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**Extra info for A treatise on the analytical geometry of the point, line, circle, and conical sections**

**Example text**

Theoretically, the above observations agree with Fig. 16 where results are shown from deductions of various theoretical/experimental data subsequently considered in this book. 3 MM-Composites To get a full picture of how shape factors vary with phase geometries we need information with respect to MM-composites, which are transition materials where geometries change from being DC to CD. Such information cannot be obtained that easily. In this book we will establish some coarse rules based on deductions from stiﬀness results obtained in the FEM analysis presented in Sect.

4. 10) and Fig. 4 that DC, MM, and CD composite geometries are described by shape functions of magnitudes (µP k , µSk ) = (+, −), (µP k , µSk ) = (+, +), and (µP k , µSk ) = (−, +) respectively. 9), that a special composite is considered. Shape Factors Shape function values of special interest are the so-called shape factors, µ0P k , µ0Sk = µP k (0), µSk (0) quantifying the composite geometry of a P-S mixture with a dilute (vanishing) amount of phase P. In the subsequent text the term “shape factor” is also used for µ1P k , µ1Sk = µP k (1), µSk (1).

The meaning of shapes, however, are re-interpreted according to the following: Flat phase P particles with low aspect ratios are understood as organic particles preparing to serve as an enveloping matrix for ﬂat phase S particles. Long 7 Quantiﬁcation of Geometry 45 Fig. 3. FEM-model used for shape factor analysis of MM-composites. Shape factors indicated are deduced from a volumetric analysis (k) phase P particles with high aspect ratios are understood as organic particles preparing to serve as an enveloping matrix for long phase S shapes.