not true visual ground truth\. We address this through*physical information isolation*by splitting the Measurer into two stages\. The Task Extractor reads only the problem text and outputs a structured measurement instruction – specifying what to measure and which entities to locate – that contains no equations or coordinates\. The Visual Measurer receives only this instruction and the diagram image; it never sees the problem text or equations\. The Visual Measurer operates in a ReAct loop with a suite of computer\-vision tools that cover coordinate mapping, distance and angle measurement, curve probing, and intersection detection\. Because we rendered the diagram ourselves, the viewport metadata \(coordinate bounds, pixel\-to\-coordinate mapping\) is known exactly, eliminating the axis\-reading and scale\-detection errors common in diagram understanding\.

### 3\.7\.Quality Gates

Quality Verifier\.A dedicated Quality Verifier enforces end\-to\-end data quality by independently verifying each stage’s output at three gate positions using task\-specific tools\. On failure, it returns structured feedback that pinpoints the error and suggests a correction; the upstream component retries \(up to two additional attempts\) with this feedback appended to its context, transforming quality control from a passive filter into an active closed\-loop correction process\. The three gates are:

- •Gate 1\(after problem generation\): verifies solvability by attempting a symbolic solution and confirms a finite answer exists\.
- •Gate 2\(after CDL annotation\): validates CDL syntax and checks that all geometric objects mentioned in the problem text are encoded in the CDL\.
- •Gate 3\(after visual measurement\): checks the answer range for plausibility and cross\-validates against a text\-only symbolic solution\.

Problems that fail all three attempts at any gate are discarded, ensuring that only verified data enters the final dataset\.

## 4\.AnalyticGeo7k

We apply the framework described in Section[3](https://arxiv.org/html/2607.12982#S3)to generate the AnalyticGeo7k dataset\. This section reports dataset statistics, quality, and diversity\.

![Refer to caption](https://arxiv.org/html/2607.12982v1/case.001.jpeg)Figure 3\.Three representative problems generated by FormalAnalyticGeo, with Measurer answers and analytical solutions\.### 4\.1\.Generation Setup

All four components \(Generator, Formalizer, Measurer, and Quality Verifier\) use Gemini\-3\.1\-Pro\-Preview via an OpenAI\-compatible API endpoint\. The framework runs in streaming mode: each problem is generated, annotated, rendered, measured, and verified end\-to\-end before the next problem begins\. Target curve types cycle uniformly among ellipse, hyperbola, parabola, and circle; goal types cycle among chord length, area, distance, angle, and perimeter\.

The framework produced a total of 7,823 candidate problems\. Of these, 780 were discarded because they exhausted all quality\-gate retries without passing\. This was typically due to persistent CDL annotation errors or rendering failures, leaving a final dataset of 7,043 verified problems\. Each problem takes∼\{\\sim\}5 minutes to generate on average\. As the framework is fully automatic and requires no human annotation, the dataset can be scaled to any target size by running additional generation cycles; the 7,043\-problem release reported here represents an initial demonstration of the framework’s capability\.

### 4\.2\.Dataset Statistics

Table[1](https://arxiv.org/html/2607.12982#S4.T1)summarizes the dataset composition\. The 7,043 problems span four curve types and five goal types\. Each curve type contributes 18–32% of the total, ensuring broad coverage across conic sections\. Area and chord length are the most frequent goal types, while distance, angle, and perimeter provide complementary geometric reasoning challenges\.

Table 1\.Curve type×\\timesgoal type distribution \(number of problems\)\.AreaChordDist\.AnglePerim\.TotalCircle4855373314654362,254Ellipse5582034062213021,690Hyperbola4924282193523401,831Parabola2483072791392951,268Total1,7831,4751,2351,1771,3737,043

### 4\.3\.Quality Analysis

We evaluate dataset quality along two dimensions: framework reliability and ground\-truth accuracy\.

Framework Reliability\.Table[2](https://arxiv.org/html/2607.12982#S4.T2)reports pass rates and retry statistics at each quality gate\.All three gates achieve≥\\geq99% final pass rate\.Gate 1 reaches 99\.1% on the first try, indicating that the Generator’s built\-in validation tools effectively prevent unsolvable problems from entering the framework\. Gate 2 achieves 98\.2% first\-try, with the CDL parser’s syntactic auto\-repair resolving most issues before the Quality Verifier is invoked; only 1\.8% of problems require a Quality Verifier retry\.Gate 3 exhibits the highest retry rate \(4\.9%\), reflecting the inherent difficulty of cross\-validating visual measurements against symbolic solutions\. Problems that exhaust all retries are discarded and replaced\.

Table 2\.Quality gate pass rates and retry statistics\.Ground\-Truth Accuracy\.To evaluate the accuracy of the Measurer’s visual measurements, we sample 164 problems from the dataset \(balanced across curve and goal types\) and manually compute their exact symbolic answers\. For each sampled problem, we derive the analytical solution using standard analytic geometry techniques and compute the relative errorϵ=\|ameas−aexact\|/\|aexact\|\\epsilon=\|a\_\{\\text\{meas\}\}\-a\_\{\\text\{exact\}\}\|/\|a\_\{\\text\{exact\}\}\|\.

Across the 164 sampled problems, the Measurer achieves a median relative error of 0\.70%\(mean 2\.84%\), with 82\.3% of answers falling within 5% of the exact solution\. Figure[4](https://arxiv.org/html/2607.12982#S4.F4)shows the accuracy breakdown by curve type and goal type\. Most categories achieve\>\>95% accuracy \(measured as1−ϵ¯1\-\\bar\{\\epsilon\}\)\.Parabola–area problems are the hardest category\(93\.97% accuracy\)\. We attribute this to multi\-step visual reasoning: measuring a parabolic segment area requires identifying the tangent line, computing intercepts, and deriving the enclosed area, with each step accumulating measurement error from successive tool calls\. This category\-specific pattern suggests a fundamental trade\-off in visual ground\-truth extraction: single\-step measurements \(distances, angles\) are highly accurate, while multi\-step derivations amplify per\-step errors\. Mitigating this accumulation is a direction for future work; potential approaches include intermediate result verification between tool calls and symbolic cross\-checking of partial measurements before combining them into a final answer\. The 1% evaluation threshold in Table[5\.1](https://arxiv.org/html/2607.12982#S5.SS1)remains reliable under this noise level, as the median GT error \(0\.70%\) is well below the threshold\.

![Refer to caption](https://arxiv.org/html/2607.12982v1/radar_curve_goal_acc.png)Figure 4\.Per\-category ground\-truth accuracy \(1−ϵ¯1\-\\bar\{\\epsilon\}, zoomed to 85–100%\)\. Parabola–area is the hardest combination \(93\.97%\)\.

## 5\.Experiment

### 5\.1\.Experimental Results

We evaluate eight representative MLLMs on the AnalyticGeo7K benchmark under both image and text\-only input modes\. Table[5\.1](https://arxiv.org/html/2607.12982#S5.SS1)reports accuracy broken down by five goal types and four curve types\.

Challenging Nature of Analytic Geometry\.The best\-performing model, Gemini 3 Flash with image input, achieves an overall accuracy of 77\.6%, followed by GPT\-5\.2 \(73\.1%\) and Claude Opus 4\.5 \(72\.0%\)\. GPT\-4o reaches only 23\.2%, indicating that analytic geometry problems remain difficult even for frontier MLLMs\. Performance drops sharply in text\-only mode: the best text\-only result \(GPT\-5\.2, 42\.0%\) trails the best image result by over 35 percentage points, and weaker models such as Mistral Large \(18\.4%\) and GPT\-4o \(18\.8%\) fall below 20%\. This substantial gap suggests that jointly reasoning about coordinate systems, algebraic curves, and geometric relationships poses a significant challenge to current models\. Moreover, no single model dominates all fine\-grained categories: Claude Opus 4\.5 leads on Area \(79\.4%\) and Angle \(88\.0%\) but ranks only third on Chord, while GPT\-5\.2 leads on Distance \(81\.3%\) but falls behind on Area \(68\.0%\), indicating complementary strengths across goal types and curve families\.

Impact of Visual Input\.Image\-input accuracy substantially exceeds text\-only performance for all models, confirming that diagrams provide informative visual cues for analytic geometry reasoning\. Gemini 3 Flash exhibits the largest gap \(77\.6%→\\to36\.0%,−\-41\.6 pp\), while GPT\-5\.2 shows a smaller but still significant drop \(73\.1%→\\to42\.0%,−\-31\.1 pp\)\. The gap is especially pronounced on specific curve types: Gemini’s Circle accuracy drops from 85\.3% to 33\.0% \(−\-52\.3 pp\) without images\. Interestingly, the text\-only ranking partially reshuffles: GPT\-5\.2 overtakes Gemini 3 Flash \(42\.0% vs\. 36\.0%\), suggesting that some models rely more heavily on visual parsing while others maintain stronger algebraic reasoning in the absence of diagrams\.

Table 3\.Model performance on AnalyticGeo7K benchmark \(%\)\. Accuracy = answer within 1% relative error of ground truth\. Thebestandsecond bestresults per section are highlighted\.Goal TypeCurve TypeModelOverallAreaChordDist\.AnglePerim\.CircleEllipseHyper\.Parab\.\\rowcolororange\!10Image InputGPT\-4o\(OpenAI,[2024b](https://arxiv.org/html/2607.12982#bib.bib31)\)23\.223\.814\.324\.137\.019\.424\.520\.523\.324\.1Claude Sonnet 4\(Anthropic,[2025b](https://arxiv.org/html/2607.12982#bib.bib32)\)49\.459\.542\.941\.459\.341\.960\.459\.034\.937\.9Qwen\-VL\-Max\(Baiet al\.,[2025](https://arxiv.org/html/2607.12982#bib.bib33)\)69\.069\.653\.970\.379\.974\.966\.580\.265\.471\.9Claude Opus 4\.5\(Anthropic,[2025a](https://arxiv.org/html/2607.12982#bib.bib34)\)72\.079\.460\.968\.588\.070\.079\.082\.361\.062\.9GPT\-5\.2\(OpenAI,[2025](https://arxiv.org/html/2607.12982#bib.bib35)\)73\.168\.064\.181\.379\.876\.472\.076\.073\.470\.5Gemini 3 Flash\(Google DeepMind,[2025](https://arxiv.org/html/2607.12982#bib.bib36)\)77\.674\.371\.078\.785\.378\.185\.376\.271\.468\.3\\rowcolororange\!10Text OnlyMistral Large\(Mistral AI,[2024](https://arxiv.org/html/2607.12982#bib.bib30)\)18\.426\.112\.216\.312\.524\.218\.624\.511\.519\.6GPT\-4o\(OpenAI,[2024b](https://arxiv.org/html/2607.12982#bib.bib31)\)18\.826\.712\.516\.76\.230\.819\.025\.011\.820\.0Claude Sonnet 4\(Anthropic,[2025b](https://arxiv.org/html/2607.12982#bib.bib32)\)25\.032\.512\.117\.339\.224\.428\.339\.017\.514\.1DeepSeek\-V3\.1\(DeepSeek\-AI,[2025](https://arxiv.org/html/2607.12982#bib.bib37)\)30\.933\.914\.229\.654\.232\.530\.943\.623\.826\.2Qwen\-VL\-Max\(Baiet al\.,[2025](https://arxiv.org/html/2607.12982#bib.bib33)\)35\.031\.014\.038\.264\.735\.132\.746\.828\.234\.6Gemini 3 Flash\(Google DeepMind,[2025](https://arxiv.org/html/2607.12982#bib.bib36)\)36\.043\.719\.424\.041\.450\.933\.049\.141\.215\.9Claude Opus 4\.5\(Anthropic,[2025a](https://arxiv.org/html/2607.12982#bib.bib34)\)38\.046\.018\.029\.957\.942\.538\.758\.132\.024\.4GPT\-5\.2\(OpenAI,[2025](https://arxiv.org/html/2607.12982#bib.bib35)\)42\.044\.620\.035\.862\.850\.740\.457\.039\.329\.0

### 5\.2\.Ablation Study

We study four design decisions in FormalAnalyticGeo, each isolating one framework component \(Tables[4](https://arxiv.org/html/2607.12982#S5.T4)and[5](https://arxiv.org/html/2607.12982#S5.T5)\)\.w/o Quality Verifierdisables all three quality gates and feedback\-driven retry loops, while retaining each component’s internal self\-validation tools, so that this variant isolates the contribution of cross\-component external verification\.w/o Seed Libraryremoves theseed\_pooltool from the Generator, eliminating in\-context exposure to verified seed problems; because the reverse path accounts for only 3\.7% of baseline outputs, the primary effect is the loss of implicit structural priors for forward\-path generation\.Formalizer Few\-shot Gradientre\-runs only the Formalizer on 200 existing problems at four few\-shot counts \(N∈\{0,4,8,10\}N\\in\\\{0,4,8,10\\\}\) to measure CDL annotation quality without re\-generating problems or diagrams\.w/o CV Toolsremoves all measurement tools from the Visual Measurer, forcing visual parameter estimation from the rendered diagram alone; this variant requires no new data collection since it is evaluated on 200 problems sampled from the 7,043\-problem dataset\.

Table 4\.Ablation study results \(200 problems per variant\)\.Table 5\.Formalizer few\-shot gradient \(200 problems, balanced curve types\)\.Removing all Quality Verifier gates causes a dramatic accuracy drop: within\-5% accuracy falls from 82\.3% to 45\.5% and mean relative error rises from 2\.84% to 35\.7%\. CDL completeness remains 100% under human review, indicating the degradation stems not from missing problem structure but from compounded visual measurement errors in the Visual Measurer that the gate’ssolve\_text\_onlyverification would have caught and triggered retries for\. This confirms that cross\-component external verification provides a substantial quality lift over each component’s self\-validation alone\.

Removing seed library access from the Generator increases the Gate 3 retry rate from 4\.9% to 26\.5%, reflecting that forward\-path problems are geometrically less well\-formed without in\-context structural priors from verified examples\. Within\-5% accuracy drops 9\.4 percentage points to 72\.9% and mean error rises to 9\.92%, despite Gate 1 and Gate 2 pass rates remaining unaffected, confirming that the seed library’s contribution lies not in enabling generation but in improving the geometric coherence of generated problems\.

Without CV tools, the Visual Measurer is forced to visually estimate geometric parameters and apply analytical formulas, compounding parameter\-reading errors into the final answer: a small misread of an ellipse’s semi\-axisaa, for example, propagates through the chord\-length formula to yield errors of 30–80%\. Within\-5% accuracy drops to 49\.0% and mean relative error rises to 40\.5%, a 33\.3 percentage point decline that directly quantifies the contribution of structured pixel\-level measurement to GT extraction precision\.

Table[5](https://arxiv.org/html/2607.12982#S5.T5)shows a monotone positive trend: L1 first\-try pass rate rises from 60\.0% to 84\.0% asNNincreases from 0 to 10, and average attempts decrease from 1\.58 to 1\.18\. The largest single gain \(\+14 pp\) occurs betweenN=4N=4andN=8N=8, where most curve types first receive at least three curve\-specific examples; below this threshold, certain types have only one example, insufficient for the Formalizer to infer CDL’s predicate names and argument ordering\.

## 6\.Conclusion

We presented FormalAnalyticGeo, a scalable framework that generates multimodal analytic geometry problems without human annotation\. By unifying CDL as a formal intermediate representation, SDF\-based rendering for geometrically exact diagrams, and a four\-component generation loop supervised by a Quality Verifier with feedback\-driven retries, the framework produces over 7K verified problems with a median ground\-truth error of 0\.70%\. Ablation experiments confirm that each component contributes measurably: removing the Quality Verifier drops within\-5% accuracy from 82\.3% to 45\.5%, and disabling CV tools raises mean error from 2\.84% to 40\.5%\. The CDL specification and SDF engine are designed to be extensible; broader analytic geometry categories such as locus problems and parametric equations can be incorporated by adding corresponding primitives without modifying the core framework\.

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- R\. Zhang, D\. Jiang, Y\. Zhang, H\. Lin, Z\. Guo, P\. Qiu, A\. Zhou, P\. Lu, K\. Chang, P\. Gao,et al\.\(2024b\)MathVerse: does your multi\-modal llm truly see the diagrams in visual math problems?\.arXiv preprint arXiv:2403\.14624\.Cited by:[§2\.1](https://arxiv.org/html/2607.12982#S2.SS1.p1.1)\.
- R\. Zhang, X\. Wei, D\. Jiang, Y\. Zhang, Z\. Guo, C\. Tong, J\. Liu, A\. Zhou, B\. Wei, S\. Zhang, P\. Gao, and H\. Li \(2024c\)MAVIS: mathematical visual instruction tuning\.External Links:2407\.08739,[Link](https://arxiv.org/abs/2407.08739)Cited by:[§1](https://arxiv.org/html/2607.12982#S1.p3.1)\.
- X\. Zhang, N\. Zhu, Y\. He, J\. Zou, Q\. Huang, X\. Jin, Y\. Guo, C\. Mao, Y\. Li, Z\. Zhu, D\. Yue, F\. Zhu, Y\. Wang, Y\. Huang, R\. Wang, C\. Qin, Z\. Zeng, S\. Xie, X\. Luo, and T\. Leng \(2024d\)FormalGeo: an extensible formalized framework for olympiad geometric problem solving\.External Links:2310\.18021,[Link](https://arxiv.org/abs/2310.18021)Cited by:[§1](https://arxiv.org/html/2607.12982#S1.p3.1),[§2\.2](https://arxiv.org/html/2607.12982#S2.SS2.p3.1)\.
- C\. Zou, X\. Guo, R\. Yang, J\. Zhang, B\. Hu, and H\. Zhang \(2024\)DynaMath: a dynamic visual benchmark for evaluating mathematical reasoning robustness of vision language models\.Cited by:[§2\.1](https://arxiv.org/html/2607.12982#S2.SS1.p1.1)\.

## Appendix ACDL Specification

This appendix provides the complete operator catalog of the Condition Description Language \(CDL\) introduced in Section[3\.2](https://arxiv.org/html/2607.12982#S3.SS2)\. A CDL program is a sequence of semicolon\-separated statements that collectively specify an analytic\-geometry diagram suitable for rendering\.

#### Declarations and Equations\.

Every geometric object must be declared with a name and a type before use\. Supported types includePoint,Origin\(fixed at the coordinate origin\(0,0\)\(0,0\)\),Line,Circle,Ellipse,Hyperbola,Parabola, andNumber\(an algebraic parameter\)\. AnEquationstatement binds an algebraic equation to a declared curve or line; the parser classifies the equation by discriminant analysis of the general quadratic formA​x2\+B​x​y\+C​y2\+D​x\+E​y\+F=0Ax^\{2\}\+Bxy\+Cy^\{2\}\+Dx\+Ey\+F=0and extracts the curve parameters automatically\. ACoordinatestatement fixes a point at a specified position\.

Table[A](https://arxiv.org/html/2607.12982#A1.SS0.SSS0.Px1)lists all 35 operators, organized into three categories: derived\-object operators, predicates, and metric constraints\.

Table 6\.Complete CDL operator catalog: 12 derived\-object operators, 8 predicates, and 15 metric constraints\.#### Derived Objects\.

Derived\-object operators compute new geometric entities from existing ones\. Operators that produce points \(e\.g\.,Focus,MidPoint\) evaluate symbolically at parse time from the parent curve’s parameters\.Intersectionis resolved analytically by substitution into the curve equation and may return multiple points via set\-assignment syntax:

\{A, B\} = Intersection\(l, G\)

Focus,Vertex, andDirectrixaccept an optionaldirectionparameter drawn from \{left,right,upper,lower\}\. It may be omitted when the curve admits a unique instance, e\.g\., the single focus of a parabola\. In Table[A](https://arxiv.org/html/2607.12982#A1.SS0.SSS0.Px1), square brackets denote optional arguments\.

Note thatProjectionmaps a point onto a coordinate axis \(or an axis\-aligned line\), whereasFootPointcomputes the perpendicular foot from a point onto an arbitrary line\.

CDL distinguishes three tangent\-related operators\.TangentOnPoint\(P, G\) computes the tangent line to curveGGat a pointPPthat already lies onGG, via implicit differentiation of the curve equation\.TangentOfPoint\(P, G\) finds tangent lines from an external pointPPtoGGusing discriminant analysis and may yield multiple solutions\.TangentPoint\(l, G\) returns the contact point\(s\) where a known tangent linelltouchesGG\.

#### Predicates\.

Each predicate asserts a geometric relationship and compiles to a differentiable SDF constraint whose satisfaction corresponds to a zero loss value\. For instance,PointOnCurve\(P, G\)penalizes the signed distance fromPPtoGG, whileIsTangent\(l, G\)enforces that the discriminant of the line–curve intersection system vanishes\.IsInTangentandIsOutTangentconstrain the distance between two circle centers to equal\|r1−r2\|\|r\_\{1\}\-r\_\{2\}\|orr1\+r2r\_\{1\}\+r\_\{2\}, respectively\. Predicates carry no explicit numeric target; they are satisfied when the constraint residual reaches zero\.

#### Metric Constraints\.

A metric constraint binds a geometric measurement to a target value, creating a loss term\|fmeasured−ftarget\|\|f\_\{\\text\{measured\}\}\-f\_\{\\text\{target\}\}\|\. The measurement function is evaluated from the current configuration:Distancecomputes Euclidean distance,Areauses the cross\-product formula, andAngleOfapplies the law of cosines\. Algebraic parameters declared asNumbermay appear as targets, enabling parametric constraints such asLength\(SemiMajorAxis\(G\)\) = a\.

#### Inequality Constraints and Auxiliary Wrappers\.

Inequality constraints \(e\.g\.,a \> 0;a \> b\) restrict the admissible ranges of algebraic parameters\. CDL also provides auxiliary wrappers for composing references inside other operators:

- •LineSegmentOf\(A, B\)— line segment throughAAandBB\.
- •OverlappingLine\(…\)— the full line through two points\.
- •RightPart\(G\),LeftPart\(G\)— individual branches of a hyperbola\.
- •TriangleOf\(A, B, C\),VectorOf\(A, B\)— point\-tuple wrappers for area/perimeter and dot\-product metrics\.

#### Worked Example\.

The following CDL program encodes the problem:*“From the left focusF1F\_\{1\}of the hyperbolax216−y225=1\\tfrac\{x^\{2\}\}\{16\}\-\\tfrac\{y^\{2\}\}\{25\}=1, draw a line tangent to the circlex2\+y2=16x^\{2\}\+y^\{2\}=16, touching it at pointTT\. ExtendF1​TF\_\{1\}Tto meet the right branch atPP, and letMMbe the midpoint ofF1​PF\_\{1\}P\.”*

G:Hyperbola;H:Circle;F1:Point;Z:Line

T:Point;P:Point;M:Point;O:Origin

Equation\(G\)=x^2/16\-y^2/25=1

Equation\(H\)=x^2\+y^2=16

Focus\(G,left\)=F1

TangentOfPoint\(F1,H\)=Z

TangentPoint\(Z,H\)=T

Intersection\(OverlappingLine\(LineSegmentOf\(F1,T\)\),

RightPart\(G\)\)=P

MidPoint\(LineSegmentOf\(F1,P\)\)=M

This nine\-statement program declares four points, one line, and two curves\. It derives the left focus symbolically, computes the tangent line via discriminant analysis, intersects the extended ray with the right branch of the hyperbola, and determines the midpointMM\. The CDL parser compiles these statements into constraint tuples that are consumed by the SDF\-based rendering engine\.

## Appendix BSigned Distance Field Formulation and Constraint Details

In this appendix, we provide a comprehensive mathematical formulation of the Signed Distance Field \(SDF\) representation, the differentiable loss functions for geometric constraints, the optimization procedure, and the boundary extraction method used in our diagram synthesis engine\. This material supplements the description in Section[3\.3](https://arxiv.org/html/2607.12982#S3.SS3)of the main paper\.

### B\.1\.SDF Definition and Basic Representations

A Signed Distance Field \(SDF\) is a scalar field that assigns to every point𝐱∈ℝ2\\mathbf\{x\}\\in\\mathbb\{R\}^\{2\}the signed distance to the closest point on the boundary of a geometric shape\. Formally, for a shapeΩ⊂ℝ2\\Omega\\subset\\mathbb\{R\}^\{2\}with boundary∂Ω\\partial\\Omega, the SDFF​\(𝐱\)F\(\\mathbf\{x\}\)is defined as:

F​\(𝐱\)=\{min𝐲∈∂Ω⁡‖𝐱−𝐲‖,if​𝐱​is outside​Ω,−min𝐲∈∂Ω⁡‖𝐱−𝐲‖,if​𝐱​is inside​Ω,F\(\\mathbf\{x\}\)=\\begin\{cases\}\\displaystyle\\min\_\{\\mathbf\{y\}\\in\\partial\\Omega\}\\\|\\mathbf\{x\}\-\\mathbf\{y\}\\\|,&\\text\{if \}\\mathbf\{x\}\\text\{ is outside \}\\Omega,\\\\\[11\.99998pt\] \\displaystyle\-\\min\_\{\\mathbf\{y\}\\in\\partial\\Omega\}\\\|\\mathbf\{x\}\-\\mathbf\{y\}\\\|,&\\text\{if \}\\mathbf\{x\}\\text\{ is inside \}\\Omega,\\end\{cases\}where the sign convention follows the common “outside positive, inside negative” rule\. Equivalently, the unsigned distance is given by:

d​\(𝐱,∂Ω\)=inf𝐲∈∂Ω‖𝐱−𝐲‖,d\(\\mathbf\{x\},\\partial\\Omega\)=\\inf\_\{\\mathbf\{y\}\\in\\partial\\Omega\}\\\|\\mathbf\{x\}\-\\mathbf\{y\}\\\|,and the sign is determined by a point\-in\-polygon test for polygons or by the side of the curve for lines and circles\.

In our framework, each geometric element is represented by its own SDF, which depends on a small set of continuous parameters\. TableLABEL:tab:sdf\_primitiveslists the SDF expressions for all primitives used in our rendering engine, covering both basic elements and analytic\-geometry curves\.

Measurer answer:10\.352810\.3528\(relative error0\.38%0\.38\\%\)\. The Measurer locatesF1F\_\{1\}andF2F\_\{2\}viapixel\_to\_coordat the labelled foci, measuresAAat\(2\.496,2\.594\)\(2\.496,2\.594\), and callscompute\_areaon the three vertices\.

Both conics share foci at\(±4,0\)\(\\pm 4,0\)\(sincec=25−9=4\+12=4c=\\sqrt\{25\-9\}=\\sqrt\{4\+12\}=4\)\. Their four intersection points lie at\(±2\.5,±3​32\)\(\\pm 2\.5,\\;\\pm\\frac\{3\\sqrt\{3\}\}\{2\}\); confocal conics meet at right angles\. The CDL declares both anEllipseand aHyperbola, and uses a cross\-curveTangentOnPointconstruction\.

![Refer to caption](https://arxiv.org/html/2607.12982v1/appendix_confocal.png)Figure 5\.Confocal ellipse \(blue\) and hyperbola \(red\) with shared foci, tangent atAA\.
### D\.2\.Upward Parabola and Circle

> “The parabolax2=4​yx^\{2\}=4yhas focusFFand directrixy=−1y=\-1\. A circle centred atM​\(0,3\)M\(0,3\)with radius33intersects the parabola atAAandBB\. Find the distance\|A​B\|\|AB\|\.”

#### CDL annotation\.

G:Parabola;C:Circle;F:Point;M:Point

A:Point;B:Point;ld:Line;O:Origin

Equation\(G\)=x^2=4\*y

Equation\(C\)=x^2\+\(y\-3\)^2=9

Focus\(G\)=F;Center\(C\)=M

Coordinate\(A\)=\(2\.828,2\)

Coordinate\(B\)=\(\-2\.828,2\)

PointOnCurve\(A,G\);PointOnCurve\(B,G\)

Equation\(ld\)=y=\-1

Analytical ground truth:Distance=4​2≈5\.6569=4\\sqrt\{2\}\\approx 5\.6569\.Measurer answer:5\.64375\.6437\(relative error0\.23%0\.23\\%\)\. The Measurer usesintersect\_curvesto locate the two parabola–circle crossings at\(2\.824,1\.997\)\(2\.824,1\.997\)and\(−2\.831,2\.003\)\(\-2\.831,2\.003\), then callscompute\_distanceon the pair\.

The circle and parabola intersect wherex2\+\(y−3\)2=9x^\{2\}\+\(y\-3\)^\{2\}=9andx2=4​yx^\{2\}=4y, givingy​\(y−2\)=0y\(y\-2\)=0; the non\-trivial rooty=2y=2yieldsAAandBBat\(±2​2,2\)\(\\pm 2\\sqrt\{2\},\\,2\)\. The directrixy=−1y=\-1is rendered as an additional horizontal line\.

![Refer to caption](https://arxiv.org/html/2607.12982v1/appendix_parab_circ.png)Figure 6\.Upward parabola \(green\) and circle \(magenta\), with directrix and intersection points\.

### D\.3\.Ellipse and Auxiliary Circle

> “The ellipsex216\+y29=1\\frac\{x^\{2\}\}\{16\}\+\\frac\{y^\{2\}\}\{9\}=1and its auxiliary circlex2\+y2=16x^\{2\}\+y^\{2\}=16share centreOO\. The ellipse has fociF1F\_\{1\}andF2F\_\{2\}\. A tangent to the ellipse atP​\(2,2\.598\)P\(2,\\,2\.598\)intersects the auxiliary circle atAAandBB\. Find the area of△​F1​P​F2\\triangle F\_\{1\}PF\_\{2\}\.”

#### CDL annotation\.

E:Ellipse;C:Circle;P:Point;l:Line

A:Point;B:Point;O:Origin;F1:Point;F2:Point

Equation\(E\)=x^2/16\+y^2/9=1

Equation\(C\)=x^2\+y^2=16

Focus\(E,left\)=F1;Focus\(E,right\)=F2

Coordinate\(P\)=\(2,2\.598\)

PointOnCurve\(P,E\)

TangentOnPoint\(P,E\)=l

Intersection\(l,C\)=\{A,B\}

Analytical ground truth:Area=3​212≈6\.8738=\\frac\{3\\sqrt\{21\}\}\{2\}\\approx 6\.8738\.Measurer answer:6\.84926\.8492\(relative error0\.36%0\.36\\%\)\. The Measurer measuresF1F\_\{1\}at\(−2\.644,0\.003\)\(\-2\.644,0\.003\)andF2F\_\{2\}at\(2\.641,−0\.002\)\(2\.641,\-0\.002\)viapixel\_to\_coord, readsPPat\(1\.998,2\.595\)\(1\.998,2\.595\), and callscompute\_areaon the triangle\.

The auxiliary circlex2\+y2=a2x^\{2\}\+y^\{2\}=a^\{2\}circumscribes the ellipse along the major axis\. TheIntersectionoperates*across*two curve types: the tangent to the ellipse is intersected with the circle\.

![Refer to caption](https://arxiv.org/html/2607.12982v1/appendix_aux_circ.png)Figure 7\.Ellipse \(blue\) inscribed in auxiliary circle \(magenta\), with cross\-curve tangent\.

### D\.4\.Hyperbola and Circle with Asymptotes

> “The hyperbolax24−y2=1\\frac\{x^\{2\}\}\{4\}\-y^\{2\}=1has vertexVV, fociF1F\_\{1\},F2F\_\{2\}, and asymptotesy=±x2y=\\pm\\frac\{x\}\{2\}\. The circlex2\+y2=9x^\{2\}\+y^\{2\}=9intersects the right branch atAAandBB\. Find the length\|A​B\|\|AB\|\.”

#### CDL annotation\.

H:Hyperbola;C:Circle;V:Point;F1:Point

F2:Point;a1:Line;a2:Line

A:Point;B:Point;O:Origin

Equation\(H\)=x^2/4\-y^2=1

Equation\(C\)=x^2\+y^2=9

Vertex\(H,right\)=V

Focus\(H,left\)=F1;Focus\(H,right\)=F2

Equation\(a1\)=y=0\.5\*x

Equation\(a2\)=y=\-0\.5\*x

Coordinate\(A\)=\(2\.828,1\)

Coordinate\(B\)=\(2\.828,\-1\)

PointOnCurve\(A,H\);PointOnCurve\(B,H\)

Analytical ground truth:\|A​B\|=2\|AB\|=2\.Measurer answer:1\.99631\.9963\(relative error0\.19%0\.19\\%\)\. The Measurer usesintersect\_curvesin the right half\-plane to locateAAat\(2\.831,0\.999\)\(2\.831,0\.999\)andBBat\(2\.827,−0\.997\)\(2\.827,\-0\.997\), then callscompute\_distance\.

This is the most element\-rich example: aHyperbola\(both branches\), aCircle, two asymptote lines, a vertex, and both foci—12 CDL statements producing 6 labelled points alongside two curves\.

![Refer to caption](https://arxiv.org/html/2607.12982v1/appendix_hyp_circ.png)Figure 8\.Hyperbola \(red\) and circle \(magenta\) with asymptotes, foci, and vertex\.