Quantitative Analysis of Surface Growth

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Abstract
Characterization of the growth of a small patch of surface, e.g., the outer face of a growing cell [animal or plant] during a fixed period of time, can take the form of a strain cross. The long arm of the cross shows the maximum relative extension in the patch. The length of the short arm indicates the minimum extension. With the conventions used here, relative extension less than one is a contraction. When these perpendicular maximum and minimum extensions are parallel to the borders of the patch, as when a square deforms to a rectangle, quantitation of the cross is simple. When, however, the patch is irregular in outline and the directions of the extremal extensions are not obvious, much more computation is required. A 2 .times. 2 matrix of numbers summarizes the deformation of the coordinates of "landmark" points in the initial figure to their final configuration. The matrix can be decomposed to yield the two essential types of information: (1) the magnitude of each arm of the strain cross, as above, and (2) the angular relation of the cross to the orientation of the patch (e.g., a recognizable edge). This angle has one value before growth and another afterward. Using three landmarks on a patch, the matrix deforms the initial figure to the final figure exactly. For more than three landmarks, one finds the best-fit matrix that can produce the deformation.

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