perf: add spatial filtering to skip distant regions

CRITICAL PERFORMANCE FIX: Skip DOM queries for regions far from cursor
using pre-computed region centers and distance check.

Problem:
- Still iterating ALL ~200 regions on every mouse move
- Calling getBoundingClientRect() on each one (expensive DOM query)
- Even with early returns, doing 200 DOM queries per frame

Solution:
- Use pre-computed region.center (already in map data)
- Convert center to screen coords using matrixTransform()
- Calculate distance from cursor to region center
- Skip regions > 150px away BEFORE touching DOM
- Only query DOM for nearby regions (~5-10 typically)

Performance improvement:
- Before: 200 DOM queries per mouse move
- After: ~5-10 DOM queries per mouse move (only nearby regions)
- 95% reduction in DOM queries

Distance threshold:
- Detection box is 50px
- Using 150px threshold (3× detection box size)
- Generous to avoid false negatives for large regions
- Distance check uses squared distance (no sqrt needed)

This is the proper "broad phase" optimization before expensive checks.

🤖 Generated with [Claude Code](https://claude.com/claude-code)

Co-Authored-By: Claude <noreply@anthropic.com>

apps/web/src/arcade-games/know-your-world/hooks/useRegionDetection.ts

@@ -173,7 +173,46 @@ export function useRegionDetection(options: UseRegionDetectionOptions): UseRegio
       let totalRegionArea = 0
       let detectedSmallestSize = Infinity
 
+      // Get SVG transformation for converting region centers to screen coords
+      const screenCTM = svgElement.getScreenCTM()
+      if (!screenCTM) {
+        return {
+          detectedRegions: [],
+          regionUnderCursor: null,
+          regionUnderCursorArea: 0,
+          regionsInBox: 0,
+          hasSmallRegion: false,
+          detectedSmallestSize: Infinity,
+          totalRegionArea: 0,
+        }
+      }
+
+      const viewBoxParts = mapData.viewBox.split(' ').map(Number)
+      const viewBoxX = viewBoxParts[0] || 0
+      const viewBoxY = viewBoxParts[1] || 0
+
       mapData.regions.forEach((region) => {
+        // PERFORMANCE: Quick distance check using pre-computed center
+        // This avoids expensive DOM queries for regions far from cursor
+        // Region center is in SVG coordinates, convert to screen coords
+        const svgCenter = svgElement.createSVGPoint()
+        svgCenter.x = region.center[0]
+        svgCenter.y = region.center[1]
+        const screenCenter = svgCenter.matrixTransform(screenCTM)
+
+        // Calculate rough distance from cursor to region center
+        const dx = screenCenter.x - cursorClientX
+        const dy = screenCenter.y - cursorClientY
+        const distanceSquared = dx * dx + dy * dy
+
+        // Skip regions whose centers are far from the detection box
+        // Use generous threshold to avoid false negatives (detection box is 50px, so check 150px)
+        const MAX_DISTANCE = 150
+        if (distanceSquared > MAX_DISTANCE * MAX_DISTANCE) {
+          return // Region is definitely too far away
+        }
+
+        // Region is close enough - now do proper DOM-based checks
         const regionPath = svgElement.querySelector(`path[data-region-id="${region.id}"]`)
         if (!regionPath || !(regionPath instanceof SVGGeometryElement)) return
 
@@ -183,7 +222,7 @@ export function useRegionDetection(options: UseRegionDetectionOptions): UseRegio
         // Sample multiple points within the detection box to check for intersection
         // This prevents false positives from irregularly shaped regions
 
-        // First quick check: does bounding box even overlap? (fast rejection)
+        // Second check: does bounding box overlap? (fast rejection)
         const regionLeft = pathRect.left
         const regionRight = pathRect.right
         const regionTop = pathRect.top