2. Ellipses are useful in drawing because they are perspective views of circles.
However, the perspective projections may be difficult to draw, because major
and minor axes do not project as major and minor axes, centres do not project
to centres (in general), concentric circles do not project as concentric ellipses,
and so on. Ellipses may be drawn from the definitions above by calculating
and plotting points, or directly by precise or approximate constructions. The
concentric-circle method introduced above is one practial method for
constructing an accurate ellipse when the axes are known.
A construction based on the focal property may be used for laying out small
ellipses in the field, called gardeners' ellipses. Place two stakes in the ground
with tacks in their tops marking the desired foci. Then make a loop of stout
cord of length 2a + 2c, and loop it around the tacks. Taking something like a
chaining pin or pointed metal rod, put it in the loop and draw the loop taut.
Everywhere the pin may be will be a point on the ellipse, and as many points
as desired may be marked. This method works quite well and is rapid. Stakes
and string make it possible to lay out many decorative curves: circles, ellipses,
parabolas and spirals, as the constructors of corn circles know.

3. The simplest way to draw an ellipse on the drawing board is by using a
trammel, as shown at the right. This is simply a straight strip of cardboard,
with distances o-a = a and o-b = b marked on it. The perpendicular axes are
drawn where the ellipse is required, and the position of point "o" when the
other points are carefully held on the axes is a point on the ellipse. There are
machines for drawing ellipses based on this principle, but they are expensive
and probably no longer manufactured. The trammel requires care, but is dead
easy to use.
A standard method is called the parallelogram method, and is applicable to
conjugate diameters as well as to the usual perpendicular axes. OE and EA
are divided into N equal intervals. In the figure, N = 2 for simplicity. Then draw
CB and intersect with DG extended to determine a point P on the ellipse. A
similar method can be used to draw a parabola--it is all in how the intersecting
lines are drawn. Note that we could draw circles with this method or the
trammel, but there are easier ways!

4. The theory behind the parallelogram method can be found from the diagram
at the right. The point P on the ellipse is the intersection of lines AB and CD.
s is a parameter that runs from 0 to 1 as P goes from the end of the minor
axis to the end of the major axis. Line AB has the equation y = b - sbx/a,
while line CD has the equation x = sa(b + y)/b. Eliminating s between these
two equations, we find quite readily that x2
/a2
+ y2
/b2
= 1, which is the
equation of an ellipse with semimajor axis a and semiminor axis b.
If you have the major minor axes of an ellipse, it is easy to find the foci by
swinging an arc of radius equal to the semimajor axis from the end of a
minor axis. This arc will cut the major axis at the focal points. Conversely,
if you know the foci and the major axis, the intersection of arcs drawn from
the foci with radius equal to the semimajor axis will determine the ends of
the minor axis.

5. The above methods all make accurate ellipses. It is much more convenient to
represent an ellipse by circular arcs that can be drawn with a compass than to
connect points laboriously determined. The simplest case is shown in the
figure at the left, called a three-center arch. To a draftsman, it is a "four-center
ellipse." The centers are the symmetrically-placed D, C, D' and C'. On the line
AB joining the ends of the major and minor axes, lay off distance BM = a - b,
the difference in the semiaxes. Now draw the perpendicular bisector of the
remainder AM, the line L, which determines the centers D and C, as well as
the point G where the two arcs meet. Now the symmetrical points D' and C'
can be laid out. Arcs of radii DG and CG are then drawn to complete the arch
or ellipse. This method is excellent for representing ellipses on drawings, but
may be a little crude for an actual arch, where the approximation may give an
uneasy feeling. It would probably be better in any case to calculate an exact
ellipse and lay it out by coordinates for the actual structure, while the three-
centered approximate ellipse will always do for a drawing.

6. Stevens' Method gives a somewhat better three-centred ellipse useful in isometric
projection. In the figure at the right, an ellipse is to be drawn in the rhombus ABCD,
which is the isometric projection of a square. Using two triangles, first find the tangent
point E, and draw an arc of radius R equal to ED, cutting the minor axis at Q. Do the
same thing with C as centre. With radius OQ, draw an arc locating points P and P' on
the major axis. Now draw PD and extend it to the arc of radius R. This is not shown to
avoid confusing the diagram. Now, draw arcs of radius r equal to the distance from P
to the curve at each end. They will be tangent to the arcs of radius R, so the
approximate ellipse has been constructed.
A five-centred arch gives a better approximation by using three different radii,
but is a little complicated to lay out. Here is the method: refer to the figure at the
left. Start with the box AFDO with width equal to the half-span and height equal
to the rise. Draw AD, and then from F draw a perpendicular to this line, which
intersects the minor axis at H. Make OK equal to OD, and then draw a circle on
AK as a diameter (Q is not the center of the line AK). Now lay off OM equal to
LD, and draw an arc with center at H and radius HM. You will not have the point
N yet, but be patient. From A, lay off AQ equal to OL, and AP equal to half of
AQ. P should be on the line FH. Draw an arc with center P and radius PQ,
intersecting the arc through M at point N. Now, P, N and H are the centers of the
arcs defining one-quarter of the approximate ellipse. The sectors are shaded to
make this clear, and to show that the arcs are tangent to each other. Find the
centers of the remaining arcs by symmetry.

7. For accurate drawing, it is useful to be able to find the directions of tangents to
an ellipse, and points of tangency. Tangents to a circle are perpendicular to the
radius at the point of tangency, so it is easy to draw a tangent at a given point on
a circle, or to construct the tangent from an external point. The principles of two
methods for drawing tangents to an ellipse are shown in the figure at the right.
The method at (a) uses the auxiliary circle. First, the point S on the auxilary circle
corresponding to the point of tangency P is found, and a tangent to the auxiliary
circle is constructed, cutting the axis at point T. The desired tangent is the line
TP. This method uses the vertical compression that turns the auxiliary circle into
the ellipse.
Method (b) uses the focal properties of the ellipse. The focal radii from P to the
foci F and F' are drawn, and the external angle is bisected. The bisector is the
desired tangent. This direction is perpendicular to the bisector of the internal
angle between the focal radii.

8. The fact that the tangent is perpendicular to the bisector of the angle between
the focal radii at P is not difficult to prove. In the figure at the right, the
neighborhood of a point A on the ellipse is shown. Suppose B is a neighboring
point, so close that the directions to F and F' are not sensibly changed. For B to
be a neighboring point on the ellipse, the sum of the focal radii must remain
unchanged (and equal to the major axis). In the direction shown, it is clear that
the lengthening BC of the focal radius to F' is equal to the shortening AD of the
focal radius to F, so that their sum remains constant, and B is also a point on
the ellipse, or, in this approximation, on the tangent from A. The tangent, of
course, is the limit of the chord through BA as B approaches A.
The method for drawing a tangent from an external point P to an ellipse is shown
at the right. First, draw a circle with centre at P and radius PF. The intersection of
this circle with an arc of a radius equal to the major axis (2a) is point E. Point R is
the intersection of radius F'E and the ellipse. In the triangles PER and PFR,
corresponding sides are equal, since FR + RF' = 2a = ER + RF', or FR = ER. PE
and PF are equal by construction, while the side PR is common. Therefore, PR
bisects the external angle and is thus the tangent. A similar construction gives
the other tangent from point P.

9. The line PR also bisects the line FE. In the diagram, the triangles do not appear
exactly congruent because the foci are not accurately located. If you make a
careful drawing, the triangles will be congruent, and PR will be an accurate
perpendicular bisector of FE. Since FE is perpendicular to the tangent, if we want
a tangent with a certain direction (and there is no point P given), a line is drawn
in the perpendicular direction from F, and E is the intersection with an arc of
length 2a from F. Now EF is bisected, and the perpendicular bisector is the
desired tangent. We can now find tangents at a point on the ellipse, from an
external point, and in a certain direction.
If the focal points F and F' approach one another, direction of the tangent is more
and more restricted. In the limit as the ellipse becomes a circle, F = F', and the
tangent will be perpendicular to the radius, as we know from Euclid.
The drawing utilities in Windows draw an ellipse in a circumscribed rectangle,
which made preparing the graphics for this page quite easy, compared with the
effort required for parabolas and hyperbolas, which are not easy to draw with the
Windows routines. It is annoying not to be able to draw a circle from center and
radius, as in real drafting programs, but this can be worked around by putting the
center of a square of side equal to the diameter at the desired center point. If the
Shift key is held down in a graphics program, it is only necessary to move the
cursor along one diameter to get a circle, which makes it a little easier. A
template of three small crosses can be constructed and copied to wherever a
circle is desired if you are drawing a number of circles of the same diameter. One
cross is the center, the other two the ends of a horizontal line equal to the
diameter.

10. The line PR also bisects the line FE. In the diagram, the triangles do not appear
exactly congruent because the foci are not accurately located. If you make a
careful drawing, the triangles will be congruent, and PR will be an accurate
perpendicular bisector of FE. Since FE is perpendicular to the tangent, if we want
a tangent with a certain direction (and there is no point P given), a line is drawn
in the perpendicular direction from F, and E is the intersection with an arc of
length 2a from F. Now EF is bisected, and the perpendicular bisector is the
desired tangent. We can now find tangents at a point on the ellipse, from an
external point, and in a certain direction.
If the focal points F and F' approach one another, direction of the tangent is more
and more restricted. In the limit as the ellipse becomes a circle, F = F', and the
tangent will be perpendicular to the radius, as we know from Euclid.
The drawing utilities in Windows draw an ellipse in a circumscribed rectangle,
which made preparing the graphics for this page quite easy, compared with the
effort required for parabolas and hyperbolas, which are not easy to draw with the
Windows routines. It is annoying not to be able to draw a circle from center and
radius, as in real drafting programs, but this can be worked around by putting the
center of a square of side equal to the diameter at the desired center point. If the
Shift key is held down in a graphics program, it is only necessary to move the
cursor along one diameter to get a circle, which makes it a little easier. A
template of three small crosses can be constructed and copied to wherever a
circle is desired if you are drawing a number of circles of the same diameter. One
cross is the center, the other two the ends of a horizontal line equal to the
diameter.