ROTATION THROUGH DEFORMATION

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

The wave profile combines radial and angular components:

The inclination $\gamma \left(t\right)$ transforms the radial–vertical plane. Defining the projected radius:

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

$A$ is the amplitude; $K_r$ and $K_a$ are the radial and angular cycle counts. The phases ${\phi }_r\left(t\right)$ and ${\phi }_a\left(t\right)$ and the inclination $\gamma \left(t\right)$ 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 $K_a\in {\mathbb{N}}_{>0}$: $\alpha =0$ and $\alpha =2\pi$ represent the same position. The inclined family uses $-\frac{\pi }{2}\le \gamma \left(t\right)\le \frac{\pi }{2}$.

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

Here, $\omega$ is the phase rate, $\Omega$ the oscillation angular frequency and $G$ its nonnegative angular amplitude, with $\left|{\gamma }_0\right|+G\le \frac{\pi }{2}$. The radial phase can follow the same linear law. These models describe geometric evolution, rather than contact forces.

The opening image is a visual interpretation of the concept. This figure is calculated from the equation and does not simulate the dynamics of contact with a sphere.