Pythagorean Hodograph Curve¶
A planar polynomial curve whose derivative components have a polynomial-square norm, giving exact polynomial arc length on regular intervals.
Core Idea¶
A planar polynomial curve is a Pythagorean hodograph curve when the squared components of its derivative add to a perfect polynomial square. On a regular interval, its speed and arc length can then be written polynomially. The planar unit normal and constant-distance offset have rational parametrizations. These properties follow from the derivative-norm identity, not from a general ability to draw a hodograph.[^ref-cf0e73c7acc3]
Scope of Application¶
The class is useful in a published rational cam-profile construction, exact length computation and a real-time CNC interpolation method. The explicit cubic in V2 is an author-derived algebraic witness, not a measured application. Spatial and rational PH variants need separate normal, offset and length claims; a PH curve is not automatically a good fit or safe motion path.[ref-ed64eb1bf25b][ref-4d56bb1b0033][^ref-94984233c3ff]
Clarity¶
State whether the curve is planar and polynomial. Check zero-speed points and sign changes before using one polynomial arc-length expression. “Rational offset” means a symbolic representation, not a guarantee against cusps or self-intersections.
Manages Complexity¶
The special derivative identity replaces a difficult square-root length integral with polynomial integration and simplifies planar offsets. The price is a restricted curve family and remaining fit and validity checks.
Abstract Reasoning¶
Differentiate the proposed curve, verify that its derivative squared norm is a polynomial square, partition into regular sign-consistent intervals, and only then compute length or offset. A feed profile remains a separate application decision.
Knowledge Transfer¶
The same exact-length geometry can support CAD and machine-path parameter stepping. Machine dynamics and spatial normal-frame formulas do not transfer automatically from the planar algebraic identity. This is a stricter kind of differentiable curve; stationary derivative points do not make it nondifferentiable.
[^ref-cf0e73c7acc3]: Farouki and Sakkalis, original Pythagorean-hodograph article (1990). [^ref-ed64eb1bf25b]: Farouki and collaborators, original PH rational-cam-profile article, abstract checked. [^ref-4d56bb1b0033]: Farouki and Shah, original CNC interpolator study (1996). [^ref-94984233c3ff]: Farouki, rational PH arc-length correction.
Relationships to Other Abstractions¶
Current abstraction Pythagorean Hodograph Curve Domain-specific
Parents (1) — more general patterns this builds on
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Pythagorean Hodograph Curve is a kind of Differentiable curve Domain-specific
Every polynomial Pythagorean-hodograph curve is differentiable.
Hierarchy path (1) — routes to 1 parentless root
- Pythagorean Hodograph Curve → Differentiable curve → Path → Network → Reservoir-Flux Network → Conservation Laws → Invariance
Neighborhood in Abstraction Space¶
Pythagorean Hodograph Curve sits in a sparse region of the domain-specific corpus (86th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Unclustered & Miscellaneous (2551 abstractions)
Nearest neighbors
- Smallest-Circle Problem — 0.82
- Skip list — 0.82
- Planar ternary ring — 0.81
- Autoregressive Integrated Moving Average — 0.81
- Càdlàg Function — 0.80
Computed from structural-signature embeddings · 2026-10-08