Flutter for OpenHarmony 可视化教学：Graham Scan 凸包算法的交互式演示

在计算几何中，凸包（Convex Hull） 是一个基础而关键的概念——它是指包含给定点集中所有点的最小凸多边形。从机器人路径规划、碰撞检测，到地理围栏、图像处理，凸包算法无处不在。然而，其背后的数学逻辑（如极角排序、叉积判断转向）对初学者而言常显抽象。

本文将深入剖析一段完整的 Flutter 应用代码，该应用不仅实现了经典的 Graham Scan 算法，更通过精心设计的 UI/UX，将其转化为一场生动的可视化教学体验。你将看到：如何用 Dart 优雅地表达几何逻辑，如何用 CustomPaint 绘制动态图形，以及如何构建一个兼具教育性与美感的交互式学习工具。

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完整效果

一、整体架构：以教学为核心的交互设计

该应用围绕 “理解 > 操作 > 验证” 的学习闭环构建：

• 实时反馈：滑动调节点数，即时生成新点集；
• 过程可视化：逐步动画展示凸包构建的四个核心阶段；
• 数据驱动：实时显示点集规模、凸包顶点数、计算耗时等统计信息；
• 上下文帮助：清晰标注算法复杂度、图例说明、执行步骤。

💡 设计哲学：不让用户“猜”，而是通过视觉、文本、交互三位一体，降低认知负荷。

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二、核心算法：Graham Scan 的 Dart 实现

1. 数据结构：不可变的 Point 类

@immutable
class Point {
  final double x, y;
  const Point(this.x, this.y);

  double distanceTo(Point other) {
    final dx = x - other.x;
    final dy = y - other.y;
    return math.sqrt(dx * dx + dy * dy);
  }
}

• 使用 @immutable 注解强调安全性；
• 手动展开 dx*dx + dy*dy 避免 pow() 函数调用开销；
• 为后续排序和距离比较提供基础。

2. 四步 Graham Scan 流程

▶ 步骤 1：寻找基准点（Pivot）

Point _findBottomLeftPoint(List<Point> points) {
  var bottomLeft = points[0];
  for (final p in points) {
    if (p.y < bottomLeft.y || (p.y == bottomLeft.y && p.x < bottomLeft.x)) {
      bottomLeft = p;
    }
  }
  return bottomLeft;
}

• 选择 y 最小（若相同则 x 最小） 的点作为起点；
• 保证所有其他点位于其上方或右侧，简化极角范围至 [0, 2π)。

▶ 步骤 2：极角排序

sortedPoints.sort((a, b) {
  final angleA = _angle(pivot, a);
  final angleB = _angle(pivot, b);
  if (angleA == angleB) {
    return pivot.distanceTo(a).compareTo(pivot.distanceTo(b)); // 近者优先
  }
  return angleA.compareTo(angleB);
});

• 使用 atan2(dy, dx) 计算精确极角；
• 共线处理：角度相同时，近点排在前面，避免远点遮挡近点导致错误剔除。

▶ 步骤 3 & 4：栈式构建与凹点剔除

while (stack.length >= 2 &&
    _crossProduct(stack[stack.length - 2], stack.last, point) <= 0) {
  stack.removeLast(); // 剔除造成“右转”的凹点
}
stack.add(point);

• 叉积公式：(a.x - o.x)*(b.y - o.y) - (a.y - o.y)*(b.x - o.x)
  • > 0：逆时针（左转，保留）
  • ≤ 0：顺时针或共线（右转/直行，剔除）

✅ 关键洞察：凸包的本质是“永不右转”的路径。Graham Scan 通过维护一个单调栈，确保每一步都符合这一几何约束。

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三、UI/UX 设计：沉浸式学习体验

1. 视觉层次：深色主题 + 渐变卡片

• 主背景：顶部透明靛蓝渐变至纯黑，营造科技感；
• 功能卡片：使用蓝-青、绿、紫等色系区分“控制面板”、“统计”、“算法信息”，符合 Material Design 色彩语义。

2. 核心组件详解

▶ 动态画布 (_buildCanvas)

• 尺寸固定：400×400 像素，保证坐标一致性；
• 网格背景：40px 间隔灰色细线，辅助空间感知；
• 裁剪圆角：ClipRRect 配合阴影，提升视觉精致度。

▶ 算法步骤追踪 (_buildAlgorithmSteps)

// 实时高亮当前执行阶段
final isCurrent = index == _currentStep - 1;
...
decoration: BoxDecoration(
  color: isCurrent ? Colors.blue.withValues(alpha: 0.2) : Colors.transparent,
),

• 四个步骤对应算法四阶段；
• 当前步骤高亮，已完成步骤显示绿色对勾；
• 用户可直观看到“程序现在在做什么”。

▶ 实时统计面板 (_buildStatistics)

点集大小: 25 | 凸包顶点: 8 | 凸包占比: 32.0% | 计算时间: 12ms

• 凸包占比：揭示点集分布密度（均匀分布通常占比低，边界聚集则高）；
• 毫秒级计时：验证 O(n log n) 复杂度在小规模数据下的高效性。

▶ 图例系统 (_LegendItem)

• 数据点：绿色实心圆；
• 凸包边：蓝色粗线；
• 基准点 & 顶点：橙色圆 + 白色外圈（突出关键角色）。

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四、绘制引擎：ConvexHullPainter 的细节打磨

1. 分层绘制策略

void paint(Canvas canvas, Size size) {
  _drawGrid(canvas, size);     // 底层：网格
  _drawPoints(canvas);         // 中层：所有点
  _drawHull(canvas);           // 上层：凸包边与顶点
  _drawPivotPoint(canvas);     // 顶层：基准点（覆盖 hull 绘制）
}

• 层级分明：避免视觉干扰，确保关键元素（如基准点）始终可见。

2. 动态高亮关键点

• 基准点：独立绘制，即使不在 _hull 列表中也能被识别（修复了上一版潜在 bug）；
• 凸包顶点：比普通点更大，并带白色描边，在蓝色边线上清晰可辨。

3. 性能优化

• 智能重绘：shouldRepaint 仅在点数、凸包或运行状态变化时触发；
• 避免冗余计算：基准点在绘制时重新查找，而非依赖状态变量，保证一致性。

---

五、交互逻辑：流畅的用户体验

1. 异步动画流程

Future<void> _computeHull() async {
  // ...
  for (int i = 0; i <= hull.length; i++) {
    setState(() { _hull = hull.sublist(0, i); });
    await Future.delayed(const Duration(milliseconds: 200));
  }
}

• 分帧更新：每次只显示部分凸包，形成“生长”动画；
• 防内存泄漏：全程检查 mounted，避免页面销毁后 setState。

2. 状态管理

• 禁用按钮：计算过程中，“随机生成”和“计算凸包”按钮置灰；
• 加载指示器：计算按钮内嵌 CircularProgressIndicator，提供即时反馈。

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六、教育价值：为什么这个演示器有效？

教学痛点	本应用解决方案
算法抽象	将叉积、极角等概念映射为可视化的“转向”和“排序”
过程黑盒	分步动画 + 文字说明，揭示内部状态变化
参数敏感	滑块调节点数，观察不同分布对凸包的影响
缺乏验证	实时统计 + 完美闭合多边形，提供正确性反馈

🎯 终极目标：让用户不仅能“看懂”Graham Scan，更能“感受”到计算几何的优雅与力量。

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七、总结与延伸

这段代码远不止是一个算法实现，它是一个精心设计的认知脚手架。通过将数学、代码、视觉艺术融为一体，它降低了学习门槛，激发了探索欲。

完整代码

import 'dart:math' as math;
import 'package:flutter/material.dart';

void main() {
  runApp(const ComputationalGeometryApp());
}

class ComputationalGeometryApp extends StatelessWidget {
  const ComputationalGeometryApp({super.key});

  @override
  Widget build(BuildContext context) {
    return MaterialApp(
      title: '凸包算法可视化',
      debugShowCheckedModeBanner: false,
      theme: ThemeData(
        brightness: Brightness.dark,
        primarySwatch: Colors.indigo,
        useMaterial3: true,
        visualDensity: VisualDensity.adaptivePlatformDensity,
      ),
      home: const ConvexHullScreen(),
    );
  }
}

/// 二维点类
@immutable
class Point {
  final double x, y;

  const Point(this.x, this.y);

  double distanceTo(Point other) {
    final dx = x - other.x;
    final dy = y - other.y;
    return math.sqrt(dx * dx + dy * dy);
  }
}

class ConvexHullScreen extends StatefulWidget {
  const ConvexHullScreen({super.key});

  @override
  State<ConvexHullScreen> createState() => _ConvexHullScreenState();
}

class _ConvexHullScreenState extends State<ConvexHullScreen> {
  static const int _defaultPointCount = 20;
  static const int _minPoints = 5;
  static const int _maxPoints = 50;
  static const int _canvasSize = 400;

  List<Point> _points = [];
  List<Point> _hull = [];
  bool _isRunning = false;
  int _pointCount = _defaultPointCount;
  String _algorithmInfo = '';
  String _statistics = '点集大小: 0 | 凸包顶点: 0 | 计算时间: 0ms';
  List<String> _algorithmSteps = [];
  int _currentStep = 0;

  @override
  void initState() {
    super.initState();
    _generateRandomPoints();
    _updateAlgorithmInfo();
  }

  void _updateAlgorithmInfo() {
    setState(() {
      _algorithmInfo = '算法: Graham Scan\n时间复杂度: O(n log n)\n空间复杂度: O(n)';
    });
  }

  /// 生成随机点集
  void _generateRandomPoints() {
    final random = math.Random();
    setState(() {
      _points = List.generate(_pointCount, (index) {
        return Point(
          random.nextDouble() * _canvasSize,
          random.nextDouble() * _canvasSize,
        );
      });
      _hull = [];
      _updateStatistics();
    });
  }

  /// 寻找最下方（最左方）的点作为基准点
  Point _findBottomLeftPoint(List<Point> points) {
    var bottomLeft = points[0];
    for (final p in points) {
      if (p.y < bottomLeft.y || (p.y == bottomLeft.y && p.x < bottomLeft.x)) {
        bottomLeft = p;
      }
    }
    return bottomLeft;
  }

  /// 计算叉积 (判断转向)
  /// 返回值 > 0: 逆时针 (左转)
  /// 返回值 < 0: 顺时针 (右转)
  /// 返回值 = 0: 共线
  double _crossProduct(Point o, Point a, Point b) {
    return (a.x - o.x) * (b.y - o.y) - (a.y - o.y) * (b.x - o.x);
  }

  /// 计算相对于基准点的角度
  double _angle(Point pivot, Point p) {
    return math.atan2(p.y - pivot.y, p.x - pivot.x);
  }

  /// Graham Scan 核心算法
  List<Point> _grahamScan() {
    if (_points.length < 3) return [];

    // 1. 找到基准点 (最下方最左方)
    final pivot = _findBottomLeftPoint(_points);

    // 2. 按照相对于基准点的极角进行排序
    final sortedPoints = List<Point>.from(_points);
    sortedPoints.sort((a, b) {
      final angleA = _angle(pivot, a);
      final angleB = _angle(pivot, b);
      if (angleA == angleB) {
        // 如果角度相同，距离近的排在前面
        return pivot.distanceTo(a).compareTo(pivot.distanceTo(b));
      }
      return angleA.compareTo(angleB);
    });

    // 3. 使用栈构建凸包
    final stack = <Point>[];
    for (final point in sortedPoints) {
      // 如果栈中至少有两个点，检查是否为右转
      while (stack.length >= 2 &&
          _crossProduct(stack[stack.length - 2], stack.last, point) <= 0) {
        stack.removeLast(); // 弹出栈顶，剔除凹点
      }
      stack.add(point);
    }

    return stack;
  }

  /// 更新统计信息
  void _updateStatistics() {
    setState(() {
      _statistics = '点集大小: ${_points.length} | '
          '凸包顶点: ${_hull.length} | '
          '凸包占比: ${_points.isEmpty ? 0 : (_hull.length / _points.length * 100).toStringAsFixed(1)}%';
    });
  }

  /// 计算凸包（带动画演示）
  Future<void> _computeHull() async {
    if (_isRunning) return;

    final stopwatch = Stopwatch()..start();

    setState(() {
      _isRunning = true;
      _hull = [];
      _currentStep = 0;
      _algorithmSteps = [
        '1. 找到基准点 (最左下角)',
        '2. 按极角对点排序',
        '3. 遍历点构建凸包',
        '4. 剔除凹点，完成构建',
      ];
    });

    try {
      // 模拟计算延迟，展示过程
      await Future.delayed(const Duration(milliseconds: 300));

      final hull = _grahamScan();

      // 逐步显示凸包形成过程
      for (int i = 0; i <= hull.length && mounted; i++) {
        setState(() {
          _hull = hull.sublist(0, i);
          _currentStep = (i * _algorithmSteps.length / hull.length).ceil();
        });
        await Future.delayed(const Duration(milliseconds: 200));
      }

      stopwatch.stop();

      if (mounted) {
        setState(() {
          _statistics += ' | 计算时间: ${stopwatch.elapsedMilliseconds}ms';
        });
      }
    } finally {
      if (mounted) {
        setState(() {
          _isRunning = false;
          _currentStep = 0;
          _updateStatistics();
        });
      }
    }
  }

  @override
  Widget build(BuildContext context) {
    return Scaffold(
      appBar: AppBar(
        title: const Text('凸包算法可视化'),
        centerTitle: true,
        elevation: 0,
      ),
      body: Container(
        decoration: BoxDecoration(
          gradient: LinearGradient(
            begin: Alignment.topCenter,
            end: Alignment.bottomCenter,
            colors: [
              Theme.of(context).primaryColor.withValues(alpha: 0.3),
              Colors.black,
            ],
          ),
        ),
        child: SafeArea(
          child: SingleChildScrollView(
            padding: const EdgeInsets.all(16.0),
            child: Column(
              crossAxisAlignment: CrossAxisAlignment.stretch,
              children: [
                // 标题卡片
                _buildInfoCard(
                  icon: Icons.polyline,
                  title: 'Graham Scan 算法',
                  subtitle: '计算几何：寻找包围所有点的最小凸多边形',
                  gradient: LinearGradient(
                    colors: [Colors.blue.shade700, Colors.cyan.shade700],
                  ),
                ),

                const SizedBox(height: 20),

                // 控制面板
                _buildControlPanel(),

                const SizedBox(height: 20),

                // 算法步骤
                if (_isRunning && _algorithmSteps.isNotEmpty)
                  _buildAlgorithmSteps(),

                // 画布
                _buildCanvas(),

                const SizedBox(height: 20),

                // 统计信息
                _buildStatistics(),

                const SizedBox(height: 20),

                // 算法信息
                _buildAlgorithmInfo(),

                const SizedBox(height: 20),

                // 图例
                _buildLegend(),
              ],
            ),
          ),
        ),
      ),
    );
  }

  Widget _buildInfoCard({
    required IconData icon,
    required String title,
    required String subtitle,
    required Gradient gradient,
  }) {
    return Container(
      padding: const EdgeInsets.all(20.0),
      decoration: BoxDecoration(
        gradient: gradient,
        borderRadius: BorderRadius.circular(16),
        boxShadow: [
          BoxShadow(
            color: Colors.black.withValues(alpha: 0.3),
            blurRadius: 10,
            offset: const Offset(0, 5),
          ),
        ],
      ),
      child: Row(
        children: [
          Icon(icon, size: 40, color: Colors.white),
          const SizedBox(width: 16),
          Expanded(
            child: Column(
              crossAxisAlignment: CrossAxisAlignment.start,
              children: [
                Text(
                  title,
                  style: const TextStyle(
                    fontSize: 20,
                    fontWeight: FontWeight.bold,
                    color: Colors.white,
                  ),
                ),
                const SizedBox(height: 4),
                Text(
                  subtitle,
                  style: TextStyle(
                    fontSize: 14,
                    color: Colors.white.withValues(alpha: 0.9),
                  ),
                ),
              ],
            ),
          ),
        ],
      ),
    );
  }

  Widget _buildControlPanel() {
    return Container(
      padding: const EdgeInsets.all(16.0),
      decoration: BoxDecoration(
        color: Colors.white.withValues(alpha: 0.1),
        borderRadius: BorderRadius.circular(12),
        border: Border.all(color: Colors.white.withValues(alpha: 0.2)),
      ),
      child: Column(
        crossAxisAlignment: CrossAxisAlignment.start,
        children: [
          const Text(
            '控制面板',
            style: TextStyle(fontSize: 16, fontWeight: FontWeight.bold),
          ),
          const SizedBox(height: 12),
          Row(
            children: [
              const Icon(Icons.tune, size: 20),
              const SizedBox(width: 8),
              const Text('点集大小:'),
              const SizedBox(width: 12),
              Expanded(
                child: Slider(
                  value: _pointCount.toDouble(),
                  min: _minPoints.toDouble(),
                  max: _maxPoints.toDouble(),
                  divisions: (_maxPoints - _minPoints) ~/ 5,
                  label: _pointCount.toString(),
                  onChanged: _isRunning
                      ? null
                      : (value) {
                          setState(() {
                            _pointCount = value.toInt();
                            _generateRandomPoints();
                          });
                        },
                ),
              ),
              Container(
                width: 40,
                alignment: Alignment.center,
                child: Text(
                  _pointCount.toString(),
                  style: const TextStyle(fontWeight: FontWeight.bold),
                ),
              ),
            ],
          ),
          const SizedBox(height: 16),
          Row(
            children: [
              Expanded(
                child: ElevatedButton.icon(
                  onPressed: _isRunning ? null : _generateRandomPoints,
                  icon: const Icon(Icons.refresh),
                  label: const Text('随机生成'),
                  style: ElevatedButton.styleFrom(
                    padding: const EdgeInsets.symmetric(vertical: 12),
                  ),
                ),
              ),
              const SizedBox(width: 16),
              Expanded(
                child: ElevatedButton.icon(
                  onPressed: _computeHull,
                  icon: _isRunning
                      ? const SizedBox(
                          width: 20,
                          height: 20,
                          child: CircularProgressIndicator(strokeWidth: 2),
                        )
                      : const Icon(Icons.polyline),
                  label: Text(_isRunning ? '计算中...' : '计算凸包'),
                  style: ElevatedButton.styleFrom(
                    padding: const EdgeInsets.symmetric(vertical: 12),
                  ),
                ),
              ),
            ],
          ),
        ],
      ),
    );
  }

  Widget _buildAlgorithmSteps() {
    return Container(
      margin: const EdgeInsets.only(bottom: 20),
      padding: const EdgeInsets.all(16.0),
      decoration: BoxDecoration(
        color: Colors.blue.withValues(alpha: 0.1),
        borderRadius: BorderRadius.circular(12),
        border: Border.all(color: Colors.blue.withValues(alpha: 0.3)),
      ),
      child: Column(
        crossAxisAlignment: CrossAxisAlignment.start,
        children: [
          Row(
            children: [
              Icon(Icons.info_outline, color: Colors.blue.shade300),
              const SizedBox(width: 8),
              const Text(
                '算法执行步骤',
                style: TextStyle(fontWeight: FontWeight.bold),
              ),
            ],
          ),
          const SizedBox(height: 12),
          ..._algorithmSteps.asMap().entries.map((entry) {
            final index = entry.key;
            final step = entry.value;
            final isActive = index < _currentStep;
            final isCurrent = index == _currentStep - 1;

            return Container(
              margin: const EdgeInsets.only(bottom: 8),
              padding: const EdgeInsets.all(8.0),
              decoration: BoxDecoration(
                color: isCurrent
                    ? Colors.blue.withValues(alpha: 0.2)
                    : Colors.transparent,
                borderRadius: BorderRadius.circular(8),
              ),
              child: Row(
                children: [
                  Container(
                    width: 24,
                    height: 24,
                    decoration: BoxDecoration(
                      shape: BoxShape.circle,
                      color: isActive ? Colors.green : Colors.grey,
                    ),
                    child: Center(
                      child: isActive
                          ? const Icon(Icons.check,
                              size: 14, color: Colors.white)
                          : Text(
                              '${index + 1}',
                              style: const TextStyle(
                                fontSize: 12,
                                color: Colors.white,
                              ),
                            ),
                    ),
                  ),
                  const SizedBox(width: 12),
                  Expanded(
                    child: Text(
                      step,
                      style: TextStyle(
                        fontWeight:
                            isCurrent ? FontWeight.bold : FontWeight.normal,
                      ),
                    ),
                  ),
                ],
              ),
            );
          }),
        ],
      ),
    );
  }

  Widget _buildCanvas() {
    return Container(
      height: _canvasSize.toDouble(),
      decoration: BoxDecoration(
        color: Colors.black.withValues(alpha: 0.3),
        borderRadius: BorderRadius.circular(12),
        border: Border.all(color: Colors.white.withValues(alpha: 0.2)),
        boxShadow: [
          BoxShadow(
            color: Colors.black.withValues(alpha: 0.3),
            blurRadius: 20,
            spreadRadius: 5,
          ),
        ],
      ),
      child: ClipRRect(
        borderRadius: BorderRadius.circular(12),
        child: CustomPaint(
          size: const Size(double.infinity, double.infinity),
          painter: ConvexHullPainter(
            points: _points,
            hull: _hull,
            isRunning: _isRunning,
          ),
        ),
      ),
    );
  }

  Widget _buildStatistics() {
    return Container(
      padding: const EdgeInsets.all(16.0),
      decoration: BoxDecoration(
        color: Colors.green.withValues(alpha: 0.1),
        borderRadius: BorderRadius.circular(12),
        border: Border.all(color: Colors.green.withValues(alpha: 0.3)),
      ),
      child: Column(
        crossAxisAlignment: CrossAxisAlignment.start,
        children: [
          Row(
            children: [
              Icon(Icons.analytics, color: Colors.green.shade300),
              const SizedBox(width: 8),
              const Text(
                '实时统计',
                style: TextStyle(fontWeight: FontWeight.bold),
              ),
            ],
          ),
          const SizedBox(height: 12),
          Text(_statistics),
        ],
      ),
    );
  }

  Widget _buildAlgorithmInfo() {
    return Container(
      padding: const EdgeInsets.all(16.0),
      decoration: BoxDecoration(
        color: Colors.purple.withValues(alpha: 0.1),
        borderRadius: BorderRadius.circular(12),
        border: Border.all(color: Colors.purple.withValues(alpha: 0.3)),
      ),
      child: Column(
        crossAxisAlignment: CrossAxisAlignment.start,
        children: [
          Row(
            children: [
              Icon(Icons.code, color: Colors.purple.shade300),
              const SizedBox(width: 8),
              const Text(
                '算法信息',
                style: TextStyle(fontWeight: FontWeight.bold),
              ),
            ],
          ),
          const SizedBox(height: 12),
          Text(_algorithmInfo),
        ],
      ),
    );
  }

  Widget _buildLegend() {
    return Container(
      padding: const EdgeInsets.all(12.0),
      decoration: BoxDecoration(
        color: Colors.white.withValues(alpha: 0.05),
        borderRadius: BorderRadius.circular(8),
      ),
      child: Row(
        mainAxisAlignment: MainAxisAlignment.spaceEvenly,
        children: const [
          _LegendItem(
            color: Colors.green,
            label: '数据点',
          ),
          _LegendItem(
            color: Colors.blue,
            label: '凸包边',
          ),
          _LegendItem(
            color: Colors.orange,
            label: '基准点',
          ),
        ],
      ),
    );
  }
}

class _LegendItem extends StatelessWidget {
  final Color color;
  final String label;

  const _LegendItem({required this.color, required this.label});

  @override
  Widget build(BuildContext context) {
    return Row(
      children: [
        Container(
          width: 16,
          height: 16,
          decoration: BoxDecoration(
            color: color,
            shape: label == '数据点' ? BoxShape.circle : BoxShape.rectangle,
            border: label != '数据点' ? Border.all(color: color, width: 3) : null,
          ),
        ),
        const SizedBox(width: 8),
        Text(label, style: const TextStyle(fontSize: 12)),
      ],
    );
  }
}

class ConvexHullPainter extends CustomPainter {
  final List<Point> points;
  final List<Point> hull;
  final bool isRunning;

  ConvexHullPainter({
    required this.points,
    required this.hull,
    required this.isRunning,
  });

  @override
  void paint(Canvas canvas, Size size) {
    // 绘制背景
    final bgPaint = Paint()..color = Colors.black;
    canvas.drawRect(Offset.zero & size, bgPaint);

    // 绘制网格
    _drawGrid(canvas, size);

    // 绘制所有数据点
    _drawPoints(canvas);

    // 绘制凸包连线
    _drawHull(canvas);

    // 绘制基准点
    _drawPivotPoint(canvas);
  }

  void _drawGrid(Canvas canvas, Size size) {
    final gridPaint = Paint()
      ..color = Colors.grey.withValues(alpha: 0.1)
      ..strokeWidth = 0.5;

    const gridSize = 40.0;

    // 垂直线
    for (double x = 0; x < size.width; x += gridSize) {
      canvas.drawLine(Offset(x, 0), Offset(x, size.height), gridPaint);
    }

    // 水平线
    for (double y = 0; y < size.height; y += gridSize) {
      canvas.drawLine(Offset(0, y), Offset(size.width, y), gridPaint);
    }
  }

  void _drawPoints(Canvas canvas) {
    final pointPaint = Paint()
      ..color = Colors.green
      ..style = PaintingStyle.fill;

    for (final point in points) {
      canvas.drawCircle(
        Offset(point.x, point.y),
        6,
        pointPaint,
      );
    }
  }

  void _drawHull(Canvas canvas) {
    if (hull.isEmpty) return;

    final hullPaint = Paint()
      ..color = Colors.blue.withValues(alpha: 0.8)
      ..style = PaintingStyle.stroke
      ..strokeWidth = 4
      ..strokeCap = StrokeCap.round;

    final path = Path();
    path.moveTo(hull[0].x, hull[0].y);

    for (int i = 1; i < hull.length; i++) {
      path.lineTo(hull[i].x, hull[i].y);
    }

    // 闭合凸包
    if (hull.length > 2) {
      path.close();
    }

    canvas.drawPath(path, hullPaint);

    // 绘制顶点
    final vertexPaint = Paint()
      ..color = Colors.orange
      ..style = PaintingStyle.fill;

    for (final point in hull) {
      canvas.drawCircle(
        Offset(point.x, point.y),
        8,
        vertexPaint,
      );

      // 绘制外圈
      final outlinePaint = Paint()
        ..color = Colors.white
        ..style = PaintingStyle.stroke
        ..strokeWidth = 2;

      canvas.drawCircle(
        Offset(point.x, point.y),
        10,
        outlinePaint,
      );
    }
  }

  void _drawPivotPoint(Canvas canvas) {
    if (points.isEmpty) return;

    // 找到基准点
    var pivot = points[0];
    for (final p in points) {
      if (p.y > pivot.y || (p.y == pivot.y && p.x < pivot.x)) {
        pivot = p;
      }
    }

    // 绘制基准点
    final pivotPaint = Paint()
      ..color = Colors.orange
      ..style = PaintingStyle.fill;

    canvas.drawCircle(
      Offset(pivot.x, pivot.y),
      8,
      pivotPaint,
    );

    // 绘制外圈
    final outlinePaint = Paint()
      ..color = Colors.white
      ..style = PaintingStyle.stroke
      ..strokeWidth = 2;

    canvas.drawCircle(
      Offset(pivot.x, pivot.y),
      10,
      outlinePaint,
    );
  }

  @override
  bool shouldRepaint(covariant ConvexHullPainter oldDelegate) {
    return oldDelegate.points.length != points.length ||
        oldDelegate.hull.length != hull.length ||
        oldDelegate.isRunning != isRunning;
  }
}