--- jupyter: description: 'Complete examples with planex2d: mechanisms, gears, antennas, heat sinks, toolpaths and more, each checked against a formula or a measured property.' --- # Examples These examples show complete parts, from the sketch to the solid, the mesh and a simulation. Every example is a real part or a classic problem with its own numbers, and each checks its result against a formula, a textbook value or a measured property. ## Machines and vehicles - [Toy plane](toy-plane.md) - [Model rocket](rocket.md) - [Rear wing with DRS](rear-wing.md) - [Suspension kinematics](suspension-kinematics.md) - [Excavator arm](excavator-arm.md) - [Spur gears under load](spur-gears.md) - [Planetary gearbox](planetary-gearbox.md) - [Wankel engine](wankel-engine.md) - [Idler plate](idler-plate.md)

Toy planeFuselage frames, NACA wings and engines, lofted and revolved.

Model rocketA von Kármán nose cone as a spline, revolved, with swept fins.

Rear wing with DRSA three-element Formula Student wing, in 3D and meshed in 2D.

Suspension kinematicsDouble wishbones in bump and rebound, with their instant and roll centers.

Excavator armA 20 t arm as one constraint sketch, with its working range and a trench.

Spur gears under loadInvolute flanks as splines, meshed and loaded, with the von Mises stress.

Planetary gearboxSun, three planets and ring gear, 15 mm wide, with backlash.

Wankel engineEpitrochoid housing, a rotor that fits in every position, three chambers.

Idler plateA classic drafting exercise, projected, joined by tangents and extruded.

## Mechanisms - [Four-bar linkage](four-bar-linkage.md) - [Hood hinge](hood-hinge.md) - [Jansen linkage](jansen-linkage.md) - [Straight-line linkages](straight-line-linkages.md) - [Geneva drive](geneva-drive.md) - [Cycloidal drive](cycloidal-drive.md) - [Belt drives](belt-drives.md) - [Drilling a square hole](square-hole.md)

Four-bar linkageA crank-rocker and its coupler curve through 72 positions.

Hood hingeThree-position synthesis in a constraint sketch, checked against the textbook.

Jansen linkageThe Strandbeest leg with eleven links on one crank and the foot's flat step.

Straight-line linkagesWatt, Chebyshev and the exact Peaucellier-Lipkin linkage.

Geneva driveA quarter turn per revolution, with the locking disc cut by what the wheel sweeps.

Cycloidal driveThe disc cut by every ring pin at every crank angle.

Belt drivesThe belt length as one expression, with the pulley and tensioner placed by the solver.

Drilling a square holeA Reuleaux triangle covers exactly 98.77 % of the square.

## Physics - [Isospectral drums](isospectral-drums.md) - [Boat stability](boat-stability.md) - [Flow through sand](flow-through-sand.md) - [Solar shading](solar-shading.md)

Isospectral drumsTwo drums with the same eigenvalues, so the shape cannot be heard.

Boat stabilityThe waterline at constant displacement and the righting lever curve.

Flow through sandStokes flow through packed grains against the Kozeny-Carman law.

Solar shadingThe shaded share of a roof panel, hour by hour, through the year.

## Fields and simulation - [Surface magnet motor](motor.md) - [Heat sink](heat-sink.md) - [Hilbert heater](hilbert-heater.md) - [Parallelogram flexure](parallelogram-flexure.md) - [Vivaldi antenna](vivaldi-antenna.md) - [Patch antenna](patch-antenna.md) - [Voronoi bone](voronoi-bone.md) - [Lens ray tracing](lens-ray-tracing.md) - [Airplane windows](airplane-windows.md)

Surface magnet motorA lamination glued into regions, its field, and a 12-slot design under load.

Heat sinkConduction and convection in an extruded profile, one fin against fin theory.

Hilbert heaterA heater track along a Hilbert curve, with current, Joule heat and temperature.

Parallelogram flexureA flexure stage from one block, pushed sideways in a 3D elasticity solve.

Vivaldi antennaAn exponential tapered slot with a round cavity and a radial stub.

Patch antennaA 2.45 GHz inset-fed patch from the design formulas, and its potential.

Voronoi boneSome 6000 struts united at once, pressed against a scaling law.

Lens ray tracingA biconvex lens as two intersected discs, with Snell's law and chief and marginal rays.

Airplane windowsRound, oval and square windows in a pressurized skin, against Kirsch, Inglis and Williams.

## Manufacturing - [Gear cutting](gear-cutting.md) - [Pocket milling](pocket-milling.md) - [A slicer for 3D printing](slicer.md) - [Laser-cut box](laser-cut-box.md) - [Stencil lettering](stencil-lettering.md)

Gear cuttingA rack cutter with one subtraction per position, from which involutes and undercut emerge.

Pocket millingOffsets for a 6 mm end mill and the material left in the corners.

A slicer for 3D printingA STEP part cut into layers, with perimeters, infill and G-code.

Laser-cut boxFinger joints with the kerf compensated by offsets, and the cut path as DXF.

Stencil letteringIslands inside letters found and bridged so that nothing falls out.

## Geometry and design - [Penrose tiling](penrose-tiling.md) - [Rose window](rose-window.md) - [Path planning](path-planning.md) - [The moving sofa](moving-sofa.md) - [Prince Rupert's cube](prince-ruperts-cube.md) - [The coastline paradox](coastline-paradox.md) - [Drawing a logo](drawing-a-logo.md)

Penrose tilingGolden triangles deflated, with the area kept exactly and the thick-to-thin ratio tending to φ.

Rose windowGothic tracery from circles that touch circles, bars as offsets.

Path planningObstacles grown by offsets, and the shortest path through a visibility graph.

The moving sofaThe largest sofa around a corner as one intersection, Hammersley's exactly and Gerver's within 0.0001.

Prince Rupert's cubeA cube through a hole in a cube of its size, the largest one 3√2/4 to twelve digits.

The coastline paradoxRichardson's dividers on the Koch snowflake and on the coast of Great Britain.

Drawing a logoThe planex2d mark from its own SVG: Bézier curves as exact B-splines, a Reuleaux triangle of constant width.