CREATIO · CT–01

CT–01 · TECHNICAL NOTE

Mathematical formulation

Version 0.1 · 19 September 2026 · Preliminary formulation for review

This note organizes the CREATIO parametrization and proposes an initial analysis framework. Kinematic relations follow from the chain rule; the dynamic model below approximates a point mass on a surface with prescribed motion.

1. Geometry and parameters

The CREATIO surface is described by the Cartesian parameterization $c (α, R, t)$ . The parameter $R$ measures distance from the inner radius in the reference surface; $α$ is the angular coordinate and $t$ denotes time.

0 < R_{0} < R_{1}, Δ R = R_{1} - R_{0}

R \in [0, Δ R], α \in [0, 2 π]

A \geq 0, K_{r}, K_{a} \in N_{>0}, t \geq 0

The wave profile combines radial and angular components:

W (α, R, t) = A \sin (\frac{2 π K_{r} R}{Δ R} + φ_{r} (t)) \cos (K_{a} α + φ_{a} (t))

(1)

The inclination $γ (t)$ transforms the radial–vertical plane. Defining the projected radius:

ρ (α, R, t) = R_{0} + R \cos (γ (t)) - W (α, R, t) \sin (γ (t))

(2)

c (α, R, t) = (\begin{array}{c} ρ (α, R, t) \cos (α) \\ ρ (α, R, t) \sin (α) \\ R \sin (γ (t)) + W (α, R, t) \cos (γ (t)) \end{array})

(3)

In the reference case $γ \equiv 0$ , this reduces to:

c (α, R, t) = (\begin{array}{c} (R_{0} + R) \cos (α) \\ (R_{0} + R) \sin (α) \\ W (α, R, t) \end{array})

(4)

$A$ is the amplitude; $K_{r}$ and $K_{a}$ are the radial and angular cycle counts. The phases $φ_{r} (t)$ and $φ_{a} (t)$ and the inclination $γ (t)$ are real-valued functions of time. All angles in the equations are in radians. The visualization controls display degrees and convert them internally.

Angular periodicity follows from $K_{a} \in N_{>0}$ : $α = 0$ and $α = 2 π$ represent the same position. The inclined family uses $- \frac{π}{2} \leq γ (t) \leq \frac{π}{2}$ .

In the animation, a phase can vary linearly, while inclination can oscillate sinusoidally:

φ_{a} (t) = φ_{a} (0) + ω t

(5)

γ (t) = γ_{0} + G \sin (Ω t)

(6)

Here, $ω$ is the phase rate, $Ω$ the oscillation angular frequency and $G$ its nonnegative angular amplitude, with $| γ_{0} | + G \leq \frac{π}{2}$ . The radial phase can follow the same linear law. These models describe geometric evolution, rather than contact forces.

2. Surface regularity and orientation

The parametrization is assumed twice continuously differentiable over the analysis domain. At regular points, the unit normal is defined by:

n = \frac{c_{α} \times c_{R}}{‖ c_{α} \times c_{R} ‖}, ‖ c_{α} \times c_{R} ‖ > 0

(7)

The α and R subscripts denote partial derivatives. The normal can be reversed to point towards the side accessible to the mass. Local regularity, absence of self-intersections and deformation limits must be checked for each parameter set.

3. Kinematics: surface, pattern and mass

Let q = (α, R) and the trajectory x(t) = c(q(t), t). Below, q₁ = α and q₂ = R; repeated indices i, j denote sums from 1 to 2. Partial time derivatives hold q fixed.

\dot{x} = c_{t} + c_{i} \dot{q_{i}}

(8)

\ddot{x} = c_{tt} + 2 c_{ti} \dot{q_{i}} + c_{ij} \dot{q_{i}} \dot{q_{j}} + c_{i} \ddot{q_{i}}

(9)

Here cᵢ = ∂c/∂qᵢ, cᵢⱼ = ∂²c/(∂qᵢ∂qⱼ), cₜᵢ = ∂²c/(∂t∂qᵢ) and cₜₜ = ∂²c/∂t². Dots on q denote total time derivatives. The velocity cₜ describes the parametrization at fixed coordinates; it equals membrane material velocity only if these coordinates label material points.

For constant Kₐ and a constant angular phase of the pattern, Kₐαₚ + φₐ(t) = constant, one obtains:

{\dot{α}}_{p} = - \frac{\dot{φ_{a}}}{K_{a}}

(10)

This is the angular velocity of the phase pattern. It does not determine a sphere’s speed or guarantee that it follows a cavity. Its trajectory depends on initial conditions, forces and contact.

4. Initial dynamic model

Assumptions: inertial frame; prescribed surface; constant point mass m; maintained contact in a regular region; ideal normal reaction. Load-induced membrane deformation and actuator dynamics are excluded from this first approximation.

T = \frac{m}{2} {‖ c_{t} + c_{i} \dot{q_{i}} ‖}^{2}, L = T - V

(11)

\frac{d}{d t} (\frac{\partial L}{\partial \dot{q_{i}}}) - \frac{\partial L}{\partial q_{i}} = Q_{i}

(12)

V(q,t) is potential energy evaluated on the surface. Qᵢ = fₙ꜀ · cᵢ is the nonconservative generalized force. For ideal frictionless sliding with no other nonconservative forces, Qᵢ = 0. In uniform gravity, V = −m g · c. Because the surface moves, the mass’s mechanical energy is generally not conserved: actuation performs work.

m \ddot{x} = m g + N n + f_{τ}, N \geq 0

(13)

Here N is the normal reaction and fτ is the tangential contact force; only gravity and contact are included. If the solution requires N < 0, unilateral contact is no longer admissible and separation must be handled. A finite-radius sphere requires an offset surface for its centre, rotational inertia and a friction/rolling law: the sphere centre is not identified with c.

5. Aerospace extension

\dot{H_{O}} = τ_{ext,O}

(14)

Hₒ is total angular momentum and τext,O is external torque, both about an origin O fixed in an inertial frame. The full system must include structure, membrane, actuators and moving masses. With zero external torque, Hₒ is conserved; internal motion can redistribute angular momentum and change orientation. Attitude prediction requires a coupled model, beyond the geometric animation.

In microgravity, the mechanism maintaining mass contact must be specified — for example, mechanical confinement or a modelled magnetic interaction. The assumptions in this note are working proposals, not CREATIO validation results.