Precision Loss in SQLite R-Tree Coordinates: Causes and Solutions

Understanding R-Tree Coordinate Storage Limitations in SQLite

Issue Overview
The core issue revolves around precision loss when storing high-precision numerical values (e.g., GPS coordinates) in SQLite’s R-Tree virtual tables. Users report that values with 7 decimal places are truncated or rounded starting at the 6th decimal. This occurs because R-Tree modules (rtree and rtree_i32) use 32-bit storage formats (floats or integers) for coordinates, which inherently limit precision.

For example, a latitude value like -33.1234567 stored in an rtree table might be persisted as -33.123456 (loss at the 7th decimal). This truncation affects applications requiring sub-meter accuracy, such as geospatial tracking, CAD systems, or scientific simulations. The problem is exacerbated when users misinterpret R-Tree as a primary data store rather than an indexing tool. SQLite’s documentation clarifies that R-Tree is designed to filter search results approximately, with final precision checks delegated to auxiliary data sources.

Underlying Causes of Precision Loss

1. 32-Bit Float Storage in rtree:
   The default rtree module stores coordinates as 32-bit IEEE 754 floating-point numbers. These have a 23-bit mantissa, providing ~7 significant decimal digits. However, decimal precision depends on the magnitude of the value. For example:

   • Values < 10: 7 decimals preserved.
   • Values ≥ 100: Only 5 decimals preserved.

   Latitude (-90 to 90) and longitude (-180 to 180) coordinates often exceed 100 in magnitude (e.g., 179.1234567), causing precision loss at the 5th decimal.

2. Integer Scaling in rtree_i32:
   The rtree_i32 variant stores coordinates as 32-bit signed integers. While integers avoid floating-point rounding errors, they require scaling to preserve decimal precision. For instance, multiplying 179.1234567 by 10^7 converts it to 1791234567, which exceeds the 32-bit integer limit (2,147,483,647), leading to overflow and incorrect storage.

3. Misuse of R-Tree as Primary Storage:
   Users often attempt to retrieve exact coordinates directly from R-Tree columns, unaware that its primary purpose is spatial indexing. Auxiliary columns or external tables are necessary for full precision.

4. Internal Computation Precision:
   Even with rtree_i32, SQLite performs internal calculations (e.g., bounding box comparisons) using 64-bit floats. While 64-bit floats preserve 32-bit integer precision, improper scaling during input/output can reintroduce errors.

Resolving Precision Loss: Strategies and Implementation
1. Use Auxiliary Columns for Exact Values

• Step 1: Create an R-Tree table alongside a regular table storing exact coordinates:

  CREATE VIRTUAL TABLE geospatial_index USING rtree(
      id, 
      minx, maxx,  -- 32-bit floats
      miny, maxy
  );
  CREATE TABLE geospatial_data(
      id INTEGER PRIMARY KEY,
      exact_minx REAL,  -- 64-bit double
      exact_maxx REAL,
      exact_miny REAL,
      exact_maxy REAL
  );
  

• Step 2: Insert data into both tables:

  INSERT INTO geospatial_index VALUES (1, -33.123456, -33.123450, 150.123456, 150.123460);
  INSERT INTO geospatial_data VALUES (1, -33.1234567, -33.1234501, 150.1234567, 150.1234603);
  

• Step 3: Query using R-Tree for approximate matches, then filter with exact values:

  SELECT data.* 
  FROM geospatial_index idx
  JOIN geospatial_data data ON idx.id = data.id
  WHERE idx.minx <= -33.1234567 AND idx.maxx >= -33.1234567
    AND idx.miny <= 150.1234567 AND idx.maxy >= 150.1234567
    AND data.exact_minx = -33.1234567;
  

2. Scale Coordinates for rtree_i32

• Step 1: Choose a scaling factor that avoids overflow. For GPS coordinates (max ±180°), scaling by 10^7 works:

  CREATE VIRTUAL TABLE geospatial_scaled USING rtree_i32(
      id, 
      minx, maxx,  -- 32-bit integers
      miny, maxy
  );
  

• Step 2: Convert coordinates to integers during insertion:

  # Python example
  lat = -33.1234567
  scaled_lat = int(lat * 10**7)  # -331234567
  

  INSERT INTO geospatial_scaled VALUES (1, -331234567, -331234501, 1501234567, 1501234603);
  

• Step 3: Reconvert during queries:

  SELECT id, 
         minx * 1e-7 AS exact_minx, 
         maxx * 1e-7 AS exact_maxx 
  FROM geospatial_scaled;
  

• Critical Consideration: Ensure scaled values fit within [-2,147,483,648, 2,147,483,647]. For example, 180 * 10^7 = 1,800,000,000, which is within the 32-bit limit.

3. Hybrid Approach: Indexing with R-Tree and Storing Exact Values in Auxiliary Columns
SQLite allows attaching auxiliary columns to R-Tree tables for storing exact values:

CREATE VIRTUAL TABLE geospatial_hybrid USING rtree(
    id, 
    minx, maxx, 
    miny, maxy,
    exact_minx REAL,  -- Auxiliary columns
    exact_maxx REAL,
    exact_miny REAL,
    exact_maxy REAL
);

During queries, use the auxiliary columns for exact comparisons:

SELECT * FROM geospatial_hybrid
WHERE minx <= -33.1234567 AND maxx >= -33.1234567
  AND exact_minx = -33.1234567;

4. Understanding Internal Computations

• Myth: rtree_i32 uses 32-bit floats for computations.
• Reality: SQLite uses 64-bit floats internally, preserving the precision of 32-bit integers. For example, a scaled integer 1791234567 (1,791,234,567) is converted to a 64-bit float during computation. Since 64-bit floats have a 52-bit mantissa, they can exactly represent all integers up to 2^53 (9,007,199,254,740,992), far exceeding 32-bit limits.

5. Precision Thresholds and Error Margins

• GPS at Equator:
  • 1° ≈ 111 km.
  • 1e-7° ≈ 1.11 cm.
• 32-bit Float Precision:
  • At 180°, the smallest representable difference is ~0.0000305° (3.4 meters).
• 32-bit Integer with Scaling (1e7):
  • Smallest difference: 1e-7° (1.11 cm).

6. Debugging Overflow in rtree_i32
If INSERT INTO rtree_i32 fails silently:

• Step 1: Check scaled values:

  SELECT minx, maxx FROM geospatial_scaled WHERE id = 1;
  

• Step 2: Verify integer limits:

  scaled = int(180.1234567 * 10**7)  # 1,801,234,567 (overflows)
  scaled = int(179.9999999 * 10**7)  # 1,799,999,999 (safe)
  

7. Best Practices

• Always store exact coordinates externally or in auxiliary columns.
• Use R-Tree for fast approximate searches, not direct value retrieval.
• Prefer rtree_i32 with scaling for applications requiring <1 cm precision.
• Validate coordinate ranges before insertion to prevent overflow.

By combining these strategies, developers can mitigate precision loss while leveraging R-Tree’s performance benefits for spatial indexing.