If the circle lies in a plane, inclined
to the plane of the drawing, its projection is an ellipse. 95. Perpendicular parallel
projection
of the circle
If the circle lies in a plane which is parallel to the plane of the drawing, its perpendicular projection is a congruent circle. If it lies in a plane, perpendicular to the plane of the drawing, it falls on the trace line and has the length of the diameter. If you imagine the circle to be specified by projection and height of its centre, you can flip it into the plane of the drawing; you thus discover that, in general, to every point of the projection correspond two points of its circumference.
If the circle lies in a plane, inclined
to the plane of the drawing, its projection is an ellipse.
If you project the ends of any diameter and its centre, you obtain two congruent strips; the image of the centre bisects the projection of the diameter. The projection of the centre is therefore also the centre of the ellipse, the projections of its diameters are diameters of the ellipse.
The diameter of the circle, which is parallel to the plane of the drawing, appears in its true length, the one perpendicular to it is a fall line and its projection is perpendicular to that of the parallel one. The first of these diameters is the longest, the second the shortest. These projections are the large and the small axes of the ellipse. Since pairs of diameters of the circle are equally inclined to the parallel diameter and to thar perpendicular to it, also their projections lie symmetrically with respect to the axes. These axes are the axes of symmetry of the ellipse.
Construction
Since the projection of a circle does
not change on parallel displacement in the direction of
projection, you can assume, that the centre lies in the centre of
the plane of the drawing.
Let there be given
a circle by its main
axis and angle of inclination and assuming that you have flipped it into the plane of the drawing. You then find the length of the small
axis by
transferring the radius to one leg of the angle of inclination
and projecting the end point on to the other leg. You can
construct in this manner the projection of any point X.
All chords, parallel to the large axis are reproduced unchanged.
The line linking (C) and (X) 
intersects the extension of the large
axis at the trace
point of the line Y;
hence also C'X' must pass through Y;
but, by the generalization
of the Ray Theorem, you
will have X'Q : (XQ) = b : a.
The lengths of all chords, which are parallel to the small axis,
are reduced in the ratio b : a.
Hence one has the following simple construction of the points of an ellipse. You draw about M the large vertex circle with the radius a and the small vertex circle with the radius b. If you now draw any radius, which intersects the circles in E and F and draw through E the parallel to the main axis, through F that to the other axis, then their point of intersection P lies on the ellipse, because, by the Ray Theorem,
QP : QF = ME : MF = b : a.
You can draw any points of the ellipse in this way.
Having seen that the vertical projection of a circle is an ellipse, you can also draw axonometric images of the cone and cylinder.