Start with a curve and surface.
The surface sets the domain. A smaller closed curve provides the starting condition.
Computational design / 2024–2026
Differential growth.
From curve to canopy.
A parametric design system that turns one closed curve into looping patterns on a chosen surface. Adjustable controls shape the folds; comparative studies reveal how the geometry responds.
Rhino · Grasshopper · Kangaroo · Python / Digital design study
- My role
- Research & computational modelling
- System
- Geometry, controls & simulation
- Controls
- Local spacing & folding behaviour
- Studies
- Growth & density comparisons
- Application
- COVE pavilion design study
Tools & technologies
Geometry modelling, growth simulation and documentation of the tests.
Programming
Python
Spatial fields, seed preparation & checks
Design & geometry
Rhino · Grasshopper
Parametric modelling & connected data flow
Simulation
Kangaroo
Length, bending, collision & surface goals
Documentation
Illustrator
Vector comparisons & technical diagrams
01 / Project overview
Controlling how
a curve folds.
University of Toronto · ARC3201
2024 research, developed in 2026
The investigation centres on control: how to vary the spacing and tightness of folds while keeping the curve on a chosen surface.
The challenge: produce complex, organic geometry without specifying every loop by hand, while retaining useful control over its spacing and shape.
My contribution: I investigated seed preparation, local density controls, and surface adaptation through comparative geometry studies, and brought those studies together in a documented workflow and pavilion proposal.
The result: a configurable Grasshopper definition, comparative studies showing how inputs affect the geometry, and COVE, a conceptual architectural application.
Develop a controlled pattern.
Competing constraints create folds. Local settings influence their spacing and tightness.
Carry the pattern into a form.
COVE applies the approach to a rounded canopy, giving the curve a tubular volume.
02 / System architecture
How the pattern
is generated.
The curve is divided into connected segments. Four constraints act together: extend the segments, separate them, resist sharp bends, and keep them on the surface.
A “host” is the surface the curve follows. The solver repeatedly adjusts the connected segments to balance the constraints. The pattern emerges from those adjustments.
A / Prepare
Host + closed seed
Scale the host boundary. Rebuild, pull to the face, then divide into a fixed segment network.
Host mesh OnMesh + field domain
B / Assign local behaviour
One field. Three settings.
Sample initial midpoints · smooth around the chain
Radius rᵢRest length ℓᵢBend strength kᵢ
01 / Expansion
Length(Line)
Set the length each segment tries to reach.
02 / Separation
Collider
Resist overlap in the segment network.
03 / Bending
Angle
Control resistance to tight folding.
04 / Conformity
OnMesh
Keep the curve’s points on the host surface.
BouncySolver
Repeated updates to particle positions + momentum
C / Inspect
Reconstruct the curve
Solver vertices become a closed NURBS. Pause before checking and saving.
Check + export 3DM · JSON · CSV
Separate preview branch
The same field function is sampled across the full host mesh. Preview colours show the input field; they do not feed the solver.
View the complete source-derived system diagram
03 / Recorded evolution
One curve.
Seven recorded states.
Compare seven saved curves as the seed folds, spreads across the surface and continues to rearrange.
Select a state to compare the actual exported curve and its measured length. Each drawing uses the same boundary and scale.
A closed curve establishes the starting condition.
Small deformations appear in four regions of the seed.
Distinct lobes extend toward the corners of the domain.
Larger folds spread through the available square.
Successive folds form a denser spatial pattern.
The curve occupies the square with pronounced local bands.
The longer-run export records a rearranged, shorter curve.
Measured from the exports
Folds emerge.
The pattern rearranges.
19.77×curve length / seed length
State 05 reaches 24.39 times the seed length. State 06 is 18.94% shorter, despite representing a longer run.
This measures curve length, not coverage, elapsed time, or convergence.
Download measurements · CSVThese are seven recorded exports, not equal time intervals or a live simulation. Curve lengths were measured using 32-point quadrature of 1,500 cubic Bézier segments per export. Full-precision values are in the CSV.
04 / Local controls
Shape local spacing
and folding.
A density field describes where tighter or looser patterns are intended. Uniform values, gradients, points, and guide curves translate that intent into three local controls.
What the controls change
Spacing between segments, their target length, and their resistance to bending.
Uniform & gradient
Input field
Uniform
Input field
Low intensity
Input field
High intensity
Point fields
Input field
Centre
Input field
Left edge
Input field
Upper-left corner
Curve fields
Input field
Vertical guide
Input field
Diagonal guide
Input field
Curved guide
How the field is used. Attractors assign local settings rather than pulling the curve. Values are sampled on the starting segments and travel with them during relaxation.
How to read the studies. Each coloured input field is paired with its corresponding growth export. Colour indicates relative influence, not a measured final density.
Read the implemented parameter mapping
Bound the effect of each control.
The smoothed field becomes a positive multiplier. Each output has its own exponent and clamp. Setting blend to zero restores the scalar baseline.
These are implementation-derived relationships, not measurements of the final pattern.
Dᵢ = contrast ^ (blend × fieldᵢ)
rᵢ / r₀ = clamp(Dᵢ ^ −s, 0.3, 3.0)
ℓᵢ / ℓ₀ = clamp(Dᵢ ^ g, 0.65, 1.5)
kⱼ / k₀ = clamp(D̂ⱼ ^ (−2q), 0.12, 6.0)
D̂ⱼ = √(Dᵢ × Dpaired)
s: spacing contrast
g: local packing boost
q: loop-size contrastA second control / Collision radius
Change the spacing scale.
Compare the spacing and folding in four radius settings.
Radius sets a solver constraint; it is not a measured clearance. The Collider goal does not guarantee that the final smooth curve is free of intersections.
05 / Boundary & surface tests
Testing different outlines
and curved surfaces.
Four flat outlines and three surface conditions test how the growth method responds to a change in its boundary or supporting surface.
The planar tests change the outline; the surface tests add curvature and twist. Together they show the range of domains represented in the study.
Design decision
Keep the host definition separate from the growth logic. Geometry supplies the domain; the constraint system develops the curve within it.
06 / Implementation
Preparing inputs.
Checking the output.
The implementation repairs the starting curve, keeps segment connectivity fixed and checks the reconstructed output separately from the solver.
- 01
Repair the seed before solving.
Rebuilding can move the seed away from the surface. Pulling it back before division restores placement before the solver starts.
- 02
Keep connectivity fixed.
The saved definition begins with 1,500 segments. Local variation comes from goal settings rather than adding vertices, keeping the control logic separate from network resolution.
- 03
Check the output independently.
Constraints act on the segment network, but the final output is a smooth curve. Paused diagnostics check that output for curvature and mesh deviation without modifying the simulation.
- 04
Preserve the state behind the result.
The optional export branch saves geometry, settings, metrics, and curve samples, making it possible to retain the context of a result. The matching Grasshopper definition is archived separately.
Inspect the definition and saved controls
| Control | Archived study | Current file |
|---|---|---|
| Seed segments / scale | 1,500 / 0.50 | 1,500 / 0.50 |
| Rest length / radius | 10.000 / 1.127 | 10.000 / 1.000 |
| Density blend / contrast | 0.00 / 2.00 | 0.65 / 2.00 |
| Gradient bias / axis | 0.00 / Z | 1.00 / Z |
| Momentum / iterations per update | 0.60 / 30 | 0.60 / 30 |
All four embedded scripts match between the two inspected snapshots. These saved controls have not been verified for every image. The file comparison did not include a new Rhino/Kangaroo run.
07 / Architectural application
COVE.
A pavilion application.
For COVE, the growth curve follows a rounded canopy and gains a tubular volume. The digital proposal applies the pattern controls at architectural scale.
A digital design proposal. Model-derived geometry shows the development; conceptual renders explore light, shadow, and human scale.
Context, people, material appearance, and rendered joints are illustrative. These images do not establish structural capacity or construction detail.
Inspect the supplied model-derived geometry
08 / Findings & reflection
What the studies show.
What remains untested.
The studies compare how local controls and host geometry affect the curve. They support a digital design workflow, but do not yet establish structural or fabrication performance.
The definition connects geometry preparation, local parameters and simulation. The exported curves and measurements make the differences between results available for inspection.
Delivered outcomes
Local controls and comparative geometry.
- Configurable behaviour: spatial fields translate design intent into bounded spacing, length, and bending controls.
- Geometric adaptation: four boundaries and three surface conditions document the method across different domains.
- Comparative evidence: growth states, density studies, and measurements make differences between outputs visible.
- Architectural application: COVE connects the pattern-generation method to a complete digital form.
What needs further validation
Checks still needed.
- Clearance: test smooth-curve intersections and minimum non-adjacent distance.
- Conformity: measure maximum and RMS curve-to-host deviation.
- Density & stability: compare spatial length distributions and repeat at several resolutions.
- Fabrication: define material, joints, supports, and manufacturing constraints.
Existing paused diagnostics sample 1,000 intervals. Automated coverage measurement, verified clearance, convergence and fabrication feasibility have not yet been established.
Project evidence
Explore the full record.
The complete document includes mechanics, all study plates, file-state comparisons, and source references.