Mathematics & Spline Trajectories

Mathematical Rigor Behind Arc-Length LUTs & Aerodynamic Flares

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This document provides a mathematical breakdown of the trajectory and flight mechanics implemented in Decnet.Perch.Runtime.Motion.

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1. Cubic Hermite Spline Formulation

An approach curve connects the creature’s current spatial state $(\mathbf{P}0, \mathbf{V}_0)$ to the target landing pose $(\mathbf{P}_1, \mathbf{N}{\text{normal}})$ over parametric domain $u \in [0, 1]$:

\[\mathbf{S}(u) = h_{00}(u)\mathbf{P}_0 + h_{10}(u)\mathbf{M}_0 + h_{01}(u)\mathbf{P}_1 + h_{11}(u)\mathbf{M}_1\]

Where the cubic Hermite basis functions are:

• $h_{00}(u) = 2u^3 - 3u^2 + 1$
• $h_{10}(u) = u^3 - 2u^2 + u$
• $h_{01}(u) = -2u^3 + 3u^2$
• $h_{11}(u) = u^3 - u^2$

Tangents $\mathbf{M}_0$ and $\mathbf{M}_1$ are scaled by curve chord length $|\mathbf{P}_1 - \mathbf{P}_0|$ and creature cruise speed to ensure continuous curvature and zero second-derivative discontinuities at boundary conditions.

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2. Numerical Arc-Length Integration (DEC-004)

Because $|\mathbf{S}’(u)|$ varies along the curve, evaluating the spline at uniform parameter steps $\Delta u$ results in severe physical velocity fluctuations.

PERCH solves this by constructing a 32-sample cumulative arc-length look-up table:

\[s_k = \sum_{i=1}^{k} \frac{1}{2} \left( \|\mathbf{S}'(u_{i-1})\| + \|\mathbf{S}'(u_i)\| \right) \cdot \Delta u\]

Given a required travel distance $s \in [0, s_{\text{total}}]$:

1. Binary search finds index $k$ such that $s_k \le s < s_{k+1}$.
2. Linear interpolation resolves normalized factor $\alpha = \frac{s - s_k}{s_{k+1} - s_k}$.
3. The true parametric coordinate is resolved as: \(u = u_k + \alpha (u_{k+1} - u_k)\)

This guarantees constant, controllable physical deceleration ($a \le a_{\text{max}}$) along the entire flight path.

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3. Dynamic Banking Physics

When a flying creature navigates a curved trajectory, aerodynamic lift must tilt to balance centrifugal force:

\[\tan(\phi) = \frac{v^2}{g \cdot R} = \frac{v^2 \cdot \kappa}{g}\]

Where:

• $\phi$ is the roll banking angle.
• $v$ is the creature’s forward airspeed.
• $\kappa$ is the instantaneous spline curvature: \(\kappa(u) = \frac{\|\mathbf{S}'(u) \times \mathbf{S}''(u)\|}{\|\mathbf{S}'(u)\|^3}\)
• $g$ is gravitational acceleration ($9.81\text{ m/s}^2$).

PERCH caps $\phi$ against the profile’s MaxBankAngle (default: $45^\circ$) to prevent excessive or unnatural inverted roll attitudes during sharp approach turns.