Notes on Elements of (Analytical) Solid Geometry (Classic Reprint)

Excerpt from Notes on Elements of (Analytical) Solid Geometry

1. We have seen how the position of a point in a plane with reference to a given origin O is determined by means of its distances from two axes Ox, Oy meeting in O. In space, as there are three dimensions, we must add a third axis Oz. So that each pair of axes determines a plane, Ox and Oy determining the plane xOy; Ox and Oz the plane xOz ; Oy and Oz the plane yOz. And the position of the point P with reference to the origin O is determined by its distances PM, PN, PR from the zOy, zOx, xOy respectively, these distances being measured on lines parallel to the axes Ox, Oy and Oz respectively. This system of coordinates in space is called The System of Triplanar Coordinates, and the transition to it from the System of Rectilinear Plane Coordinates is very easy. We can best conceive of these three coordinates of P by conceiving O as the corner of a parallelopipedon of which OA, OB, OC are the edges, and the point P is the opposite corner, so that OP is one diagonal of the parallelopipedon.

2. If PM= OA =a, PN =OB = b, PR = OC = c, the equations of the point P are x= a, y = b, z = c, and the point given by these equations may be found by the following construction: Measure on OX the distance OA =a, and through A draw the plane PNAR parallel to the plane yOz. Measure on Oy the distance OB= b, and draw the plane PMBR parallel to xOz, and finally lay off OC and draw the plane PMCN parallel to xOy. The intersection of these three planes is the point P required. (Fig. 1.)

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