Rectilinear Polygon - Representation and Algorithms

---
count: false
class: nord-dark, middle, center

# 📐 Rectilinear Polygon: Representation and Algorithms

### VLSI Physical Design Computational Geometry

@luk036 👨‍💻 · 2026 📅

---

### 📋 Agenda

**Part 1: Introduction & Representation** 🎯
- What is a Rectilinear Polygon? 🤔
- Rectilinear vs General 📐
- Origin + Vectors Representation 📦

**Part 2: Basic Operations** ⚙️
- Creating RPolygons 🔧
- Translation Operations ➕➖
- Signed Area & Orientation 📐
- Convert to Standard Polygon 🔃

**Part 3: Monotone & Convex Hull** 🛡️
- X/Y-Monotone Creation 🔄
- Convex Hull Algorithm 🛡️
- Hull Visualizations 📊

]
.pull-right[

**Part 4: Point Inclusion & Cutting** 🔍
- Ray-Casting Algorithm 📐
- Convex Decomposition ✂️

**Part 5: Code Demo** 💻
- Basic Usage 🐍
- Monotone & Point Inclusion 🧪
- Convex Decomposition 🔪

**Part 6: Analysis & Applications** 📊
- Complexity 📈
- Related Modules 🔄
- File Structure 📚
- VLSI Applications 🎯
- Summary ✅

**Appendices** 📎
- Monotone Detection & Formulas 🧮

]

---

## Part 1: Introduction & Representation 🎯

---

### 🎯 What is a Rectilinear Polygon?

> A **rectilinear polygon** (or orthogonal polygon) is a polygon whose edges are **either horizontal or vertical**.

.mermaid[
<pre>
graph TD
    A[Polygon Types] --> B[General Polygon]
    A --> C[Rectilinear Polygon]
    B --> D[Any angle edges]
    C --> E[Only H/V edges]

style A fill:#e1f5fe,stroke:#01579b,stroke-width:3px
    style C fill:#e1f5fe,stroke:#01579b,stroke-width:3px
    style E fill:#e1f5fe,stroke:#01579b,stroke-width:3px
</pre>
]
]
.pull-right[
#### Key Properties

-   ✅ Edges alternate between horizontal and vertical
-   ✅ Interior angles are either 90° or 270°
-   ✅ Common in **VLSI layout**, **floorplanning**, **routing**
]

---

### 📐 Example: Rectilinear vs General

```text
General Polygon:          Rectilinear Polygon:

/\
     /  \                 ┌──────┐
    /    \                │      │
   /      \               │      └─────┐
  /        \              │            │
 /__________\             └────────────┘

Any angles               Only 90°/270° corners
```

#### Why Rectilinear?

-   ✅ Matches Manhattan geometry of VLSI circuits
-   ✅ Simplified collision detection
-   ✅ Efficient area calculation
-   ✅ Natural representation for routing grids

---

### 📦 Representation: Origin + Vectors

#### Internal Data Structure

```python
class RPolygon:
    _origin: Point[int, int]           # Reference point
    _vecs: List[Vector2[int, int]]      # Edge vectors from origin
```

#### Visualization

```text

(origin) +──v1──>┐
              │     │
              │     v2
              v0<──┘

Origin: (400, 500)
   Vectors: [(0,-4), (0,-1), (3,-3), (5,1), ...]
```

#### Key Insight

> Vectors are stored **relative to origin**, enabling efficient translation.

---

## Part 2: Basic Operations ⚙️

---

### 🔧 Creating RPolygons

#### Method 1: From Point Set

.font-sm.mb-xs[
```python
@classmethod
def from_pointset(cls, pointset: PointSet) -> "RPolygon":
    """Create polygon from a list of points"""
    origin = pointset[0]
    vecs = [vtx.displace(origin) for vtx in pointset[1:]]
    return cls(origin, vecs)
```
]

#### Method 2: Direct Construction

.font-sm.mb-xs[
```python
def __init__(self, origin: Point[int, int], vecs: List[Vector2[int, int]]):
    self._origin = origin
    self._vecs = vecs
```
]

#### Example

.font-sm.mb-xs[
```python
coords = [(0,0), (0,1), (1,1), (1,0)]
points = [Point(x, y) for x, y in coords]
poly = RPolygon.from_pointset(points)
```
]

---

### ➕➖ Translation Operations

#### In-Place Addition/Subtraction

```python
def __iadd__(self, rhs: Vector2[int, int]) -> "RPolygon":
    """Translate polygon by adding vector to origin"""
    self._origin += rhs
    return self

def __isub__(self, rhs: Vector2[int, int]) -> "RPolygon":
    """Translate polygon by subtracting vector from origin"""
    self._origin -= rhs
    return self
```

```python
poly += Vector2(10, 5)
poly -= Vector2(3, 0)
```
]
.pull-right[
#### Why in-place?

-   ✅ Memory efficient
-   ✅ Useful for incremental placement
]

---

### 📐 Signed Area Calculation

#### Using the Shoelace-like Formula

.font-sm.mb-xs[

```python
@cached_property
def signed_area(self) -> int:
    """Calculate signed area of the polygon"""
    if len(self._vecs) < 1:
        return 0
    vec0 = self._vecs[0]
    return sum(
        [v1.x * (v1.y - v0.y) for v0, v1 in zip(self._vecs[:-1], self._vecs[1:])],
        vec0.x * vec0.y,
    )
```
]

#### Example

```python
coords = [(0,0), (0,1), (1,1), (1,0)]
P = RPolygon.from_pointset([Point(x,y) for x,y in coords])
print(P.signed_area)  # -1 (clockwise)
```

---

### 🔄 Orientation Detection

#### Clockwise vs Anticlockwise

```python
def is_anticlockwise(self) -> bool:
    """Check if polygon is anticlockwise"""
    pointset = [Vector2(0, 0)] + self._vecs

# Find minimum coordinate point (bottom-left)
    min_index, min_point = min(enumerate(pointset),
                               key=lambda it: (it[1].x, it[1].y))

prev_point = pointset[(min_index - 1) % len(pointset)]
    return prev_point.y > min_point.y
```

#### Key Insight

> Uses the **bottom-leftmost vertex** as reference to determine orientation.

---

### 🔃 Convert to Standard Polygon

#### RPolygon → Polygon

```python
def to_polygon(self) -> Polygon[int]:
    """Convert rectilinear polygon to standard polygon"""
    new_vecs = []
    current_pt = Vector2(0, 0)

for next_pt in self._vecs:
        if current_pt.x != next_pt.x and current_pt.y != next_pt.y:
            # Add intermediate point for diagonal
            new_vecs.append(Vector2(next_pt.x, current_pt.y))
        new_vecs.append(next_pt)
        current_pt = next_pt

# Handle closing segment
    first_pt = Vector2(0, 0)
    if current_pt.x != first_pt.x and current_pt.y != first_pt.y:
        new_vecs.append(Vector2(first_pt.x, current_pt.y))

return Polygon(self._origin, new_vecs)
```

---

## Part 3: Monotone & Convex Hull 🛡️

---

### 🔄 Monotone Polygon Creation

#### What is a Monotone Polygon?

> A polygon is **monotone** if a line in a given direction intersects it at most twice.

.pull-left[
.mermaid[
<pre>
graph LR
    subgraph "X-Monotone"
        A[Leftmost] --> B[Rightmost]
    end

style A fill:#c8e6c9,stroke:#2e7d32,stroke-width:3px
    style B fill:#c8e6c9,stroke:#2e7d32,stroke-width:3px
</pre>
]
Every vertical line cuts at most twice
]
.pull-right[
.mermaid[
<pre>
graph LR
    subgraph "Y-Monotone"
        C[Bottommost] --> D[Topmost]
    end

style C fill:#e3f2fd,stroke:#1565c0,stroke-width:3px
    style D fill:#e3f2fd,stroke:#1565c0,stroke-width:3px
</pre>
]
Every horizontal line cuts at most twice
]

---

### 🔧 Creating X-Monotone Polygon

#### Algorithm

```python
def create_xmono_rpolygon(lst: PointSet) -> Tuple[PointSet, bool]:
    """
    Create x-monotone rectilinear polygon.
    Returns: (pointset, is_anticlockwise)
    """
    return create_mono_rpolygon(
        lst,
        lambda pt: (pt.xcoord, pt.ycoord),  # Sort by x, then y
        lambda a, b: a < b
    )
```

#### Process

1. Find **leftmost** and **rightmost** points
2. Partition points into **upper** and **lower** chains
3. Sort each chain in the appropriate direction
4. Concatenate: **lower chain** + **upper chain**

---

![X-Monotone Hull](./outputs/rpolyon_xmono_hull.svg)

*Output: `rpolyon_xmono_hull.svg`*
]
.pull-right[

### 📊 Y-Monotone Hull Visualization

![Y-Monotone Hull](./outputs/rpolyon_ymono_hull.svg)

*Output: `rpolyon_ymono_hull.svg`*

]

---

### 🛡️ Convex Hull Algorithm

#### Two-Step Process

```python
def rpolygon_make_convex_hull(
    pointset: PointSet,
    is_anticlockwise: bool
) -> PointSet:
    """Create the convex hull of a rectilinear polygon"""
    # Step 1: Make X-monotone hull
    S = rpolygon_make_xmonotone_hull(pointset, is_anticlockwise)
    # Step 2: Make Y-monotone hull
    return rpolygon_make_ymonotone_hull(S, is_anticlockwise)
```

#### Why Two Steps?

1. **First pass**: Reduce to X-monotone (removes some concavities)
2. **Second pass**: Reduce to Y-monotone (removes remaining concavities)
3. **Result**: Fully convex polygon

#### Complexity: ${\color{green} O(n)}$ using doubly-linked list

---

### 📊 Convex Hull Visualization

#### Rectilinear Polygon Convex Hull

![Convex Hull](./outputs/rpolyon_convex_hull.svg)

*Output: `rpolyon_convex_hull.svg`*
]
.pull-right[
![img](./outputs/rpolyon_convex_hull13.svg)

*Output: `rpolyon_convex_hull13.svg`*
]

---

## Part 4: Point Inclusion & Cutting 🔍

---

### 🔍 Point Inclusion Test

#### Ray-Casting Algorithm

.font-sm.mb-xs[

```python
def point_in_rpolygon(pointset: PointSet, ptq: Point[int, int]) -> bool:
    """
    Determine if a point is inside a rectilinear polygon.
    Uses ray-casting algorithm.
    """
    res = False
    pt0 = pointset[-1]
    for pt1 in pointset:
        if (pt1.ycoord <= ptq.ycoord < pt0.ycoord) or \
           (pt0.ycoord <= ptq.ycoord < pt1.ycoord):
            if pt1.xcoord > ptq.xcoord:
                res = not res
        pt0 = pt1
    return res
```
]

#### Algorithm

1. Cast ray from point to infinity (positive x-direction)
2. Count intersections with polygon edges
3. **Odd** = inside, **Even** = outside

---

### 📐 Ray-Casting Visual

```text

│     │                │    │    │       │
   │     │  o────────┐    │    │    │       │
   │     │  │        │    │    │    │       │
   │     │  │    q───T────F────T────F───────T──────►
   │     │  └──o     │    │    │    │       │
   │     │     │     │    │    │    │       │
   │     │     │     o────┘    │    │       │
   │     │     │               │    │       │

T = Toggle (intersection)
   F = No toggle (ray passes through vertex)
```

---

### ✂️ Polygon Cutting - Convex Decomposition

#### Problem

> Convert a **concave** rectilinear polygon into a set of **convex** polygons.

#### Algorithm: `rpolygon_cut_convex()`

.font-sm.mb-xs[

```python
def rpolygon_cut_convex(lst: PointSet, is_anticlockwise: bool) -> List[PointSet]:
    """
    Recursively cut a rectilinear polygon into convex pieces.

Steps:
    1. Find a concave vertex
    2. Find nearest vertex to cut to
    3. Make the cut (add new edge)
    4. Recursively process both resulting polygons
    """
    rdll = RDllist(len(lst))
    vertices_list = rpolygon_cut_convex_recur(rdll[0], lst, is_anticlockwise, rdll)
    # Convert index lists back to point sets
```
]

---

### 🔍 Finding Concave Vertices

#### Detection Logic

```python
def _find_concave_point(vcurr, cmp2):
    while True:
        vnext = vcurr.next
        vprev = vcurr.prev
        prev_point = lst[vprev.data]
        curr_point = lst[vcurr.data]
        next_point = lst[vnext.data]

vec1 = curr_point.displace(prev_point)
        vec2 = next_point.displace(curr_point)

# Check if edges turn in opposite directions
        if vec1.x_ * vec2.x_ < 0 or vec1.y_ * vec2.y_ < 0:
            area_diff = (curr_point.ycoord - prev_point.ycoord) * \
                        (next_point.xcoord - curr_point.xcoord)
            if cmp2(area_diff):
                return vcurr  # Found concave!
        vcurr = vnext
    return None  # Convex
```

---

### 🎯 Cutting Strategy

#### Find Minimum Distance Point

.font-sm.mb-xs[

```python
def find_min_dist_point(lst: PointSet, vcurr: Dllink[int]) -> Tuple[Dllink[int], bool]:
    """
    Find the closest vertex to connect for cutting.
    Returns: (vertex, is_vertical)
    """
    min_value = math.inf
    vertical = True
    v_min = vcurr
    pcurr = lst[vcurr.data]

# Search through all vertices
    vi = vnext.next
    while id(vi) != id(vprev):
        # Check horizontal alignment
        if (prev_point.ycoord <= pcurr.ycoord <= curr_point.ycoord):
            if abs(vec_i.x_) < min_value:
                min_value = abs(vec_i.x_)
                v_min = vi
                vertical = True
        # Check vertical alignment
        if (next_point.xcoord <= pcurr.xcoord <= curr_point.xcoord):
            if abs(vec_i.y_) < min_value:
                min_value = abs(vec_i.y_)
                v_min = vi
                vertical = False
        vi = vi.next
    return v_min, vertical
```
]

---

### 📊 Convex Cut Visualization

#### After Decomposition

![Cut Convex](./outputs/rpolyon_cut_convex.svg)

*Output: `rpolyon_cut_convex.svg`*

]
.pull-right[

![Cut Convex 2](./outputs/rpolyon_cut_convex2.svg)

*Output: `rpolyon_cut_convex2.svg`*

]

---

## Part 5: Code Demo 💻

---

### 💻 Code Demo - Basic Usage

#### Create a Rectilinear Polygon

```python
from physdes.point import Point
from physdes.rpolygon import RPolygon

# Define vertices (must be rectilinear!)
coords = [
    (0, 0), (0, 2), (1, 2), (1, 1),
    (2, 1), (2, 0)
]
points = [Point(x, y) for x, y in coords]

# Create polygon
poly = RPolygon.from_pointset(points)

# Properties
print(f"Signed area: {poly.signed_area}")
print(f"Is anticlockwise: {poly.is_anticlockwise()}")
```

---

#### Create Monotone Polygons

.font-sm.mb-xs[

```python
from physdes.rpolygon import (
    create_xmono_rpolygon,
    create_ymono_rpolygon,
    rpolygon_make_xmonotone_hull,
    rpolygon_make_convex_hull
)

# Random points (need sorting first)
coords = [(0, -4), (0, -1), (3, -3), (5, 1), (2, 2), (3, 3), (1, 4)]
points = [Point(x, y) for x, y in coords]

# Create x-monotone polygon
xmono, is_anticlockwise = create_xmono_rpolygon(points)
print(f"X-monotone: {len(xmono)} vertices, anticlockwise: {is_anticlockwise}")

# Create y-monotone polygon
ymono, is_clockwise = create_ymono_rpolygon(points)
print(f"Y-monotone: {len(ymono)} vertices, clockwise: {is_clockwise}")
```
]

---

#### Point Inclusion Test

```python
from physdes.rpolygon import point_in_rpolygon

# Define polygon
coords = [(0,0), (0,2), (2,2), (2,0)]
points = [Point(x, y) for x, y in coords]

# Test points
test_points = [
    Point(1, 1),   # Inside
    Point(1, 3),   # Outside
    Point(0, 1),   # On edge
    Point(-1, 1),  # Outside
]

for pt in test_points:
    result = point_in_rpolygon(points, pt)
    print(f"Point {pt}: {'Inside' if result else 'Outside'}")
```

---

#### Convex Decomposition

```python
from physdes.rpolygon import rpolygon_cut_convex

# Define a concave polygon
coords = [
    (0, 0), (0, 3), (1, 3), (1, 1),
    (2, 1), (2, 0)
]  # L-shaped polygon
points = [Point(x, y) for x, y in coords]

# Cut into convex pieces
convex_pieces = rpolygon_cut_convex(points, is_anticlockwise=False)

print(f"Number of convex pieces: {len(convex_pieces)}")
for i, piece in enumerate(convex_pieces):
    print(f"Piece {i+1}: {len(piece)} vertices")
```

---

## Part 6: Analysis & Applications 📊

---

### 📈 Algorithm Complexity

.font-sm[
| Operation | Time Complexity |
|-----------|----------------|
| Signed area | ${\color{green} O(n)}$ |
| Is anticlockwise | ${\color{green} O(n)}$ |
| Create X/Y-monotone | ${\color{orange} O(n \log n)}$ (sorting) |
| Convex hull (X+Y) | ${\color{green} O(n)}$ |
| Point inclusion | ${\color{green} O(n)}$ |
| Convex decomposition | ${\color{red} O(n^2)}$ worst case |
]

Space Complexity: ${\color{blue} O(n)}$ for all operations

---

### 🔄 Related Modules

.font-sm.mb-xs[

| Module | Purpose |
|--------|---------|
| `point.py` | Point and Interval classes |
| `vector2.py` | 2D vector operations |
| `rdllist.py` | Doubly-linked list for polygon traversal |
| `polygon.py` | Standard polygon (general edges) |
| `mywheel.dllist` | C++ dllist for performance |
]

]
.pull-right[
#### Data Structures Used

style RPolygon fill:#fff3e0,stroke:#e65100,stroke-width:3px
</pre>
]
]

---

### 📚 File Structure

```text
physdes-py/
├── src/physdes/
│   ├── rpolygon.py           # Main RPolygon class ⭐
│   ├── rpolygon_cut.py       # Convex decomposition ⭐
│   ├── point.py              # Point/Interval classes
│   ├── vector2.py            # Vector operations
│   ├── polygon.py            # General polygon
│   └── rdllist.py            # Doubly-linked list
├── tests/
│   ├── test_rpolygon.py     # RPolygon tests
│   └── test_rpolygon_*.py   # Specific algorithm tests
├── outputs/
│   ├── rpolyon_*.svg        # Visualization outputs
│   └── polygon_*.svg        # General polygon outputs
└── README.md
```

---

### 🎯 Applications in VLSI

#### Where RPolygons are Used

1. **Floorplanning** - Represent macro blocks
2. **Placement** - Legalization regions
3. **Global Routing** - Obstacle avoidance (keepouts)
4. **Clock Tree** - Buffer insertion areas
5. **Power Grid** - Power stripe planning

#### Advantages in VLSI

-   ✅ Matches Manhattan routing style
-   ✅ Efficient overlap detection
-   ✅ Simple area/perimeter calculations
-   ✅ Natural representation for standard cells

---

### 🔮 Future Enhancements

#### Potential Improvements

.font-sm[
| Feature | Description | Difficulty |
|---------|-------------|------------|
| **Boolean operations** | Union, intersection of polygons | ⭐⭐⭐ |
| **Minkowski sum** | Polygon offsetting | ⭐⭐ |
| **Skeleton/Medial axis** | Center-line extraction | ⭐⭐⭐ |
| **Grid-based operations** | Pixel-level processing | ⭐⭐ |
| **Spatial indexing** | R-tree for large datasets | ⭐⭐ |
]

---

### ✅ Summary

#### What We Covered

1. **Representation** - Origin + relative vectors
2. **Basic operations** - Area, orientation, conversion
3. **Monotone creation** - X/Y-monotone algorithms
4. **Convex hull** - Two-pass X+Y algorithm
5. **Point inclusion** - Ray-casting algorithm
6. **Convex decomposition** - Recursive cutting

#### Key Takeaways

-   ✅ Efficient **origin + vectors** representation
-   ✅ Monotone polygons simplify many operations
-   ✅ Convex decomposition enables geometric algorithms
-   ✅ Used extensively in **VLSI physical design**

---

# 🙋 Q&A

### Questions? Discussion? 💬

#### Contact

-   GitHub: [luk036/physdes-py](https://github.com/luk036/physdes-py)
-   Documentation: [physdes-py.readthedocs.io](https://physdes-py.readthedocs.io/)

---

## Appendices 📎

---

### 📎 Appendix: Monotone Detection

#### Check if Polygon is Monotone

```python
def rpolygon_is_xmonotone(lst: PointSet) -> bool:
    """Check if rectilinear polygon is x-monotone"""
    return rpolygon_is_monotone(lst, lambda p: (p.xcoord, p.ycoord))

def rpolygon_is_ymonotone(lst: PointSet) -> bool:
    """Check if rectilinear polygon is y-monotone"""
    return rpolygon_is_monotone(lst, lambda p: (p.ycoord, p.xcoord))

def rpolygon_is_convex(lst: PointSet) -> bool:
    """Check if rectilinear polygon is convex"""
    return rpolygon_is_xmonotone(lst) and rpolygon_is_ymonotone(lst)
```

---

### 📎 Appendix: Helper Functions

#### Partition Function

```python
def partition(pred, iterable) -> Tuple[List[Any], List[Any]]:
    """Split iterable into True/False groups based on predicate"""
    iter1, iter2 = tee(iterable)
    return list(filter(pred, iter1)), list(filterfalse(pred, iter2))

# Usage:
is_odd = lambda x: x % 2 != 0
partition(is_odd, range(10))  # ([1,3,5,7,9], [0,2,4,6,8])
```

---

### 📎 Appendix: Mathematical Formulas

#### Manhattan Distance

$$d((x_1, y_1), (x_2, y_2)) = {\color{green} |x_1 - x_2|} + {\color{orange} |y_1 - y_2|}$$

#### Cross Product (2D)

$$\text{cross}(v_1, v_2) = {\color{green} v_1.x \cdot v_2.y} - {\color{orange} v_1.y \cdot v_2.x}$$

---

# 👏 Thank You

#### Thank you! 🎉

**physdes-py** - VLSI Physical Design Python Library

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