edu-analytic-geometry

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🇺🇸

Original

English

🇨🇳

Translation

Chinese

解析几何解题 → 交互网页

Analytic Geometry Problem Solving → Interactive Webpage

这个技能产出什么

What This Skill Produces

一个可直接用浏览器打开的单页 HTML（三栏）：

左栏：题面 + 动态控制台 —— 一个可变参数滑块（如直线倾斜角 θ / 动点参数 t）驱动实时重算的几何量（交点坐标、斜率、数量积、弦长、面积…），以及"理论范围条"或"定值指示"。
中栏：分步解析（公式用 KaTeX），可一键收起把空间让给画板。
右栏：2D Canvas 动态几何画板（圆锥曲线 + 动直线/动点 + 向量 + 点标注 + 网格坐标轴），叠加画笔涂鸦工具栏。

形态与目标模板

/Users/wuyi/code/code2026/6/template/code_artifact.html

一致。

A single-page HTML file that can be directly opened in a browser (three-column layout):

Left Column: Problem statement + dynamic console — a variable parameter slider (such as line inclination angle θ / moving point parameter t) drives real-time recalculation of geometric quantities (intersection coordinates, slopes, dot products, chord lengths, areas…), along with a "theoretical range bar" or "fixed value indicator".
Middle Column: Step-by-step explanations (formulas rendered with KaTeX), which can be collapsed with one click to make space for the drawing board.
Right Column: 2D Canvas dynamic geometry drawing board (conic sections + moving lines/moving points + vectors + point annotations + grid coordinate axes), with an overlay brush doodling toolbar.

It has the same form as the target template

/Users/wuyi/code/code2026/6/template/code_artifact.html

依赖（重要）

Dependencies (Important)

计算核心

lib/analytic_kernel.py

依赖 sympy。运行脚本前先确认有能 import sympy 的解释器：

python3 -c "import sympy"

（本机用

/opt/homebrew/bin/python3.11

，sympy 1.14）。

缺库时：若 import 报错（sympy 或后续任何库），先询问用户是否安装，同意后再装（

python3 -m pip install <库名>

）或换一个已装该库的解释器；不要未经询问直接装。下文

python3

均指这个能跑通依赖的解释器。

The calculation core

lib/analytic_kernel.py

depends on sympy. Before running the script, confirm you have an interpreter that can import sympy:

python3 -c "import sympy"

(local environment uses

/opt/homebrew/bin/python3.11

, sympy 1.14).

When libraries are missing: If an import error occurs (sympy or any subsequent library), first ask the user for permission to install, then install it (

python3 -m pip install <library name>

) or switch to an interpreter that already has the library installed; do not install without asking. The term

python3

below refers to this dependency-compatible interpreter.

工作流程

Workflow

第 1 步：得到 problem spec（三入口归一）

Step 1: Obtain problem spec (unify three entry points)

把题目整理成结构化 spec（曲线类型与参数、已知点/条件、所求类型与对象、语言）。

文字题：直接抽取。
图片：视觉读图抽取，并把识别到的题目回显给用户确认（题面/曲线/参数/所求/语言）再继续。
随机出题：选曲线 + 题型，随机参数 → kernel 求解，用
```
analytic_kernel.is_clean(...)
```
判答案是否规整，不规整就重抽。

输出语言跟随提示词语言：英文提示 → 英文网页，中文 → 中文。spec 记下
language
。

Organize the problem into a structured spec (curve type and parameters, known points/conditions, target type and object, language).

Text problems: Extract directly.
Images: Extract via visual reading, and echo the recognized problem to the user for confirmation (problem statement/curve/parameters/target/language) before proceeding.
Random problem generation: Select curve + problem type, generate random parameters → solve with kernel, use
```
analytic_kernel.is_clean(...)
```
to check if the answer is neat, re-select parameters if not.

Output language follows prompt language: English prompts → English webpage, Chinese prompts → Chinese webpage. Record
language
in the spec.

第 2 步：用 kernel 精确计算（不要心算）

Step 2: Perform precise calculations with kernel (do not calculate manually)

按

references/conventions.md

的解法配方，调用

lib/analytic_kernel.py

与

lib/conics.py

：

```
conics.ellipse/hyperbola/parabola/circle(...)
```
得曲线对象（精确 a,b,c、焦点、顶点、准线、渐近线、
```
eq_latex
```
、以及给前端引擎的
```
board
```
dict）。
```
chord_setup(conic, through)
```
联立含参直线
```
x=my+c
```
得 y 的二次方程 + 韦达量（精确）。

目标量：

dot_product_expr

chord_len_sq_expr

triangle_area_expr

slope_product_central

…

取值范围：
```
range_over_m(expr, horizontal_valid=?)
```
—— 含开闭端点判定（关键正确性点，见下）。
定值：
```
is_constant_in_m(expr)
```
。

可命令行自检 kernel：

bash

python3 lib/analytic_kernel.py      # 旗舰题内置断言自检

⚠️ 端点开闭 = 正确性命门：过焦点的弦，水平线（x 轴，θ=0）与竖直线（θ=90）都是合法直线，它们取到的端点要计入。例：椭圆 MA·MB 题，x 轴取到 −3、竖直线取到 7/4，故答案是闭区间
[-3, 7/4]
（很多教辅误写成开的
(-3, 7/4]
）。
range_over_m
已据此判定，且这样答案与交互工具一致——拖滑块到 0° 就读到 −3。抛物线焦点弦的"轴方向"是退化线（只交一点），其极限端点不计入（
horizontal_valid=False
或限制 param 范围）。

Follow the solution recipe in

references/conventions.md

, call

lib/analytic_kernel.py

and

lib/conics.py

```
conics.ellipse/hyperbola/parabola/circle(...)
```
to get curve objects (precise a,b,c, foci, vertices, directrices, asymptotes,
```
eq_latex
```
, and
```
board
```
dict for front-end engine).
```
chord_setup(conic, through)
```
to set up simultaneous parametric line
```
x=my+c
```
and obtain quadratic equation in y + Vieta quantities (precise).

Target quantities:

dot_product_expr

chord_len_sq_expr

triangle_area_expr

slope_product_central

…

Value ranges:
```
range_over_m(expr, horizontal_valid=?)
```
— includes open/closed endpoint determination (key correctness point, see below).
Fixed values:
```
is_constant_in_m(expr)
```
.

Self-check the kernel via command line:

bash

python3 lib/analytic_kernel.py      # Built-in assertion self-check for flagship problems

⚠️ Endpoint open/closed = correctness critical point: For chords passing through foci, horizontal lines (x-axis, θ=0) and vertical lines (θ=90) are valid lines, and their endpoints should be included. Example: For an ellipse MA·MB problem, the x-axis gives −3 and the vertical line gives 7/4, so the answer is the closed interval
[-3, 7/4]
(many teaching materials incorrectly write it as the open interval
(-3, 7/4]
).
range_over_m
has already made this determination, and the answer is consistent with the interactive tool — drag the slider to 0° to read −3. The "axis direction" of a parabola's focal chord is a degenerate line (intersects at only one point), and its limit endpoints are not included (
horizontal_valid=False
or restrict param range).

第 3 步：组装数据并注入模板

Step 3: Assemble data and inject into template

📍 输出位置 & 唯一产物（最重要）：交付给用户的只有一个
.html
，写到当前工作目录（
Path.cwd()
）（除非用户显式指定路径）。cwd 里不要留任何别的文件——构建脚本（
.py
）、
__pycache__
、自检截图（
.png
）、临时文件都不是交付物，一律放
/tmp
或用完即删。也绝不要写进技能自身目录（
skills/edu-analytic-geometry/output/
是技能内部样例）。

把"组装数据 + 注入模板"的构建脚本写到临时目录（如

/tmp/ag_build.py

），让它只把
.html
写到 cwd；脚本拼出

lesson

steps

board

数据（schema 见

references/problem-schema.md

），调用

generate.render_html(data, out)

注入

template/board.html

，跑完即删脚本：

python

undefined

📍 Output Location & Unique Product (Most Important): The only deliverable to the user is a single
.html
file, written to the current working directory (
Path.cwd()
) (unless the user explicitly specifies a path). Do not leave any other files in the cwd — build scripts (
.py
),
__pycache__
, self-check screenshots (
.png
), temporary files are not deliverables; place them in
/tmp
or delete them after use. Never write into the skill's own directory (
skills/edu-analytic-geometry/output/
is for internal skill samples).

Write the "assemble data + inject template" build script to a temporary directory (e.g.,

/tmp/ag_build.py

), make it only write the
.html
file to cwd; the script constructs

lesson

steps

board

data (schema see

references/problem-schema.md

), calls

generate.render_html(data, out)

to inject into

template/board.html

, delete the script immediately after running:

python

undefined

构建脚本放 /tmp（不要放 cwd）：/tmp/ag_build.py

Build script placed in /tmp (not cwd): /tmp/ag_build.py

import sys; sys.dont_write_bytecode = True # 不生成 pycache sys.path.insert(0, "<技能目录>/scripts") import generate from pathlib import Path data = {"lesson": {...}, "steps": [...], "board": {...}} out = Path.cwd() / "solution-<题目简述>.html" # 唯一产物，落在用户当前目录 generate.render_html(data, out)


```bash
python3 -B /tmp/ag_build.py && rm -f /tmp/ag_build.py   # -B 不写字节码；跑完删临时脚本，cwd 只剩 .html

```
steps[*].content
```
里的数值直接引用 kernel 结果（用
```
K.tex(...)
```
输出 LaTeX），模型只负责组织讲解文字（按目标语言）。
```
board
```
用 kernel 给的曲线
```
board
```
dict、精确点坐标、
```
param
```
、
```
derived
```
构造序列、
```
readouts
```
、
```
rangeBar
```
（范围题）/
```
constant
```
（定值题）/
```
answerBand
```
（形状参数题，如离心率范围）。
形状参数题（滑块=离心率 e 等）：自然动态量是曲线本身的形状而非动直线/动点时，让滑块=该参数，把曲线
```
a/b/c
```
、焦点、动点坐标写成
```
@param
```
的表达式字符串（引擎每帧重绘曲线/焦点/渐近线），配
```
status
```
读数显示不等式状态、
```
answerBand
```
在参数轴高亮答案区间。见 conventions「形状参数题」。
可直接照抄的范本：
```
scripts/generate.py
```
里 6 个
```
build_*
```
覆盖各类交互范式：
```
ellipse_dot_range
```
（范围条）、
```
ellipse_chord_range
```
、
```
ellipse_area_max
```
、
```
ellipse_slopeprod_const
```
（定值·中心对称）、
```
parabola_dot_const
```
（定值·抛物线）、
```
hyperbola_ecc_range
```
（形状参数：滑块=e，曲线随之重绘 +
```
status
```
+
```
answerBand
```
）。

已注册题直接出（

-B

不写字节码；不传路径默认写技能 output，交付给用户时务必改成 cwd 下的

.html

）：

bash

python3 -B scripts/generate.py list                      # 列出题型
python3 -B scripts/generate.py ellipse_dot_range ./sol.html
python3 -B scripts/generate.py all ./out_dir             # 全部题型

import sys; sys.dont_write_bytecode = True # Do not generate pycache sys.path.insert(0, "<skill directory>/scripts") import generate from pathlib import Path data = {"lesson": {...}, "steps": [...], "board": {...}} out = Path.cwd() / "solution-<problem brief>.html" # Unique product, located in user's current directory generate.render_html(data, out)


```bash
python3 -B /tmp/ag_build.py && rm -f /tmp/ag_build.py   # -B does not write bytecode; delete temporary script after running, only .html remains in cwd

Numerical values in
```
steps[*].content
```
directly reference kernel results (use
```
K.tex(...)
```
to output LaTeX), the model is only responsible for organizing explanatory text (in the target language).
```
board
```
is constructed using the curve's
```
board
```
dict from kernel, precise point coordinates,
```
param
```
,
```
derived
```
sequences,
```
readouts
```
,
```
rangeBar
```
(range problems)/
```
constant
```
(fixed value problems)/
```
answerBand
```
(shape parameter problems, such as eccentricity ranges).
Shape parameter problems (slider = eccentricity e, etc.): When the natural dynamic quantity is the curve's own shape rather than moving lines/moving points, set the slider to this parameter, write the curve's
```
a/b/c
```
, foci, moving point coordinates as expression strings with
@param
(the engine redraws the curve/foci/asymptotes every frame), match with
```
status
```
reading to display inequality status, and
```
answerBand
```
to highlight the answer interval on the parameter axis. See conventions "Shape Parameter Problems".
Copyable templates: The 6
```
build_*
```
functions in
```
scripts/generate.py
```
cover various interaction paradigms:
```
ellipse_dot_range
```
(range bar),
```
ellipse_chord_range
```
,
```
ellipse_area_max
```
,
```
ellipse_slopeprod_const
```
(fixed value · central symmetry),
```
parabola_dot_const
```
(fixed value · parabola),
```
hyperbola_ecc_range
```
(shape parameter: slider = e, curve redraws accordingly +
```
status
```
+
```
answerBand
```
).

Generate registered problems directly (

-B

does not write bytecode; default writes to skill output if no path is passed, must change to

.html

in cwd when delivering to users):

bash

python3 -B scripts/generate.py list                      # List problem types
python3 -B scripts/generate.py ellipse_dot_range ./sol.html
python3 -B scripts/generate.py all ./out_dir             # All problem types

第 4 步：自检（正确性方案）

Step 4: Self-check (Correctness Plan)

kernel 答案 == 答案卡
```
lesson.answer
```
== 末步骤展示值 == JS 标准位/扫段重算值，四者一致（
```
build_*
```
内已加
```
assert
```
）。
```
rangeBar
```
端点来自 kernel 的
```
range_over_m
```
；
```
constant
```
值来自 kernel 的定值。
起本地静态服务（服务输出文件所在目录，即 cwd）用预览检查：无控制台报错、KaTeX 正常、滑块实时重算正确、范围条/定值/定点/轨迹行为符合、画笔与收起面板可用。（技能仓库内开发时可用
```
.claude/launch.json
```
的
```
ag-preview
```
，端口 4601；别处运行就对 cwd 起一个临时静态服务。）
自检截图只给你自己看：preview 工具直接返回图像，不要把
.png
存到 cwd；本地静态服务只读不写、不产生文件。自检产生的任何临时文件（构建脚本
```
.py
```
、截图
```
.png
```
、
```
__pycache__
```
等）交付前一律清掉。

⚠️ 必须关闭你开过的端口/服务：预览一结束立即停掉，绝不留占用端口的进程。
preview 工具开的：
preview_stop
（传 serverId）。
直接起的
http.server
：用完
kill
，或
lsof -nP -iTCP:<port> -sTCP:LISTEN
确认已释放。
交付前确认端口已释放再告诉用户。开了不关 = 未完成自检。

Kernel answer == answer card
```
lesson.answer
```
== final step display value == JS standard bit/segment scan recalculation value, all four are consistent (
```
assert
```
has been added in
```
build_*
```
).
```
rangeBar
```
endpoints come from kernel's
```
range_over_m
```
;
```
constant
```
value comes from kernel's fixed value.
Start a local static server (serving the directory where the output file is located, i.e., cwd) to preview and check: no console errors, KaTeX renders normally, slider recalculates in real-time correctly, range bar/fixed value/fixed point/locus behavior is correct, brush and panel collapse are usable. (When developing in the skill repository, use
```
ag-preview
```
in
```
.claude/launch.json
```
, port 4601; when running elsewhere, start a temporary static server for cwd.)
Self-check screenshots are for your own viewing only: The preview tool returns images directly, do not save
.png
to cwd; the local static server is read-only, no files are generated. Delete any temporary files generated during self-check (build scripts
```
.py
```
, screenshots
```
.png
```
,
```
__pycache__
```
, etc.) before delivery.

⚠️ Must close any ports/services you opened: Stop the server immediately after preview, never leave port-occupying processes running.
Opened by preview tool:
preview_stop
(pass serverId).
Directly started
http.server
: Use
kill
after use, or
lsof -nP -iTCP:<port> -sTCP:LISTEN
to confirm release.
Confirm ports are released before notifying the user. Not closing ports = incomplete self-check.

第 5 步：交付

Step 5: Delivery

成品写在用户当前工作目录（cwd），命名形如

solution-<题目简述>.html

，把路径告诉用户，可直接浏览器打开。交付前确认：(1) 成品在 cwd、不在技能目录；(2) 没有遗留本次预览开启的本地服务/端口；(3) cwd 里只新增了这一个
.html
——没有

.py

.png

__pycache__

/ 临时文件（用

git status

或

ls

核一眼，有就删掉）。

The finished product is written to the user's current working directory (cwd), named like

solution-<problem brief>.html

. Inform the user of the path, which can be directly opened in a browser. Before delivery, confirm: (1) The finished product is in cwd, not in the skill directory; (2) No local services/ports opened during this preview are left running; (3) Only this single
.html
file is newly added in cwd — no

.py

.png

__pycache__

/ temporary files (check with

git status

ls

, delete if any).

扩展

Expansion

加题型：在
```
analytic_kernel.py
```
加目标量函数（写成 m 的表达式）+ 复用
```
range_over_m
```
/
```
is_constant_in_m
```
；在
```
generate.py
```
加一个
```
build_*
```
，选定交互范式（范围条 / 定值 / 定点 / 轨迹 trace / 形状参数 answerBand）。见
```
references/conventions.md
```
配方表。
加曲线：
```
conics.py
```
已有椭圆/双曲线/抛物线/圆；前端
```
board.html
```
引擎已支持四类渲染、渐近线、准线方向。新曲线在两处各加一份即可。
加交互构造：
```
board.html
```
的
```
buildScene
```
switch 是构造库（
```
line_through_angle
```
、
```
intersect_line_conic
```
、
```
point_on_conic
```
、
```
point_reflect
```
、
```
tangent_at
```
、
```
foot_perp
```
…），按需扩充并在 schema 文档登记。

Add problem types: Add target quantity functions (written as expressions of m) in
```
analytic_kernel.py
```
+ reuse
```
range_over_m
```
/
```
is_constant_in_m
```
; add a
```
build_*
```
function in
```
generate.py
```
, select an interaction paradigm (range bar / fixed value / fixed point / locus trace / shape parameter answerBand). See the recipe table in
```
references/conventions.md
```
.
Add curves:
```
conics.py
```
already has ellipse/hyperbola/parabola/circle; the front-end
```
board.html
```
engine already supports rendering of four types, asymptotes, and directrix directions. Just add new curves in both places.
Add interactive constructions: The
```
buildScene
```
switch in
```
board.html
```
is a construction library (
```
line_through_angle
```
,
```
intersect_line_conic
```
,
```
point_on_conic
```
,
```
point_reflect
```
,
```
tangent_at
```
,
```
foot_perp
```
…), expand as needed and register in the schema document.

edu-analytic-geometry

Original

Translation

解析几何解题 → 交互网页

Analytic Geometry Problem Solving → Interactive Webpage

这个技能产出什么

What This Skill Produces

依赖（重要）

Dependencies (Important)

工作流程

Workflow

第 1 步：得到 problem spec（三入口归一）

Step 1: Obtain problem spec (unify three entry points)

第 2 步：用 kernel 精确计算（不要心算）

Step 2: Perform precise calculations with kernel (do not calculate manually)

第 3 步：组装数据并注入模板

Step 3: Assemble data and inject into template

构建脚本放 /tmp（不要放 cwd）：/tmp/ag_build.py

Build script placed in /tmp (not cwd): /tmp/ag_build.py

第 4 步：自检（正确性方案）

Step 4: Self-check (Correctness Plan)

第 5 步：交付

Step 5: Delivery

扩展

Expansion

目录

Directory