513 lines
20 KiB
Python
513 lines
20 KiB
Python
"""初等 / 高等数学标准知识图谱（v1）。
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知识图谱作为全局地基：
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- 每个知识点是唯一 Knowledge 节点；
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- 不同教材/章节通过 ChapterKnowledge 引用节点；
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- 题目通过 QuestionKnowledge 引用节点；
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- 用户掌握度 UserKnowledge 绑定 knowledge_id；
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- 节点间关系（包含 / 前置 / 相关）存于 knowledge_relations，
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供后续图谱展示、推荐与路径规划使用。
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"""
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from __future__ import annotations
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from sqlalchemy.orm import Session
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from models import Knowledge
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from services.knowledge_service import (
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ensure_knowledge_relation,
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)
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LEGACY_KNOWLEDGE_MAP = {
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"直角三角形": "三角比",
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"同角关系": "同角三角函数关系",
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"弧长与扇形": "弧度制",
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"和角公式": "和角与差角公式",
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"差角公式": "和角与差角公式",
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"周期函数": "三角函数图像",
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"斜率": "直线方程",
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"一次函数图像": "一次函数",
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"图像平移": "函数图像与变换",
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}
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# 常用知识点的定义；仅在节点描述为空时写入，不覆盖管理员维护的内容。
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KNOWLEDGE_DEFINITIONS = {
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"函数与图像": "以图像直观呈现自变量与函数值对应关系的分支：先“看见”图像，再回到公式与性质。",
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"函数概念": "设 x 取数集 D 中的值，若按确定的对应法则，每个 x 都有唯一确定的 y 与之对应，则称 y 是 x 的函数，记作 y=f(x)，x∈D。",
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"函数值": "当自变量取定 x=a 时，按对应法则算出的 f(a) 就是函数在 a 处的函数值，它是图像上点 (a, f(a)) 的纵坐标。",
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"定义域": "使函数表达式有意义的所有自变量取值组成的集合，常见限制有分母不为零、偶次根号下非负、对数的真数为正等。",
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"值域": "自变量取遍定义域时，所有函数值组成的集合，即 {f(x) | x∈D}。",
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"零点": "使 f(x)=0 的自变量 x 的值，几何上对应函数图像与 x 轴交点的横坐标。",
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"正负性": "函数在区间上取正值或负值的性质，由 f(x)>0 与 f(x)<0 的解集刻画，对应图像位于 x 轴上方或下方。",
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"单调性": "在区间 I 上，若 x₁<x₂ 时恒有 f(x₁)<f(x₂)（或恒有 f(x₁)>f(x₂)），则称函数在 I 上单调递增（或递减）。",
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"最值": "函数在给定区间上取得的最大值与最小值，可结合单调性、图像或配方等方法求得。",
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"奇偶性": "若定义域关于原点对称，且对任意 x 有 f(-x)=f(x)，则为偶函数；若恒有 f(-x)=-f(x)，则为奇函数。",
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"偶函数": "满足 f(-x)=f(x) 的函数，其图像关于 y 轴对称。",
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"奇函数": "满足 f(-x)=-f(x) 的函数，其图像关于原点中心对称；若 0 在定义域内，则 f(0)=0。",
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"图像对称性": "函数图像的轴对称与中心对称性质，与奇偶性、平移伸缩等变换密切相关。",
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"一次函数": "形如 y=kx+b（k≠0）的函数，图像是一条直线，k 为斜率、b 为纵截距；k>0 时单调递增。",
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"二次函数": "形如 y=ax²+bx+c（a≠0）的函数，图像是抛物线，可通过配方求顶点、对称轴与最值。",
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"抛物线": "二次函数图像的几何形状，具有对称轴与顶点，开口方向由二次项系数的正负决定。",
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"幂函数": "形如 y=x^α 的函数，其定义域、图像与单调性随指数 α 的不同而明显变化。",
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"函数图像与变换": "由基本函数的图像出发，经平移（左加右减、上加下减）、伸缩与对称得到新图像的方法。",
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"直线方程": "用方程表示直线的形式，如斜截式 y=kx+b、点斜式 y-y₀=k(x-x₀)；两直线平行则斜率相等。",
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"三角函数": "以角为自变量、以三角比为函数值的函数，包括正弦、余弦、正切等，具有周期性与有界性。",
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"任意角": "由射线绕顶点旋转生成的角，按旋转方向分为正角、负角与零角，并用终边位置统一刻画。",
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"弧度制": "用弧长与半径之比度量角的大小，π 弧度 = 180°，使弧长与扇形面积公式的形式更简洁。",
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"三角比": "直角三角形中边与边的比值（正弦、余弦、正切），并可借助单位圆推广到任意角。",
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"同角三角函数关系": "同一个角的正弦、余弦、正切之间的基本关系，如 sin²α+cos²α=1、tanα=sinα/cosα。",
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"和角与差角公式": "描述两角和或差的三角函数公式，如 sin(α±β)=sinαcosβ±cosαsinβ，是推导倍角公式的基础。",
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"三角恒等式": "对定义域内任意角都成立的三角关系式，包括平方关系、和差角、倍角与半角公式等。",
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"三角函数图像": "正弦、余弦、正切函数的图像及其周期、振幅、相位与图像变换规律。",
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"反三角函数": "三角函数在限定单调区间上的反函数，如反正弦 arcsin、反余弦 arccos、反正切 arctan。",
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"三角方程": "含有未知角的三角函数的方程，通常先化为基本三角方程，再结合周期性写出通解。",
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"正弦定理": "在 △ABC 中 a/sinA = b/sinB = c/sinC = 2R（R 为外接圆半径），用于已知边角组合解三角形。",
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"余弦定理": "在 △ABC 中 a²=b²+c²-2bc·cosA，用于已知两边及夹角或三边求解三角形。",
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"解三角形": "利用正弦定理、余弦定理与内角和关系，由已知元素求三角形其余边角的过程。",
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}
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GRAPH = [
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{
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"name": "初等数学",
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"domain": "初等",
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"description": "中学与竞赛基础的数学知识总图",
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"children": [
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{
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"name": "数与式",
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"children": [
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"自然数与整数",
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"有理数",
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"实数",
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"数的整除",
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"代数式",
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"因式分解",
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"根式与无理式",
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"幂与指数",
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],
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},
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{
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"name": "方程与不等式",
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"children": [
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"一元一次方程",
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"一元二次方程",
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"二元一次方程组",
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"分式方程",
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"一元一次不等式",
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"一元二次不等式",
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"不等式的性质与证明",
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],
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},
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{
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"name": "函数与图像",
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"children": [
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"函数概念",
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"函数值",
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"定义域",
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"值域",
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"零点",
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"正负性",
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"单调性",
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"最值",
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"奇偶性",
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"偶函数",
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"奇函数",
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"图像对称性",
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"一次函数",
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"二次函数",
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"幂函数",
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"指数函数",
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"对数函数",
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"函数图像与变换",
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],
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},
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{
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"name": "三角函数",
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"children": [
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"任意角",
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"弧度制",
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"三角比",
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"同角三角函数关系",
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"诱导公式",
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"和角与差角公式",
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"倍角公式",
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"三角恒等式",
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"三角函数图像",
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"反三角函数",
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"三角方程",
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"正弦定理",
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"余弦定理",
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"解三角形",
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],
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},
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{
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"name": "平面几何",
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"children": [
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"点线面的基本关系",
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"平行与垂直",
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"三角形",
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"全等",
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"相似",
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"四边形",
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"圆",
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"面积与周长",
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"几何变换",
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],
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},
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{
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"name": "坐标与解析几何",
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"children": [
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"数轴",
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"平面直角坐标系",
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"两点间距离",
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"直线方程",
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"圆的标准方程",
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"椭圆",
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"双曲线",
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"抛物线",
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"曲线与方程",
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],
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},
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{
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"name": "数列与证明",
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"children": [
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"等差数列",
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"等比数列",
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"数列通项与求和",
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"递推数列",
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"数学归纳法",
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"逻辑与证明方法",
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],
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},
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{
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"name": "概率统计",
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"children": [
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"计数原理",
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"排列组合",
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"古典概型",
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"条件概率",
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"随机变量",
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"统计图表",
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"均值与方差",
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],
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},
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],
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},
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{
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"name": "高等数学",
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"domain": "高等",
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"description": "大学基础数学知识总图",
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"children": [
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{
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"name": "分析与微积分",
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"children": [
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"集合与映射",
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"极限与连续",
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"导数与微分",
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"微分中值定理",
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"一元函数积分",
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"无穷级数",
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"多元函数微分",
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"重积分与曲线曲面积分",
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"微分方程",
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],
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},
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{
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"name": "线性代数",
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"children": [
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"行列式",
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"矩阵",
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"线性方程组",
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"向量空间",
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"特征值与特征向量",
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"二次型",
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],
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},
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{
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"name": "概率论与数理统计",
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"children": [
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"概率空间",
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"随机变量及其分布",
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"数字特征",
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"大数定律与中心极限定理",
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"参数估计",
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"假设检验",
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"回归分析",
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],
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},
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{
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"name": "离散与优化",
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"children": [
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"图论基础",
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"组合优化",
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"最优化方法",
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],
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},
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],
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},
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]
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PREREQUISITES: list[tuple[str, str]] = [
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("自然数与整数", "有理数"),
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("有理数", "实数"),
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("自然数与整数", "数的整除"),
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("实数", "代数式"),
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("代数式", "因式分解"),
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("一元一次方程", "一元二次方程"),
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("一元二次方程", "二次函数"),
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("幂与指数", "指数函数"),
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("幂与指数", "幂函数"),
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("函数概念", "定义域"),
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("函数概念", "函数值"),
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("函数概念", "偶函数"),
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("函数概念", "奇函数"),
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("偶函数", "奇函数"),
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("定义域", "零点"),
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("一次函数", "二次函数"),
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("二次函数", "函数图像与变换"),
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("三角比", "任意角"),
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("任意角", "弧度制"),
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("弧度制", "三角函数图像"),
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("和角与差角公式", "倍角公式"),
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("倍角公式", "三角恒等式"),
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("函数与图像", "三角函数"),
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("平面直角坐标系", "直线方程"),
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("直线方程", "圆的标准方程"),
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("圆的标准方程", "抛物线"),
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("极限与连续", "导数与微分"),
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("导数与微分", "微分中值定理"),
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("导数与微分", "一元函数积分"),
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("一元函数积分", "多元函数微分"),
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("矩阵", "行列式"),
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("矩阵", "线性方程组"),
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("线性方程组", "特征值与特征向量"),
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("随机变量及其分布", "数字特征"),
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("概率空间", "随机变量及其分布"),
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]
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def _create(
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db: Session,
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name: str,
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domain: str,
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category: str = "",
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description: str = "",
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) -> Knowledge:
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row = db.query(Knowledge).filter(Knowledge.name == name).first()
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if row is None:
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row = Knowledge(
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name=name,
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domain=domain,
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category=category,
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description=description,
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)
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db.add(row)
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db.flush()
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else:
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if row.domain != domain:
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row.domain = domain
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if category and not row.category:
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row.category = category
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if description and not row.description:
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row.description = description
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return row
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def _insert_tree(db: Session, node: dict, parent: Knowledge | None = None) -> None:
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if isinstance(node, str):
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current = _create(
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db,
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str(node),
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parent.domain if parent else "初等",
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parent.category if parent else "",
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)
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if parent is not None:
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ensure_knowledge_relation(db, parent, current, "包含")
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return
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name = node["name"]
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if parent is None:
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category = "总纲"
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elif parent.name in {"初等数学", "高等数学"}:
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category = name
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else:
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category = node.get("category") or (parent.category or "")
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current = _create(
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db,
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name,
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node.get("domain") or (parent.domain if parent else "初等"),
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category,
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node.get("description", ""),
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)
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if parent is not None:
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ensure_knowledge_relation(db, parent, current, "包含")
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for child in node.get("children", []):
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_insert_tree(db, child, current)
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def ensure_knowledge_graph(db: Session) -> None:
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if db.query(Knowledge).count() == 0:
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for node in GRAPH:
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_insert_tree(db, node)
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else:
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_repair_existing_categories(db)
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_cleanup_legacy_knowledge(db)
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for name, text in KNOWLEDGE_DEFINITIONS.items():
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row = db.query(Knowledge).filter(Knowledge.name == name).first()
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if row is not None and not (row.description or "").strip():
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row.description = text
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for source_name, target_name in PREREQUISITES:
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source = db.query(Knowledge).filter(Knowledge.name == source_name).first()
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target = db.query(Knowledge).filter(Knowledge.name == target_name).first()
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if source is None or target is None:
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continue
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ensure_knowledge_relation(db, source, target, "前置")
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db.commit()
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def _cleanup_legacy_knowledge(db: Session) -> None:
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"""把早期自由生成的节点合并到标准图谱，并重挂章节/题目/用户掌握度。"""
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from models import (
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ChapterKnowledge,
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KnowledgeRelation,
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QuestionKnowledge,
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UserKnowledge,
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)
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for legacy_name, canonical_name in LEGACY_KNOWLEDGE_MAP.items():
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legacy = db.query(Knowledge).filter(Knowledge.name == legacy_name).first()
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if legacy is None:
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continue
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canonical = db.query(Knowledge).filter(Knowledge.name == canonical_name).first()
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if canonical is None:
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canonical = Knowledge(
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name=canonical_name,
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domain=legacy.domain,
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category=legacy.category or "三角函数",
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)
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db.add(canonical)
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db.flush()
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old_chapter_links = (
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db.query(ChapterKnowledge)
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.filter(ChapterKnowledge.knowledge_id == legacy.id)
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.all()
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)
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old_question_links = (
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db.query(QuestionKnowledge)
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.filter(QuestionKnowledge.knowledge_id == legacy.id)
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.all()
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)
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old_user_links = (
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db.query(UserKnowledge)
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||
.filter(UserKnowledge.knowledge_id == legacy.id)
|
||
.all()
|
||
)
|
||
db.query(ChapterKnowledge).filter(
|
||
ChapterKnowledge.knowledge_id == legacy.id
|
||
).delete(synchronize_session=False)
|
||
db.query(QuestionKnowledge).filter(
|
||
QuestionKnowledge.knowledge_id == legacy.id
|
||
).delete(synchronize_session=False)
|
||
db.query(UserKnowledge).filter(
|
||
UserKnowledge.knowledge_id == legacy.id
|
||
).delete(synchronize_session=False)
|
||
|
||
for link in old_chapter_links:
|
||
exists = (
|
||
db.query(ChapterKnowledge)
|
||
.filter(
|
||
ChapterKnowledge.chapter_id == link.chapter_id,
|
||
ChapterKnowledge.knowledge_id == canonical.id,
|
||
)
|
||
.first()
|
||
)
|
||
if exists is None:
|
||
db.add(
|
||
ChapterKnowledge(
|
||
chapter_id=link.chapter_id,
|
||
knowledge_id=canonical.id,
|
||
)
|
||
)
|
||
for link in old_question_links:
|
||
exists = (
|
||
db.query(QuestionKnowledge)
|
||
.filter(
|
||
QuestionKnowledge.question_id == link.question_id,
|
||
QuestionKnowledge.knowledge_id == canonical.id,
|
||
)
|
||
.first()
|
||
)
|
||
if exists is None:
|
||
db.add(
|
||
QuestionKnowledge(
|
||
question_id=link.question_id,
|
||
knowledge_id=canonical.id,
|
||
)
|
||
)
|
||
for link in old_user_links:
|
||
exists = (
|
||
db.query(UserKnowledge)
|
||
.filter(
|
||
UserKnowledge.user_id == link.user_id,
|
||
UserKnowledge.knowledge_id == canonical.id,
|
||
)
|
||
.first()
|
||
)
|
||
if exists is None:
|
||
db.add(
|
||
UserKnowledge(
|
||
user_id=link.user_id,
|
||
knowledge_id=canonical.id,
|
||
mastery=link.mastery,
|
||
position=link.position,
|
||
)
|
||
)
|
||
db.query(KnowledgeRelation).filter(
|
||
(KnowledgeRelation.source_id == legacy.id)
|
||
| (KnowledgeRelation.target_id == legacy.id)
|
||
).delete(synchronize_session=False)
|
||
db.delete(legacy)
|
||
db.flush()
|
||
|
||
|
||
def _repair_existing_categories(db: Session) -> None:
|
||
"""存量库按种子树回填 domain/category（不删除已有节点）。"""
|
||
|
||
def visit(node: dict, parent: Knowledge | None = None) -> None:
|
||
if isinstance(node, str):
|
||
row = db.query(Knowledge).filter(Knowledge.name == node).first()
|
||
if row is not None:
|
||
row.domain = parent.domain if parent else row.domain
|
||
if parent and parent.category:
|
||
row.category = parent.category
|
||
return
|
||
domain = node.get("domain") or (parent.domain if parent else "初等")
|
||
if parent is None:
|
||
category = "总纲"
|
||
elif parent.name in {"初等数学", "高等数学"}:
|
||
category = node["name"]
|
||
else:
|
||
category = parent.category or ""
|
||
row = db.query(Knowledge).filter(Knowledge.name == node["name"]).first()
|
||
if row is None:
|
||
row = Knowledge(
|
||
name=node["name"],
|
||
domain=domain,
|
||
category=category,
|
||
description=node.get("description", ""),
|
||
)
|
||
db.add(row)
|
||
db.flush()
|
||
else:
|
||
row.domain = domain
|
||
row.category = category
|
||
if node.get("description") and not row.description:
|
||
row.description = node["description"]
|
||
for child in node.get("children", []):
|
||
visit(child, row)
|
||
|
||
for top in GRAPH:
|
||
visit(top)
|
||
db.commit()
|