Algorithms
Algorithms
A visual catalog of the algorithms behind pen-plotter art — from path optimization and vectorization to generative geometry and organic growth. Cards marked Live demo open an interactive p5.js sketch showing the algorithm in action.
Share this page:
[](https://twitter.com/intent/tweet?url=https%3A%2F%2Fdrawingbots.com%2Falgorithms%2F&text=A+visual+catalog+of+pen+plotter+algorithms)[](https://www.facebook.com/sharer/sharer.php?u=https%3A%2F%2Fdrawingbots.com%2Falgorithms%2F)
122 of 122 have live demos — more coming soon.
Live demo
2-opt local search
Repeatedly reverses sections of a route whenever the swap shortens its total length.
Live demo
Adaptive stippling
Places marks at variable spacing or size according to local image tone, detail, or an error measure.
Live demo
Affine transformation
Maps coordinates with combinations of translation, rotation, scaling, and shear while preserving straight lines and parallelism.
Live demo
Arc interpolation
Evaluates points along a circular or elliptical arc from its geometric parameters.
Live demo
Arc-length resampling
Redistributes points along a curve so consecutive samples are separated by approximately equal physical distances.
Live demo
Arc-to-Bézier conversion
Approximates circular or elliptical arcs with one or more cubic Bézier curve segments.
Live demo
Archimedean spiral
Generates a spiral whose radius increases linearly with angle, producing nearly constant spacing between turns.
Live demo
Attractor-field integration
Moves particles or curve samples through a vector field formed by attracting and repelling points or shapes.
Live demo
B-spline curves
Constructs smooth piecewise-polynomial curves controlled by points, knots, and polynomial degree.
Live demo
Bayer ordered dithering
Compares image values against a repeating threshold matrix to create a regular distribution of binary marks.
Live demo
Bentley–Ottmann sweep-line algorithm
Finds intersections among line segments by sweeping an ordered event line across the geometry.
Live demo
Bézier curves
Defines smooth parametric curves from endpoints and control points using Bernstein polynomials.
Live demo
Blue-noise sampling
Distributes samples with few low-frequency clumps while avoiding the rigid appearance of a grid.
Live demo
Bresenham line algorithm
Steps through a discrete grid to approximate a straight line using inexpensive incremental error calculations.
Live demo
Catmull–Rom splines
Interpolates a smooth curve through a sequence of control points using neighboring points to determine tangents.
Live demo
Cellular automata
Updates cells on a grid from local neighborhood rules to produce evolving spatial patterns.
Live demo
Centerline tracing
Extracts a single path along the middle of strokes or elongated regions in a raster image.
Live demo
Chaikin corner cutting
Smooths a polyline by repeatedly replacing each corner with two points placed along its adjacent edges.
Live demo
Circle packing
Places non-overlapping circles inside a boundary while satisfying size, spacing, or density constraints.
Live demo
CMYK color separation
Decomposes an image into cyan, magenta, yellow, and black coverage channels.
Live demo
Color quantization
Reduces a large set of image colors to a smaller representative palette while minimizing visual error.
Live demo
Concave hull
Builds a boundary around a point set that can bend inward and follow the set more closely than a convex hull.
Live demo
Concentric infill
Fills a region with successively inset or outset copies of its boundary.
Live demo
Contour extraction
Finds curves that follow boundaries or equal-valued levels in raster, scalar-field, or elevation data.
Live demo
Convex hull
Computes the smallest convex polygon containing a set of points.
Live demo
Cross-hatching
Overlays two or more families of hatch lines at different angles to build tone or texture.
Live demo
Curl noise
Derives a divergence-free vector field from the curl of a noise field, producing swirling motion without obvious sinks or sources.
Live demo
Curve flattening
Approximates arcs and parametric curves with polylines while keeping deviation or segment length below a tolerance.
Live demo
De Casteljau's algorithm
Evaluates or subdivides a Bézier curve through repeated linear interpolation of its control points.
Live demo
Delaunay triangulation
Connects points into triangles so no input point lies inside any triangle's circumcircle.
Live demo
Differential growth
Iteratively moves and subdivides a chain under attraction, repulsion, and smoothing forces so it grows without collapsing onto itself.
Live demo
Distance transform
Assigns each raster cell its distance to the nearest foreground, background, or boundary cell.
Live demo
Douglas–Peucker simplification
Recursively removes polyline vertices whose removal stays within a specified perpendicular-error tolerance.
Live demo
Dubins path planning
Finds a short path between oriented points using straight segments and fixed-radius turns.
Live demo
Edge detection
Locates strong changes in image intensity or color that are likely to correspond to visual boundaries.
Live demo
Edge tangent flow
Builds a smooth orientation field that follows nearby image edges while reducing local directional noise.
Live demo
Error-diffusion dithering
Quantizes each image sample and distributes its quantization error to neighboring samples that have not yet been processed.
Live demo
Fast marching method
Propagates a front through a weighted domain by solving an Eikonal equation in order of increasing arrival time.
Live demo
Fermat spiral fill
Fills a region with a Fermat (parabolic) spiral whose in-and-out arms interleave, so a single continuous path enters and exits at the boundary with near-uniform spacing.
Live demo
Flood fill
Visits connected cells or regions starting from a seed while respecting a boundary or value condition.
Live demo
Floyd–Steinberg dithering
An error-diffusion method that passes quantization error to four nearby pixels using fixed weights.
Live demo
Fourier series drawing
Approximates a periodic shape as a sum of rotating sinusoidal components or epicycles.
Live demo
Fractal noise
Combines multiple scaled octaves of coherent noise to produce detail at several spatial frequencies.
Live demo
G-code toolpath generation
Translates drawing geometry into ordered machine moves, coordinate commands, feed settings, and pen-control actions.
Live demo
Genetic path optimization
Evolves candidate route orderings through selection, crossover, and mutation according to a travel-cost score.
Live demo
Greedy nearest-neighbor path ordering
Repeatedly selects the unvisited path endpoint nearest to the current pen position, optionally reversing paths.
Live demo
Halftoning
Represents continuous tone with discrete marks whose size, spacing, shape, or density varies.
Live demo
Harmonograph curves
Combines damped sinusoidal oscillations to model the paths produced by a mechanical harmonograph.
Live demo
Hatching
Fills a region with parallel or locally oriented lines clipped to the region boundary.
Live demo
Hexagonal grid generation
Tiles the plane with regularly spaced hexagonal cells using axial, cube, or offset coordinate systems.
Live demo
Hidden-line removal
Determines which portions of projected 3D edges are occluded by surfaces and removes or splits those portions.
Live demo
Hilbert curve
Recursively constructs a continuous space-filling curve that visits every cell of a square grid.
Live demo
Homography
Maps points between two planes with a projective transformation defined by a 3-by-3 matrix.
Live demo
Image-guided flow fields
Converts image gradients, edges, brightness, or structure tensors into a vector field that guides curves.
Live demo
Inverse kinematics
Computes actuator positions or joint angles needed to place a tool at a requested Cartesian coordinate.
Live demo
Island detection
Identifies disconnected components, enclosed loops, or path groups that are isolated from the rest of the geometry.
Live demo
Isoline extraction
Traces curves through positions where a sampled scalar field equals selected values.
Live demo
k-d tree nearest-neighbor search
Partitions points recursively by coordinate axis to accelerate spatial range and nearest-neighbor queries.
Live demo
K-means color clustering
Alternates between assigning samples to the nearest color center and updating each center to its cluster mean.
Live demo
L-systems
Repeatedly rewrites symbols with grammar rules and interprets the result as drawing commands.
Live demo
Lawnmower infill
Covers a region with parallel back-and-forth passes connected in alternating directions.
Live demo
Liang–Barsky line clipping
Clips a line segment against an axis-aligned rectangle by updating the valid interval of its parametric equation.
Live demo
Line joining and merging
Connects nearby compatible segment endpoints, reversing direction where necessary, to form longer continuous polylines.
Live demo
Line-segment intersection
Tests whether two finite segments cross and, when they do, computes their intersection point and parameters.
Live demo
Linear interpolation
Computes intermediate values along a straight interval between two known values or points.
Live demo
Lissajous curves
Plots two perpendicular sinusoidal motions with independently chosen frequencies, phases, and amplitudes.
Live demo
Low-pass path smoothing
Suppresses high-frequency changes in sampled coordinates or motion commands while retaining slower shape changes.
Live demo
Marching squares
Extracts isoline segments from a two-dimensional sampled field by classifying the corners of each grid cell.
Live demo
Martinez polygon clipping
Uses a sweep-line process to compute boolean operations on simple or self-intersecting polygons.
Live demo
Maze generation
Builds a connected network of passages with methods such as depth-first backtracking, randomized Prim, or Kruskal growth.
Live demo
Medial axis transform
Finds the centers and radii of maximal disks contained inside a shape, forming a skeletal representation.
Live demo
Minimum spanning tree
Connects all vertices of a weighted graph with minimum total edge weight and no cycles.
Live demo
Morphological erosion
Shrinks foreground regions by removing pixels or cells that do not contain a chosen structuring element.
Live demo
Motion planning
Chooses a feasible sequence of movements that respects geometry, machine limits, obstacles, and tool state.
Live demo
NURBS curves
Represents smooth rational spline curves with weighted control points, a knot vector, and polynomial basis functions.
Live demo
OpenSimplex noise
Generates coherent gradient noise on a simplex-like lattice with fewer directional artifacts than classic grid noise.
Live demo
Orthographic projection
Projects 3D points onto a plane along parallel rays without perspective size reduction.
Live demo
Penrose tiling
Covers the plane aperiodically with a small set of tiles constrained by matching or substitution rules.
Live demo
Perlin noise
Interpolates pseudo-random gradients on a lattice to create smooth, spatially coherent noise.
Live demo
Perspective projection
Projects 3D points onto a plane so apparent size decreases with distance from the viewer.
Live demo
Physarum simulation
Models agents that deposit and follow a diffusing trail field, producing adaptive transport-like networks.
Live demo
Point-in-polygon test
Determines whether a point lies inside, outside, or on the boundary of a polygon using crossing or winding rules.
Live demo
Poisson-disc sampling
Generates samples with a guaranteed minimum separation while retaining an irregular, natural distribution.
Live demo
Polygon Boolean operations
Computes union, intersection, difference, or exclusive-or between polygonal regions.
Live demo
Polygon offsetting
Constructs parallel boundaries at a specified distance from polygon edges while resolving joins and self-intersections.
Live demo
Polygon packing
Places polygonal shapes within a boundary while avoiding overlap and optimizing space, density, or another objective.
Live demo
Potrace
Transforms a thresholded bitmap into smooth vector outlines through path decomposition, polygon fitting, and curve optimization.
Live demo
Quadtree spatial indexing
Recursively divides a two-dimensional region into four cells so spatial queries can ignore empty or distant areas.
Live demo
Random walk
Builds a path through successive random or probability-weighted steps from the current position.
Live demo
Raster-to-vector tracing
Converts raster regions, edges, or strokes into geometric paths such as polylines and Bézier curves.
Live demo
Ray casting
Projects rays from a viewpoint or query point and finds their first intersections with scene geometry.
Live demo
Ray hatching
Casts rays or probes across projected surfaces and converts visible intervals into hatch strokes.
Live demo
Ray marching
Advances a ray through a field in steps, often using signed distance values to choose safe step lengths.
Live demo
Ray tracing
Traces rays through a scene to evaluate intersections, visibility, lighting, and reflections.
Live demo
Reaction–diffusion
Simulates interacting substances that spread and react locally, producing spots, stripes, and labyrinthine structures.
Live demo
Rectangular packing
Arranges rectangles within a bounded area to minimize unused space or satisfy layout constraints.
Live demo
Scanbeam processing
Sweeps through ordered vertex events while maintaining active edges for the current horizontal or vertical band.
Live demo
Scanline polygon filling
Intersects regularly spaced scanlines with polygon edges and fills the alternating interior intervals according to a fill rule.
Live demo
Signed distance fields
Represent a shape with a scalar value giving distance to its boundary and a sign indicating inside or outside.
Live demo
Simplex noise
Generates coherent gradient noise on simplex cells, scaling efficiently to multiple dimensions.
Live demo
Single-line font rendering
Represents glyphs as centerline strokes rather than closed filled outlines, as in Hershey-style fonts.
Live demo
Skeletonization
Iteratively thins a binary shape to a one-pixel-wide topology-preserving skeleton.
Live demo
Space-colonization growth
Grows a branching network toward nearby attraction points while removing points reached by existing branches.
Live demo
Spiral infill
Covers a region with a continuous inward or outward spiral derived from repeated offsets or a spiral field.
Live demo
Spirograph curves
Generates hypotrochoid or epitrochoid paths from a circle rolling inside or outside another circle.
Live demo
Stippling
Represents tone or form with discrete dots whose density, size, or distribution varies spatially.
Live demo
Streamline integration
Numerically traces curves that remain tangent to a vector field from selected seed points.
Live demo
String-art line selection
Iteratively chooses chords between fixed anchor points to reduce the difference from a target image.
Live demo
Superformula
Generalizes the superellipse to a single polar equation whose few parameters sweep from circles and polygons to petals, stars, and organic blobs.
Live demo
Thresholding
Classifies image samples by comparing intensity, color, or another measure with one or more thresholds.
Live demo
Tone- and feature-aware circular scribbling
Builds a continuous path from circular scribbles whose placement, loop size, and motion respond to image tone and features.
Live demo
Traveling Salesman Problem optimization
Searches for a short route that visits every required point or path once before returning or stopping.
Live demo
Truchet tiling
Builds patterns by rotating or selecting tiles that contain compatible arcs or line fragments.
Live demo
Turtle graphics
Generates paths by moving an oriented cursor with relative commands such as forward, turn, and branch.
Live demo
Vatti polygon clipping
Computes boolean operations on general polygons using scanbeams, local minima, and an active edge structure.
Live demo
Voronoi diagram
Partitions a plane into cells containing the locations closest to each generating site.
Live demo
Wave Function Collapse
Assigns compatible tiles or states to cells by repeatedly propagating local constraints and resolving uncertainty.
Live demo
Weave path ordering
Splits and orders intersecting paths so alternating segments appear to pass over and under one another.
Live demo
Weighted Voronoi stippling
Iteratively relocates Voronoi sites toward density-weighted cell centroids derived from a target image.
Live demo
Worley noise
Measures distance to nearby random feature points to produce cellular bands, ridges, and regions.
Live demo
Z-buffer visibility
Stores the nearest depth reached at each sampled image location and rejects geometry farther from the viewer.
Live demo
Zingl–Bresenham curve algorithm
Incrementally approximates Bézier curves and conic sections on a discrete grid using error terms.