Elements of descriptive geometry, with applications to isometric projection and othe forms of one-plane projection; a text-book for colleges and ingineering schools by O. E. Randall.

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DESCRIPTIVE GEOMETRY

The isometric projection of a point whose coordinates with reference to three coordinate planes are known is the intersection of the
isometric projections of the three coordinate lines of the point.
The isometric projection of a straight line is the straight line determined by the isometric projections of two points of the given line.
Since the orthographic projections of parallel lines are parallel,
the isometric projections of parallel lines will be parallel whether
the lines themselves are parallel to the coordinate axes or not.
The isometric projection of a curved line is the line determined
by the isometric projections of the points which determine the line.
^dfo~ ~The isometric projection of a surZ^7,"\ a>fx face determined or limited by lines,
a / straight or curved, is that surface de\-,> ~termined by the isometric projections
0o~' ~ of the determining or limiting lines.
To draw the isometric projection
of a magnitude of three dimensions,
place the magnitude in such a position that its three principal dimension
Y' lines shall be either coincident with
or parallel to the three coordinate axes
FIG. 168
0-X, 0- Y, and 0-Z; then the isometric projections of the principal dimension lines of the magnitude
will be either coincident with or parallel to the isometric axes.
433. Problem 298. To find the isometric projection of a point whose
coordinates with reference to two rectangular axes are known.
In Fig. 168 let o'-x', o'-y', and o'-z' represent the isometric axes.
Place the two rectangular axes to which the point is referred
in coincidence with O-X and 0-Z respectively. Then o'-x' and
o'-z' are the isometric projections of the rectangular axes.
Make o'-a' equal to the distance of the point from the axis 0-Z,
and through a' draw a'-d' parallel to o'-z'.
Make o'-b' equal to the distance of the point from the axis
O-X, and through b' draw b'-d' parallel to o'-x'.
The lines a'-d' and b'-d' are the isometric projections of the
coordinate lines of the point, and the point d' is the isometric
projection sought.