An elementary treatise on cubic and quartic curves, by A. B. Basset.

PROJECTION OF A TRIANGLE.

235

projection to the original plane, and the angle s which their
line of intersection makes with the axis of x or y, which are the
five constants in question.
353. Any given triangle can be projected into any other given
triangle, in such a manner that any given point P in the plane
of the first triangle corresponds to any given point Q in the
plane of the second.
Let ABC be the given triangle; let A be the origin, AB
the axis of x, and let two lines through A in and perpendicular
to the plane ABC be the axes of y and z.
Let ABC' be a triangle similar to the second triangle, such
that the base AB is equal to that of the original triangle; and
place this triangle so that the bases AB coincide, whilst the
plane ABC' makes an angle e with the plane z.
Let (f, g, 0) be the coordinates of C, and (p, q, 0) those of
any point P in the plane z. Then if accented letters denote
the coordinates of the corresponding points in the plane ABC',
the coordinates of C' and P' are f', g' cos e, g' sin E, and p', q' cos e,
q' sin e respectively. Hence the equations of CC' and PP' are
x-f_ y-g z
f '-f g'cose-g g'sin
and
x-p _ y-q z
p'-p q cose-q q sine
The conditions that the projection should be possible require
that the lines CC' and PP' should intersect at a point V, which
is the vertex. Now the foregoing equations are sufficient to
determine (x, y, z), which gives the point V, and also the angle e
which the plane of projection makes with the plane of the
triangle. If therefore a line A'B' be drawn in the plane VAB
parallel to AB and equal to the side A'B' of the second triangle,
the section of the pyramid VABC by a plane through A'B'
parallel to the plane ABC' will be a triangle equal to the second
given triangle; also the point where VP cuts the plane of the
latter triangle will be the given point Q.
The preceding theorem shows that any quadrilateral can be
projected into a square. For let A BCD be the quadrilateral, and
A'B'C'D' a square; then the triangle ABC can be projected into