Fold Trick-Art — Flat Sheets that Become 3-D

---

# 🪄 Fold Trick-Art

## Flat drawings that snap into 3-D 🔮

### 🪑 🍽️ 🏠

@luk036 👨‍💻 · 2026 📅

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### 🗓️ Agenda

**1 · The Illusion** 🎩
- The goal 🎯
- One idea 💡
- The stage 📐

**2 · The Method** 🧭
- The $\color{steelblue}{\sigma}$ test ✅
- Unfolding 🗺️

**3 · Drawing It** ✂️
- Faces & clipping
- Painting order 🎨

]
.pull-right[

**4 · The Objects** 🧱
- 🪑 Chair · 🍽️ Table · 🏠 House

**5 · Engineering** 🧮
- `projgeom` + exact math
- 🅲➕➕ Port to C++ (xmake · MSVC)
- Verification 🔬

**6 · Takeaways** 🏁

]

---

# 🎩 The Illusion

---

### 🎯 The Goal

We want a **printable sheet** 📄 that lies perfectly flat…

…but when you **fold it 90°** and look from *one* spot 👁️, it snaps into a **solid 3-D object** 🪑.

- ✏️ Built by *projective geometry*, not by hand
- 📏 **Exact rational arithmetic** — no floating point
- 🖨️ **A4**, sized in millimetres, printable at 100 %

> 🤯 Straight-on it looks *wrong* — at the right angle it looks *real*.

---

### 🧩 The Trick, in One Idea

> Paint every point of the object **onto the sheet** so that, from the viewer, it lies on the *same line of sight* as the real point. 👁️ ➡️ 📍

The sheet is **not one plane** — it is a right-angled **corner**:

- a **wall** 🧱 and a **floor** 🟦
- meeting at the **crease** 📏

**Fold the sheet ⇒ get the corner.** That is the whole magic. 🪄

---

### 📐 The Stage — a cross-section

<p align="center">
<svg width="600" height="340" viewBox="0 0 600 340" xmlns="http://www.w3.org/2000/svg" style="background:#fafafa;border:1px solid #d0d0d0;border-radius:6px">
  <polygon points="200,300 600,300 600,340 200,340" fill="#dbeafe" opacity="0.75"/>
  <line x1="200" y1="300" x2="600" y2="300" stroke="#2563eb" stroke-width="3"/>
  <text x="590" y="330" font-family="sans-serif" font-size="14" fill="#2563eb" text-anchor="end">floor  z = 0 🟦</text>
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  <text x="150" y="80" font-family="sans-serif" font-size="14" fill="#64748b" text-anchor="end">wall  y = 0 🧱</text>
  <line x1="520" y1="140" x2="280" y2="300" stroke="#16a34a" stroke-width="2.5" stroke-dasharray="7 4"/>
  <circle cx="200" cy="300" r="5" fill="#dc2626"/>
  <text x="176" y="322" font-family="sans-serif" font-size="13" fill="#dc2626" text-anchor="end">crease 📏</text>
  <circle cx="520" cy="140" r="7" fill="#0ea5e9"/>
  <text x="532" y="137" font-family="sans-serif" font-size="17" font-weight="bold" fill="#0ea5e9">V 👁️</text>
  <circle cx="400" cy="220" r="7" fill="#ef4444"/>
  <text x="409" y="214" font-family="sans-serif" font-size="17" font-weight="bold" fill="#ef4444">P</text>
  <circle cx="280" cy="300" r="7" fill="#16a34a"/>
  <text x="232" y="292" font-family="sans-serif" font-size="17" font-weight="bold" fill="#16a34a">Q</text>
</svg>
</p>

- Viewer $\color{steelblue}{V}$ 👁️ sits in the room; the ray to a point $\color{salmon}{P}$ 🌸 hits the corner at $\color{darkgreen}{Q}$.

---

### 🔦 One Ray, Two Shelves

Along the ray, the two plane crossings are

$$\color{teal}{t_{\text{floor}}} = \frac{V_z}{V_z - P_z}
\qquad\qquad
\color{firebrick}{t_{\text{wall}}} = \frac{V_y}{V_y - P_y}$$

The **nearer** plane is the one the eye actually sees there — so *that* is where we record the colour. ✅

Fold it back, stand at $V$: the recorded point and $\color{salmon}{P}$ are **collinear** — same sight-line. 👁️ ✨

---

### ✅ Which Plane First? — the σ test

Floor is nearer $\iff \color{teal}{t_{\text{floor}}} < \color{firebrick}{t_{\text{wall}}}$, which collapses to a *linear* test in the point $\color{salmon}{P}$:

$$\color{purple}{\sigma(P)} \;=\; \color{steelblue}{V_z}\,\color{salmon}{P_y} \;-\; \color{steelblue}{V_y}\,\color{salmon}{P_z}$$

- $\color{teal}{\sigma > 0}$ ➡️ record on the **floor** 🟦
- $\color{firebrick}{\sigma < 0}$ ➡️ record on the **wall** 🧱

Because $\sigma$ is **linear** 🔹, a convex face lies wholly on one plane *iff its vertices agree* — and any crossing is a **single line**, easy to clip. ✂️

---

### 🗺️ Unfolding the Sheets

Rotate the wall flat into the floor about the crease:

$$\text{wall } (x,\,0,\,z) \;\mapsto\; (x,\; \color{firebrick}{z})
\qquad\qquad
\text{floor } (x,\,y,\,0) \;\mapsto\; (x,\; \color{teal}{-y})$$

So $v = 0$ is the **crease**: 🧱 wall flap *above*, 🟦 floor flap *below*.

That is exactly the layout of the reference crafts — backrest up, seat down. 🪑

---

### ✂️ Faces & Edges — the pipeline

<pre>
graph LR
    A["🧱 3-D object\nfaces + edges"] --> B["🔦 project from V\nσ = Vz·Py − Vy·Pz"]
    B --> C{"crosses σ = 0 ?"}
    C -->|no| D["map whole face\n(one homography)"]
    C -->|yes| E["✂️ clip into\nwall ∕ floor pieces"]
    D --> F["🗺️ unfold to the flat sheet"]
    E --> F
    F --> G["🎨 paint far→near\n+ black outlines"]
    G --> H["📄 printable A4 SVG"]

style A fill:#ffe0b2,stroke:#e65100,stroke-width:3px
    style B fill:#bbdefb,stroke:#1565c0,stroke-width:3px
    style C fill:#fff9c4,stroke:#f9a825,stroke-width:3px
    style D fill:#c8e6c9,stroke:#2e7d32,stroke-width:3px
    style E fill:#f8bbd0,stroke:#c2185b,stroke-width:3px
    style F fill:#d1c4e9,stroke:#5e35b1,stroke-width:3px
    style G fill:#b2dfdb,stroke:#00695c,stroke-width:3px
    style H fill:#eeeeee,stroke:#424242,stroke-width:3px
</pre>

]

---

### 🎨 Painting Order

Occlusion is the only hard part 🧱.

- Sort faces by **squared** distance from $V$ — uses $\lVert V - c \rVert^2$, **no square roots** 🚫√
- Paint **far ➡️ near**
- Then draw the **edges** on top ✏️
- The crease is marked with a red dashed line 📏

---

### 🧮 Exact Arithmetic

Every coordinate is a `fractions.Fraction` 🧩; the projective maps come from `projgeom.projective_transform`.

```python
from fractions import Fraction as F
from projgeom import PgPoint, projective_transform

def sigma(p):                       # wall/floor classifier
    return VZ * p[1] - VY * p[2]    # <0 wall, >0 floor

def project(p):
    if sigma(p) <= 0:                # wall flap (y = 0)
        t = VY / (VY - p[1])
        return VX + t*(p[0]-VX), VZ + t*(p[2]-VZ)
    t = VZ / (VZ - p[2])            # floor flap (z = 0)
    return VX + t*(p[0]-VX), -(VY + t*(p[1]-VY))
```

> 📏 Millimetre SVG, exact rounding, **zero** floating point in the geometry.

---

### 🪞 Why It Looks Wrong Flat

Flat, the sheet looks skewed 🌀 — *that is the point*.

The distortion baked into the sheet **cancels** both the foreshortening of the fold **and** the viewing angle. 📐➕🪄

> Fold it, stand at $V$ 👁️ … and it snaps into place. 🔮

---

# 🧱 The Objects

---

### 🪑 Chair — the flat sheet

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**Left:** print & fold 📄 → **Right:** what you see from $V$ 👁️

---

### ⚖️ The Crease Insight

The chair looks cleanest when the backrest plane sits exactly on $\sigma = 0$:

$$Y_0 \;=\; \frac{Z_s\,V_y}{V_z} \qquad\Longleftrightarrow\qquad t_{\text{wall}} = t_{\text{floor}}$$

- 🡄 move it **closer** ➡️ the crease crosses the *flat seat*
- 🡆 move it **away** ➡️ the green backrest **skews** 📐
- ⚠️ a clean backrest and “farther from the wall” are *mutually exclusive*

---

### 🍽️ Table

**Left:** print & fold 📄 → **Right:** the table you see from $V$ 🍽️

---

### 🏠 House — the most complex so far

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Walls 🧱 + gable 🔺 + **sloped roof** + door 🚪 + window 🪟 — non-axis-aligned faces, all handled by the same pipeline.

---

### 🎬 Watch It Fold

- 📄 flat sheet ➡️ 🪄 fold to 90° ➡️ 🪑 the object **rises out** of the drawing
- 🪑 chair · 🍽️ table · 🏠 house — one **24 s** loop 🎞️
- 📱 square, H.264 — folds onto any phone

]
.pull-right[

]

---

### 📱 The Same Content — Flat Screen

- 📄 on a **flat** screen the exact same frames read as a squashed, **warping** drawing 🌀
- 🔄 it only morphs — the chair never *stands up*
- 🧱 the 3-D reading needs the **fold** 👉 next slide

]
.pull-right[

]

---

### 📱 On a 90°-Folded Phone

- 📱 fold a phone's screen to **90°** — the two halves *become* the floor plane and the wall plane 🧱
- 🪑 the chair is projected onto **both halves**, so it spins as a genuine 3-D chair 🔄
- 👁️ seen from $V$, the fold **is** the corner of the anamorphosis — no paper needed
- 📺 on a real device the two panels just display two pre-computed frames 🎞️

]
.pull-right[

]

---

### 🧱 Under the Hood

<pre>
graph LR
    OBJ["📦 object model\nfaces(colour) + edges"] --> SIG["σ classifier\nlinear form"]
    SIG --> CLIP["✂️ Sutherland–Hodgman\nhalf-plane clip"]
    CLIP --> MAP["π projective map\nprojgeom.projective_transform"]
    MAP --> PAINT["🎨 painter sort\n‖V − c‖²"]
    PAINT --> SVG["📄 SVG writer\nmm, exact round"]

subgraph "exact rationals 🧩"
    SIG
    CLIP
    MAP
    end

style OBJ fill:#ffe0b2,stroke:#e65100,stroke-width:3px
    style SIG fill:#bbdefb,stroke:#1565c0,stroke-width:3px
    style CLIP fill:#f8bbd0,stroke:#c2185b,stroke-width:3px
    style MAP fill:#d1c4e9,stroke:#5e35b1,stroke-width:3px
    style PAINT fill:#b2dfdb,stroke:#00695c,stroke-width:3px
    style SVG fill:#eeeeee,stroke:#424242,stroke-width:3px
</pre>

]

---

# 📦 Porting to C++

---

### 🔢 How Wide an Integer?

<p align="center">
<svg width="660" height="205" viewBox="0 0 660 205" xmlns="http://www.w3.org/2000/svg" style="background:#fafafa;border:1px solid #d0d0d0;border-radius:6px">
  <text x="14" y="24" font-family="sans-serif" font-size="15" font-weight="bold" fill="#2e3440">Largest exact value vs integer width (log scale)</text>
  <line x1="246" y1="38" x2="246" y2="176" stroke="#bf616a" stroke-width="2" stroke-dasharray="6 4"/>
  <text x="246" y="194" text-anchor="middle" font-family="sans-serif" font-size="12" font-weight="bold" fill="#bf616a">int32 max · 2.1e9</text>
  <line x1="565" y1="38" x2="565" y2="176" stroke="#a3be8c" stroke-width="2" stroke-dasharray="6 4"/>
  <text x="565" y="194" text-anchor="middle" font-family="sans-serif" font-size="12" font-weight="bold" fill="#2e7d32">int64 max · 9.2e18</text>
  <g font-family="sans-serif" font-size="12" fill="#2e3440">
    <text x="64" y="57" text-anchor="end">chair</text>
    <rect x="70" y="48" width="50" height="13" fill="#5e81ac" rx="2"/>
    <text x="126" y="59" fill="#5e81ac">3.2e5</text>
    <text x="64" y="86" text-anchor="end">house</text>
    <rect x="70" y="77" width="114" height="13" fill="#5e81ac" rx="2"/>
    <text x="190" y="88" fill="#5e81ac">2.7e7</text>
    <text x="64" y="115" text-anchor="end">table</text>
    <rect x="70" y="106" width="126" height="13" fill="#5e81ac" rx="2"/>
    <text x="202" y="117" fill="#5e81ac">6.1e7</text>
    <text x="64" y="148" text-anchor="end" fill="#bf616a" font-weight="bold">product ~1e15</text>
    <rect x="70" y="139" width="383" height="13" fill="#bf616a" rx="2"/>
    <text x="459" y="150" fill="#bf616a" font-weight="bold">👉 overflows int32 ✖️</text>
  </g>
</svg>
</p>

$$|\text{num}|_{\max} \approx 6.1\times10^{7}, \qquad
\text{int32}_{\max} = 2.1\times10^{9}, \qquad
\text{int64}_{\max} = 9.2\times10^{18}$$

- ✅ **Storing** the largest coordinate fits in `int32` … but only $\sim 35\times$ under the limit
- ❌ **Computing** it multiplies $10^{7}$-scale values ➡️ products $\sim 10^{15}$ — `int32` overflows by $\sim 10^{5}$
- 🏆 **Verdict:** use $\color{darkgreen}{\text{int64}}$ — the sibling library's own choice: `fun::Frac = Fraction<std::int64_t>`

---

### 📦 The Same Method in C++

<pre>
graph LR
    PY["🐍 Python\nFraction + projgeom"] -->|line-by-line port| CPP["🅲➕➕ C++\nfun::Frac + fun::projective_transform"]
    CPP --> OUT["📄 chair · table · house\n(identical output)"]

style PY fill:#bbdefb,stroke:#1565c0,stroke-width:3px
    style CPP fill:#c8e6c9,stroke:#2e7d32,stroke-width:3px
    style OUT fill:#d1c4e9,stroke:#5e35b1,stroke-width:3px
</pre>

]

The sibling repo **`projgeom-cpp`** already ships both ingredients 🧩

```cpp
#include <projgeom/transform.hpp>
using fun::Frac;                        // Fraction<std::int64_t>

auto sigma(const Scene& s, const V3& p) -> Frac {
    return s.vz * p.y - s.vy * p.z;     // <0 wall flap, >0 floor flap
}
```

- 📁 added **`examples/fold_trick.cpp`** — one file, three objects 🪑 🍽️ 🏠
- 🔁 identical pipeline: $\color{purple}{\sigma}$ ➡️ ✂️ clip ➡️ `fun::projective_transform` ➡️ 🎨 painter ➡️ 📄 SVG

---

### 🧪 Verified: xmake + MSVC

- ⚙️ `xmake f -p windows -a x64 --toolchain=msvc` ➡️ **MSVC 14.51** (VS 18) ✅
- 🔨 `xmake build fold_trick` ➡️ **`build ok`** ⚡
- 🖼️ rendered **both** toolchains' SVGs ➡️ **pixel-identical**:

| object | Python ↔ C++ (absolute error) |
| ---------- | -------------- |
| 🪑 chair | **0** ✅ |
| 🍽️ table | **0** ✅ |
| 🏠 house | **0** ✅ |

---

### 🐛 Gotchas Found

- 🔀 `std::sort` is **unstable** ➡️ use `std::stable_sort` to match Python's stable sort, or equal-depth faces reorder and fills repaint
- 🐞 latent bug in `fractions.hpp`: `operator/=(const Z&)` does `std::swap(_den, rhs)` on a **const** reference ➡️ `frac / int` won't compile; divide by `Frac` instead
- 🧹 removed a duplicated gable edge from `house_flat.py` (the one thing that made the house differ)

---

### 🔬 Verification

Self-check folds every sheet point back to 3-D and tests collinearity with $V$:

$$\color{steelblue}{V},\;\; \color{salmon}{P},\;\; \color{darkgreen}{Q}\quad\text{collinear ?}$$

…in two coordinate projections, using `projgeom` incidence (`dot = 0`). ✅

- 🪑 chair · 🍽️ table · 🏠 house — all **verified**
- 🧪 `flake8` clean · valid XML · **no** geometry floats

---

### 🧗 Lessons & Pitfalls

- 🌀 Anamorphosis ≠ ordinary perspective
- ✂️ The crease **cuts** faces — clip, don't sample
- 📏 Exact rationals keep it crisp
- 🔢 Points at infinity ⇒ use homogeneous coords

]
.pull-right[

- 🧱 Occlusion is the hard part
- ⚖️ Alignment ⟺ the crease plane
- 👁️ **One** viewpoint only
- 🖨️ Print at 100 %, fold exactly 90°

]

---

### 🏁 Takeaways

> A flat drawing can *be* a 3-D object 🪄 — you just have to know **where to stand**. 👁️

- 📐 Two planes + a viewpoint $V$ = the whole method
- $\color{purple}{\sigma}$ is the switch: 🟦 floor or 🧱 wall
- 🧩 Exact arithmetic makes it reproducible
- 🖨️ Fold, look, enjoy 🔮

---

## Q&A 🎤

### Thank you! 🙏

🪑 🍽️ 🏠 — print, fold, and be amazed 🔮