Tensile Geometry
Measuring curves from their blueprint, without ever drawing them.
Two ways to measure a curve
Measurement by traveling
Measurement by blueprint
What this means
1.5710 is the total distance along the curve, computed by walking it. 0.2787 is the constraint intensity at the anchor points, computed from the blueprint. One asks "how far?" The other asks "how tightly does this geometry bend?"
Ratio (0.1774) is the fraction of traveled distance captured by the blueprint measure. It peaks at symmetry and falls as the axes diverge.
ρ measures imbalance between the axes on a logarithmic scale. Zero means symmetric. Higher means one axis dominates.
Regime names the shape: Circular Equilibrium when balanced, Balanced Asymmetry when skewed, Spiral Skew when one axis overwhelms the other. Drag the sliders below and watch the transitions.
Playground
The Story
This work began with a garment system called Gwendolyn, which generates clothing patterns from three body measurements. Each pattern piece is defined by cubic curves: the seam lines, the hems, the darts. To construct a garment, you need to know how long each curved seam is. But the system only has the control geometry: the endpoints and handle points defining each curve. The curve itself has not been drawn yet.
The standard approach to measuring a curve is integration: render it, walk along it, sum tiny distances. For cubic curves, this integral has no closed-form solution. It requires the curve to exist as a computed object before you can measure it. What was needed was a measure computable from the blueprint alone, before the curve is ever drawn.
So one was built. The Tensile Length Construct, Lgeo, derives its value entirely from anchor points, control points, and the geometric relationships between them. No integration. No curve evaluation. The result is a formally characterized geometric quantity with proved mathematical properties: symmetric, monotonic, homogeneous, and bounded.
Current State
Proved
Five formal properties establish Lgeo as a characterized geometric quantity. It is a convex combination of two virtual arc lengths whose weights partition unity exactly. It is symmetric under axis exchange, scales linearly with uniform scaling, and increases monotonically in each axis independently. As one axis grows without bound, the measure saturates at a fixed proportion of the shorter axis, revealing the binding constraint. The canonical quarter-circle approximation error is computed to exact closed form: √(426 − 72√2)/18 − 1 ≈ 0.027%.
Where This Leads
The proved properties of Lgeo open specific doors in domains where paired parameters jointly determine system behavior.
When a system has only endpoint and control-point geometry available (not a rendered curve), Lgeo provides a characterized measure directly from the construction blueprint. Pattern generation, prosthetic fitting, and any sparse-input design pipeline gain a scalar reflecting constraint intensity without integration.
The saturation property reveals which of two competing parameters is the binding constraint. As one parameter dominates, Lgeo converges to k times the shorter one, answering "which axis limits the system?" from the geometry alone. This transfers to resource allocation, capacity planning, and any context where two demands compete.
Lgeo evaluates in O(1) using basic arithmetic and two arctangent calls. On embedded systems, GPU shaders, or interactive tools evaluating hundreds of curves per frame, this replaces iterative quadrature with a deterministic closed-form computation. Monotonicity guarantees consistent ordering across evaluations.
The convex combination structure splits the measure into two individually meaningful per-anchor contributions: α₀r₀φ₀ from one axis, α₁r₁φ₁ from the other. No analogous decomposition exists for the integral arc length, where both axes are entangled at every point along the curve. A designer asking "is the problem the width or the depth?" gets a direct numerical answer.
Not Yet Demonstrated
No deployed production system uses Lgeo. The monotonicity proof's numerator sign and the sharp bound conjecture remain analytically open.
The Paper
Tensile Geometry: A Calculus-Free Arc Measure from Parametric Constraint: the full formal treatment with definitions, proofs, and derivations.
Download PDFIf you work with paired-parameter systems, cubic curves, or constraint-based geometry and see a connection to your own work, I would like to hear from you.