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Cell morphology is an important regulator of cell function. Many abnormalities in cellular behavior can be discerned from changes in the shape of the cell and its organelles, typically the nucleus. Two major challenges for developing such phenotypic assays are reconstructing 3D surfaces of individual cells and nuclei from confocal images and developing characterizations of these surfaces for comparisons. We demonstrate two algorithms – 3D active contours and 3D condensed-attention UNet – to segment cells and nuclei from confocal images. The cell and nuclear surfaces are then converted into vectors using a reversible, spherical transform – i.e., shapes can be recovered from the vectors. Typical methods for characterizing shapes using size, shape, and image parameters such as area, volume, shape factor, solidity, and pixel intensities are not amenable to such reverse transformation. Our vector representation's principal component analysis shows that the significant modes of variability among cell and nucleus shapes are scaling and flattening. We benchmark these modes using a known mechanical model for nucleus morphology. Subsequent modes alter the eccentricity of the nucleus and translate and rotate it with respect to the cell. Our vector-space representation of cell and nucleus shape helps physically interpret the variability sources. It may further help to guide mechanical models and identify molecular mechanisms driving cell and nuclear shape changes. [ABSTRACT FROM AUTHOR]