By Melvin Hausner
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Extra resources for A Vector Space Approach to Geometry
9 and in Chapter 2. * This may be found more directly by computing the mass at F via AD and also via EC. † It is also clear if we consider the possibility of converting ounces to pounds. * That is, an ordered pair (m, P) consisting of a positive real number m and a point P. The discerning reader will realize that the words “ordered pair” imply that we are going to be rigorous. * Statements such as these are understood to be preceded by a clause of the form, “For any mass-points mP, nQ, . ” * See M.
I have been informed that all paints (commercially) are, in fact, classified by three numbers, indicating the relative proportions of (ideal) primary pigments. * It makes no more sense to talk of the barycentric coordinates of a point, without vertices, than it makes to talk of the Cartesian coordinates of a point without coordinate axes. * Compare with the footnote on page 6. * Actually, a picture is more helpful. † Physically, the dotted areas are shadows of segments from point sources of light, while the shaded areas are partial shadows of points from line sources.
Therefore, by the device of “mass-points,” we shall be able to deal with points P on a segment AB which divide AB in a given ratio. We may easily remember the rule (1) which locates P by noting that AP (distance to A) is in the numerator and so is m2 (mass of B). Diagrammatically, we merely place the mass of B on the segment opposite B, and similarly for the mass of A. This is indicated in Fig. 2 where we read AP/PB = 5/3 and we say that P divides AB in the ratio 5/3. The numbers 5 and 3 do not represent the distances AP and PB but give the ratio AP/PB, and any proportional numbers will do as well.
A Vector Space Approach to Geometry by Melvin Hausner