  Quasi-conformal (QC) theory is an important topic in complex analysis, which
studies geometric patterns of deformations between shapes. Recently,
computational QC geometry has been developed and has made significant
contributions to medical imaging, computer graphics and computer vision.
Existing computational QC theories and algorithms have been built on
triangulation structures. In practical situations, many 3D acquisition
techniques often produce 3D point cloud (PC) data of the object, which does not
contain connectivity information. It calls for a need to develop computational
QC theories on PCs. In this paper, we introduce the concept of computational QC
geometry on PCs. We define PC quasi-conformal (PCQC) maps and their associated
PC Beltrami coefficients (PCBCs). The PCBC is analogous to the Beltrami
differential in the continuous setting. Theoretically, we show that the PCBC
converges to its continuous counterpart as the density of the PC tends to zero.
We also theoretically and numerically validate the ability of PCBCs to measure
local geometric distortions of PC deformations. With these concepts, many
existing QC based algorithms for geometry processing and shape analysis can be
easily extended to PC data.
