Computational Geometry with Shapely

When to Use

• Creating geometric shapes (points, lines, polygons)
• Boolean operations (intersection, union, difference)
• Spatial predicates (contains, intersects, within)
• Measurements (area, length, distance, centroid)
• Geometry transformations (translate, rotate, scale)
• Validating and fixing invalid geometries

Quick Reference

I want to...	Command	Example
Create geometry	create	create polygon --coords "0,0 1,0 1,1 0,1"
Intersection	op intersection	op intersection --g1 "POLYGON(...)" --g2 "POLYGON(...)"
Check contains	pred contains	pred contains --g1 "POLYGON(...)" --g2 "POINT(0.5 0.5)"
Calculate area	measure area	measure area --geom "POLYGON(...)"
Distance	distance	distance --g1 "POINT(0 0)" --g2 "POINT(3 4)"
Transform	transform translate	transform translate --geom "..." --params "1,2"
Validate	validate	validate --geom "POLYGON(...)"

Commands

create

Create geometric objects from coordinates.

# Point
uv run python scripts/shapely_compute.py create point --coords "1,2"

# Line (2+ points)
uv run python scripts/shapely_compute.py create line --coords "0,0 1,1 2,0"

# Polygon (3+ points, auto-closes)
uv run python scripts/shapely_compute.py create polygon --coords "0,0 1,0 1,1 0,1"

# Polygon with hole
uv run python scripts/shapely_compute.py create polygon --coords "0,0 10,0 10,10 0,10" --holes "2,2 8,2 8,8 2,8"

# MultiPoint
uv run python scripts/shapely_compute.py create multipoint --coords "0,0 1,1 2,2"

# MultiLineString (pipe-separated lines)
uv run python scripts/shapely_compute.py create multilinestring --coords "0,0 1,1|2,2 3,3"

# MultiPolygon (pipe-separated polygons)
uv run python scripts/shapely_compute.py create multipolygon --coords "0,0 1,0 1,1 0,1|2,2 3,2 3,3 2,3"

op (operations)

Boolean geometry operations.

# Intersection of two polygons
uv run python scripts/shapely_compute.py op intersection \
    --g1 "POLYGON((0 0,2 0,2 2,0 2,0 0))" \
    --g2 "POLYGON((1 1,3 1,3 3,1 3,1 1))"

# Union
uv run python scripts/shapely_compute.py op union --g1 "POLYGON(...)" --g2 "POLYGON(...)"

# Difference (g1 - g2)
uv run python scripts/shapely_compute.py op difference --g1 "POLYGON(...)" --g2 "POLYGON(...)"

# Symmetric difference (XOR)
uv run python scripts/shapely_compute.py op symmetric_difference --g1 "..." --g2 "..."

# Buffer (expand/erode)
uv run python scripts/shapely_compute.py op buffer --g1 "POINT(0 0)" --g2 "1.5"

# Convex hull
uv run python scripts/shapely_compute.py op convex_hull --g1 "MULTIPOINT((0 0),(1 1),(0 2),(2 0))"

# Envelope (bounding box)
uv run python scripts/shapely_compute.py op envelope --g1 "POLYGON(...)"

# Simplify (reduce points)
uv run python scripts/shapely_compute.py op simplify --g1 "LINESTRING(...)" --g2 "0.5"

pred (predicates)

Spatial relationship tests (returns boolean).

# Does polygon contain point?
uv run python scripts/shapely_compute.py pred contains \
    --g1 "POLYGON((0 0,2 0,2 2,0 2,0 0))" \
    --g2 "POINT(1 1)"

# Do geometries intersect?
uv run python scripts/shapely_compute.py pred intersects --g1 "..." --g2 "..."

# Is g1 within g2?
uv run python scripts/shapely_compute.py pred within --g1 "POINT(1 1)" --g2 "POLYGON(...)"

# Do geometries touch (share boundary)?
uv run python scripts/shapely_compute.py pred touches --g1 "..." --g2 "..."

# Do geometries cross?
uv run python scripts/shapely_compute.py pred crosses --g1 "LINESTRING(...)" --g2 "LINESTRING(...)"

# Are geometries disjoint (no intersection)?
uv run python scripts/shapely_compute.py pred disjoint --g1 "..." --g2 "..."

# Do geometries overlap?
uv run python scripts/shapely_compute.py pred overlaps --g1 "..." --g2 "..."

# Are geometries equal?
uv run python scripts/shapely_compute.py pred equals --g1 "..." --g2 "..."

# Does g1 cover g2?
uv run python scripts/shapely_compute.py pred covers --g1 "..." --g2 "..."

# Is g1 covered by g2?
uv run python scripts/shapely_compute.py pred covered_by --g1 "..." --g2 "..."

measure

Geometric measurements.

# Area (polygons)
uv run python scripts/shapely_compute.py measure area --geom "POLYGON((0 0,1 0,1 1,0 1,0 0))"

# Length (lines, polygon perimeter)
uv run python scripts/shapely_compute.py measure length --geom "LINESTRING(0 0,3 4)"

# Centroid
uv run python scripts/shapely_compute.py measure centroid --geom "POLYGON((0 0,2 0,2 2,0 2,0 0))"

# Bounds (minx, miny, maxx, maxy)
uv run python scripts/shapely_compute.py measure bounds --geom "POLYGON(...)"

# Exterior ring (polygon only)
uv run python scripts/shapely_compute.py measure exterior_ring --geom "POLYGON(...)"

# All measurements at once
uv run python scripts/shapely_compute.py measure all --geom "POLYGON((0 0,2 0,2 2,0 2,0 0))"

distance

Distance between geometries.

uv run python scripts/shapely_compute.py distance --g1 "POINT(0 0)" --g2 "POINT(3 4)"
# Returns: {"distance": 5.0, "g1_type": "Point", "g2_type": "Point"}

transform

Affine transformations.

# Translate (move)
uv run python scripts/shapely_compute.py transform translate \
    --geom "POLYGON((0 0,1 0,1 1,0 1,0 0))" --params "5,10"
# params: dx,dy or dx,dy,dz

# Rotate (degrees, around centroid by default)
uv run python scripts/shapely_compute.py transform rotate \
    --geom "POLYGON((0 0,1 0,1 1,0 1,0 0))" --params "45"
# params: angle or angle,origin_x,origin_y

# Scale (from centroid by default)
uv run python scripts/shapely_compute.py transform scale \
    --geom "POLYGON((0 0,1 0,1 1,0 1,0 0))" --params "2,2"
# params: sx,sy or sx,sy,origin_x,origin_y

# Skew
uv run python scripts/shapely_compute.py transform skew \
    --geom "POLYGON(...)" --params "15,0"
# params: xs,ys (degrees)

validate / makevalid

Check and fix geometry validity.

# Check if valid
uv run python scripts/shapely_compute.py validate --geom "POLYGON((0 0,1 0,1 1,0 1,0 0))"
# Returns: {"is_valid": true, "type": "Polygon", ...}

# Fix invalid geometry (self-intersecting, etc.)
uv run python scripts/shapely_compute.py makevalid --geom "POLYGON((0 0,2 2,2 0,0 2,0 0))"

coords

Extract coordinates from geometry.

uv run python scripts/shapely_compute.py coords --geom "POLYGON((0 0,1 0,1 1,0 1,0 0))"
# Returns: {"coords": [[0,0],[1,0],[1,1],[0,1],[0,0]], "type": "Polygon"}

fromwkt

Parse WKT and get geometry information.

uv run python scripts/shapely_compute.py fromwkt "POLYGON((0 0,1 0,1 1,0 1,0 0))"
# Returns: {"type": "Polygon", "bounds": [...], "area": 1.0, ...}

Geometry Types

• point - Single coordinate (x, y) or (x, y, z)
• line/linestring - Sequence of connected points
• polygon - Closed shape with optional holes
• multipoint, multilinestring, multipolygon - Collections

Input Formats

• Coordinates string: "0,0 1,0 1,1 0,1" (space-separated x,y pairs)
• WKT: "POLYGON((0 0, 1 0, 1 1, 0 1, 0 0))"

Output Format

All commands return JSON with:

• wkt: WKT representation of result geometry
• type: Geometry type (Point, LineString, Polygon, etc.)
• bounds: (minx, miny, maxx, maxy)
• is_valid, is_empty: Validity flags
• Measurement-specific fields (area, length, distance, etc.)

Common Use Cases

Use Case	Command
Collision detection	pred intersects
Point-in-polygon	pred contains
Area calculation	measure area
Buffer zones	op buffer
Shape combination	op union
Shape subtraction	op difference
Bounding box	op envelope or measure bounds
Simplify path	op simplify

Related Skills

• /math-mode - Full math orchestration (SymPy, Z3)
• /math-plot - Visualization with matplotlib