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Building Cyclidic Volumes

Pablo Colapinto (wolftype@gmail.com)
AGACSE 2018, University of Campinas (IMECC - UNICAMP), Brazil

Waldemar Cordeiro (1925–73) -- Idéia visível 1956

“Residência Ubirajara Keutenedjian”, São Paulo, 1955

Image From: Pengbo Bo, Helmut Pottmann, Martin Kilian, Wenping Wang, and Johannes Wallner. Circular arc structures. ACM Trans. Graph., 30(4):101:1–101:12, July 2011.

See:

Alexander I. Bobenko and Emanuel Huhnen-Venedey. Curvature line parametrized surfaces and orthogonal coordinate systems: discretization with Dupin cyclides. Geome- triae Dedicata, 159(1):207–237, 2012.
Pablo Colapinto, Composing Surfaces with Conformal Rotors, Advances in Applied Clifford Algebras, March 2017, Volume 27, Issue 1, pp 453–474
Leo Dorst. The construction of 3d conformal motions. Mathematics in Computer Science, 10(1):97–113, 2016. DOI 10.1007/s11786-016-0250-8.
Leo Dorst and Robert Valkenburg. Square root and logarithm of rotors in 3d conformal geometric algebra using polar decomposition. In Leo Dorst and Joan Lasenby, editors, Guide to Geometric Algebra in Practice, pages 81–104. Springer, 2011.
Charles Dupin. Applications de géométrie de mécanique à la marine, aux ponts et chaussées; pour faire suite aux développements de géométrie. Paris, 1822.
David Hestenes, New Foundations for Mathematical Physics, Chapter 2, available online at http://geocalc.clas.asu.edu/html/NFMP.html.
Ralph Robert Martin. Principal Patches for Computational Geometry. PhD thesis, Pembroke College, Cambridge University, 1983.
Pengbo Bo, Helmut Pottmann, Martin Kilian, Wenping Wang, and Johannes Wallner. Circular arc structures. ACM Trans. Graph., 30(4):101:1–101:12, July 2011.

$\sigma(k, \tau) = e^{-\frac{k}{2}\tau}(-\infty \cdot \tau)e^{\frac{k}{2}\tau}$

$\sigma(\sigma_1, \sigma_2) = R\sigma_1\tilde{R}$
$R = e^{\frac{t}{2}\mbox{log}(-\widehat{\frac{\large \sigma_2}{\large \sigma_1}})}$
NOTE "Hat" symbol $\widehat{x}$ is here used to mean "normalize" not "involution"

### Conformal Cast of Characters | | primitive | | infinitesimal | | infinite | |---|---|---|---|---|---| | $\sigma$ | Sphere | $p$ | Point | $\pi$| Plane | $K$ | Circle | $T$| Tangent Bivector | $\Lambda$ | Line | $\kappa$ | Pair | $\tau$ | Tangent Vector | | FlatPoint

### Duality Relationships - Sphere: $\sigma=\Sigma^{*}$ - Plane: $\pi=\Pi^{*}$ - Circle: $\kappa=K^{*}$ - Tangent: $\tau=T^{*}$

We can define six spheres at every frame

$\{\sigma\}^{(0,1,0)}$

$\{\sigma\}^{(0,0,0)}$

$\{\sigma\}^{(1,0,0)}$

$\{\sigma\}^{(1,1,0)}$

$\{\sigma\}^{(1,1,1)}$

$\{\sigma\}^{(1,0,1)}$

$\{\sigma\}^{(0,0,1)}$

$\{\sigma\}^{(0,1,1)}$

Transform constant coordinate surfaces

$\kappa^w_v=\frac{1}{2}\mbox{log}(-\widehat{\frac{\sigma^{v(0,1,0)}_u}{\sigma^{v(0,0,0)}_u}})$

$R^w_v = e^{\kappa^w_v}$

$\sigma^{u}_v$ on $K_u$ is orthogonal to $\sigma^v_u$ on $K_v$

$\large \sigma^v_u$ at $K_v(1)$

$\large \sigma^u_v$ at $K_u(1)$

$\underset{\small (1,0,0)}{\large \sigma^u_v}= - \underset{\small (0,1,0)}{\large \sigma^v_u} \cdot \underset{\small (1,0,0)}{\large \tau_u}$

Find Orthogonal Spheres $\sigma^u_v = -\sigma^v_u \cdot \tau$

Interpolate, Trilinearly, To Navigate the Coordinate System
$f (u, v, w): X \mapsto RX\tilde{R}$ where:
$R = R_w R_v R_u $
and
$R_u = e ^ {u\kappa^w_u}$
$R_v = e ^ {v\kappa^w_v}$
$R_w = e ^ {w\kappa^{u_x}_w}$
with
$\kappa^{u_x}_w = \frac{1}{2}\mbox{log}( -\widehat{[\frac{\sigma^{w(u_x,0,1)}_{v}} {\sigma^{w(u_x,0,0)}_{v}}]})$

Top Down view of $\sigma^{w(u_x,0,1)}_{v}$ and $\sigma^{w(u_x,0,0)}_{v}$ with changing $u_x$

Warp The Day Away

At border, we need an inverse mapping to connect to next coordinate system.

The Shape of Things to Come:

Inverse Mapping (Given a point, what are the coordinates -- needed for connecting adjacent coordinate systems)
Please, let us flesh out discrete differential geometry (with pictures?).
Besides "buildability", what advantages / disadvantages are there to defining deformations this way?

Cordeiro, Untitled - Estudo, 1952