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Chapter 3: Characterizing Structures Using Differential Geometry

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A primary task for structural geologists is to describe and characterize surfaces such as those of the Jurassic sandstone formations exposed on the flank of the Waterpocket monocline pictured in the Frontispiece for this chapter. This may be accomplished in a mathematically rigorous manner using concepts from differential geometry, the branch of mathematics that brings the power of vector calculus to geometry (Gauss, 1827). Here we review some of the elementary concepts of differential geometry that are helpful to quantify the departure of geological surfaces from a plane and geological lineations from a straight line. Structural data typically are gathered at scattered outcrops as point measurements of orientation and differential geometry provides the tools for the quantification and analysis of the spatial variations in orientation of geological structures.