A conic section is a curve obtained as the intersection of a cone
and a plane.
Characterization
The given domain in which this is to happen is three-dimensional. A cone
is, for these purposes, understood to be double-ended, in the sense
that it doesn't stop at its vertex (like a dunce's cap does); after the radius
of the cone has shrunk to zero at the vertex, it grows again on the other
side. Chose co-ordinates centred on the cone's vertex, using the axis of
rotation of the cone as z-axis; then the cone is the set of points with
[x,y,z] in {[r.Cos(a), r.Sin(a), k.r]: r is real, a is an angle} for some
non-zero constant real k; this cone can equally be characterized as
x2 +y2 = (z/k)2.
A general plane is characterized by u.x +v.y +w.z = h for some constants
u, v, w and h, with at least one of u, v and w non-zero, so its intersection
with the cone is given by u.r.Cos(a) +v.r.Sin(a) +w.k.r = h. However, we're
going to want to know what the intersection looks like as a sub-set of the
plane, so we'll want a description of it in terms of some co-ordinates that
make sense in the plane, chosen so as to describe the intersection reasonably
simply. Also, while we specified the z axis to be the cone's axis of
rotational symmetry, we still have freedom to chose our y and x axes within
the plane perpendicular to that through the cone's apex. So chose, as x-axis,
a line perpendicular to the plane's normal, i.e. parallel to some line in the
plane, as well as to the z axis. (In the case where the plane itself is
perpendicular to the z-axis, this is no constraint and the y-axis shall also
be perpendicular to the plane's normal, but there's no harm in that.) We can
then use x as one co-ordinate in our plane, that increases by the distance
moved when one moves in the plane parallel to the x-axis. This implies u = 0,
since we can change x within the plane without changing y or z.
We get the same cone if we replace k with −k, so
we can presume that k is positive. For v.y +w.z = h to specify a plane, at
least one of v and w must be non-zero; and applying a common non-zero scaling
to all of v, w and h makes no difference to which plane is specified, so we
can re-scale to make v = Sin(c), w = Cos(c) for some angle c. We still have
the freedom to reverse the directions of our co-ordinates, which we can use to
ensure that c lies between zero and a quarter turn (inclusive), with h
positive. The co-ordinate y.Cos(c) −z.Sin(c) will then, give or take a
suitable constant offset, constitute a suitable co-ordinate in our plane to,
along with x, provide us with an orthonormal system. The final step in our
preliminaries is then to find a suitable constant offset, by choosing one that
makes the intersection symmetric about the origin in our plane.
The cone meets the plane x = 0 in the pair of lines z = ±k.y, which
meet our plane Sin(c).y +Cos(c).z = h when (Sin(c) ±k.Cos(c)).y =
h. When Tan(c) = k, the ± = − case is 0.y = h; if h = 0, the
line z = −k.y is the entirety of the intersection, with the plane
actually tangent to the cone in this line (in the terms of what follows, this
is really a degenerate case of the parabola and actually only really wants one
half of this line to be thought of as intersection); otherwise, z = −k.y
is parallel to (so does not meet) the plane k.y +z = h/Cos(c), which meets z =
+k.y at y = h/Cos(c)/k/2, z = h/Cos(c)/2.
Otherwise, when Tan(c) is neither ±k (the Tan(c) = −k case
being ruled out by our choices that made c be in the first quadrant and k
positive), the plane meets the cone's two lines at y = h/(Sin(c)
±k.Cos(c)), Cos(c).z = h.(1 −Sin(c)/(Sin(c) ±k.Cos(c))) =
±k.h.Cos(c)/(Sin(c) ±k.Cos(c)). If Cos(c) = 0 – i.e. c =
turn/4, the case where our plane is parallel to the z-axis as well as the
x-axis – we simply have y = h and z = ±k.h; the mid-point between
the two points is [y,z] = [h,0]. Otherwise, z = ±k.h/(Sin(c)
±k.Cos(c)) and our two points are
To simplify this (and subsequent work) let Q = Sin(c).Sin(c)
−k.k.Cos(c).Cos(c), making the mid-point h.[Sin(c),
−k.k.Cos(c)]/Q. In the plane, we want an offset to include in a
co-ordinate that's otherwise y.Cos(c) −z.Sin(c); since I want symmetry
in the origin, I pick the offset that makes the co-ordinate zero at the
mid-point just obtained. To that end, define a constant H by:
−H
= Sin(c).Cos(c).(1+k.k).h/Q
which is clearly the value of y.Cos(c) −z.Sin(c) at our
mid-point.
= (1+k.k).h.Tan(c)/(Tan(c).Tan(c) −k.k)
when Cos(c) is non-zero.
so the co-ordinate I'll use as complement to x in the plane is Y =
y.Cos(c) −z.Sin(c) +H. We can re-construct our original co-ordinates y
and z from Y and h = Sin(c).y +Cos(c).z, treating h for these purposes as if
it were a variable; we trivially obtain
y = (Y−H).Cos(c) +h.Sin(c)
z = h.Cos(c) −(Y−H).Sin(c)
and so can substitute these into x2 +y2 =
(z/k)2 to obtain:
Given that c is in the first quadrant, so neither Sin(c) nor Cos(c)
is negative, the common co-efficient Q has the same sign as Tan(c).Tan(c)
−k.k and hence as Tan(c) −k. The further analysis divides into
three cases, according as Tan(c) is less than, equal to or greater than k;
i.e. Q is negative, zero or positive.
Note that, given only the geometry of the intersection, as seen in the
plane, absent knowledge of the cone or the co-ordinates used above, we can
only discover the ratios among the coefficients in the equation, k.h/Q and
k/√Q, whence we can obtain h/√Q and k/h; we cannot determine the
absolute values of h, k and Q only from the curve we see in the plane,
although any one of them (or c) would suffice to imply the others, given these
ratios. The Q = 0 case yields only one constant, k/h, which still doesn't
suffice to determine h, k or c, although any one of them does (with the help
of Tan(c) = k) then suffice to tell us the rest. All the same, the general
shape of the intersection does suffice to tell us which of the three cases
we're dealing with, i.e. the sign of Q.
Hyperbola
When Tan(c) > k, the equation characterizing the intersection equates
the square of a multiple of Y with the sum, with positive coefficients, of the
squares of x and of the constant h; this cannot possibly be smaller than its
term in h alone. Consequently, Y is bounded away from zero; there is a range
of values Y cannot take, for all that it can (for suitable x) take arbitrarily
large values outside that range.
For large x and Y, the terms in these shall dominate that in h and the
intersection's equation is reasonably well approximated by
±k.x.√Q = Q.Y, which describes a pair of straight lines; if we
introduce co-ordinates u, v defined by
h.k.u = Q.Y +k.x.√Q
h.k.v = Q.Y −k.x.√Q
the (unapproximated) equation reduces to u.v = 1; thus the
intersection is just the result of applying a linear transformation to the
familiar y = 1/x hyperbola.
Parabola
When Tan(c) = k we have Q = 0, so the term in Y2 has zero
co-efficient and the equation above is simply:
which, aside from some simple scaling and a translation, is simply
the familiar quadratic y = x2. Substituting k = Tan(c) in this
and re-arranging, we get:
When Tan(c) < k we have Q < 0 and our equation can be re-arranged
as
(k.h)2 = (Q.Y)2 −Q.(k.x)2
in which each term is a positive co-efficient times a square; a
simple re-scaling of each of x and Y will reduce this to x2
+y2 = 1, which describes the unit circle; a re-scaled circle is
known as an ellipse.
A classical reprise
The other way conic sections are characterized in classical mathematics is
in terms of some two-dimensional geometry. Given a straight line and a point,
called focus, not on that line, any point in the plane has a distance
from the focus and a distance from the nearest point on the straight line; we
can compute the ratio between these two distances so we can define a function
on the plane that maps each point to this ratio. We can then ask for the sets
of points in the plane with particular values for this function; it turns out
that these sets of points are exactly the same conic section curves we've just
seen.
Use the line as x-axis and chose, as origin, the point on it closest to
the focus; orient the y-axis (which, by now, inevitably passes through the
focus) so as to make the focus be the point [x,y] = [0,L], where L is the
distance from the y-axis to the focus. The ratio of distance from focus
divided by distance from x-axis is then simply (x2
+(y−L)2)1/2/y; to characterize the set of points
on which this takes some fixed value, say a, we must study the
equation
0
= x.x +(y−L).(y−L) −a.a.y.y
= x.x +(1−a.a).y.y −2.L.y +L.L
When a = ±1 this is the simple quadratic y = (x.x+L.L)/2/L,
hence a parabola as above for Tan(c) = k. Otherwise, we
have
x.x
= (a.a−1).y.y +2.L.y −L.L
= (a.a−1).(y +L/(a.a−1))2
−L.L/(a.a −1) −L.L
= (a.a−1).(y +L/(a.a−1))2
−a.a.L.L/(a.a −1)
which we can re-write as
a.a.L.L +(a.a −1).x.x
= (L +(a.a−1).y)2, or as
1 +(1 −1/a/a).(x/L)2
= (1/a +(a−1/a).y/L)2
and I hope it is, in light of the above, clear that a.a > 1 yields
a hyperbola while a.a < 1 yields an ellipse. The case a = 0 is a circle,
albeit degenerately – it's the circle of radius zero centred on
the focus.
Comparing this with 1 +Q.(x/h)2 = (Q.Y/k/h)2, Q/k/h
corresponds to (a.a−1)/a/L and Q/h/h to (1−1/a/a)/L/L, hence a.L
corresponds to h/k and a.a−1 corresponds to Q/k/k. Looking at 1+Q/k/k =
1 +Sin(c).Sin(c)/k/k −Cos(c).Cos(c) = Sin(c).Sin(c).(1 +1/k/k) we can
identify a (give or take a sign) with Sin(c).√(1 +1/k/k) hence L with
h/Sin(c)/√(k.k+1). We can infer a and L from the intersection's
geometry in the plane, but not c (or Q), k and h.
Skew cones
Technically, the cone discussed above is a right circular cone
– that is, one which meets the planes at right-angles to its
centre-line in circles. More generally, we can define a cone by any curve in a
plane and any point not in the plane; the straight lines through the given
point that intersect the given curve then form a surface which we can think of
as a generalized cone – the classic pyramid is a square cone in
this sense, while the tetrahedron is a triangular cone. The given
point is called the vertex or apex of the cone; if the normal from it to the
plane of the original figure meets the plane at the figure's centroid, we can
describe it as a right cone on the given figure; otherwise, it is a
skew cone. Let us now consider a circular skew cone – that is, the
surface obtained by connecting the points of a circle to a single point
neither in the plane of the circle, nor in that plane's normal through the
circle's centre.
By applying a shear transformation relative to the figure's plane –
that is, for some co-vector q which annihilates all displacements in this
plane and some vector v parallel to the plane, the transformation is (: x
+v.q(x−p) ←x :) for an arbitrary point p in the plane (it doesn't
matter which; changing choice of p only adds a displacement in the plane to
x−p; and q annihilates this change) – we can transform any skew
cone to a right cone. The intersection of a plane and a skew circular cone
can thus be obtained by straightening the cone in this way, intersecting with
the sheared plane (which is itself a plane) and then reversing the shear to
recover the original plane. The action of the shear on the intersection is a
linear map (within the plane) which depends on the relationship between the
direction (v, in the characterization just given) of the shear and the normal
to the intersecting plane; however, at worst, it combines a shear within the
plane and a scaling of one direction without change to the perpendicular
direction. When the plane contains a direction perpendicular to the direction
of the shear, the above analysis will make this direction the x-axis and it
shall be preserved by the shear; the effect of reversing the shear will only
be to scale the Y co-ordinate.
Viewing a tilted disk
When an astronomer looks through a telescope at a distant spiral galaxy,
the galaxy is reasonably well characterized as a flat circular disk; the shape
it makes on our sky (and hence in the astronomer's photograph) is the result
of intersecting a skew cone – with the galaxy's perimeter as figure and
the astronomer as apex – with the telescope's focal plane, which is
(unlike the plane of the figure itself) perpendicular to the line from the
cone's apex to the figure's centroid; this ensures that it contains a
direction perpendicular to the direction of the shear needed to right the
cone. From the shape of this intersection, we can hope to infer the angle
between the galactic plane and our line of sight to the galaxy.
The intersection of the righted cone and sheared focal plane will be an
ellipse and unshearing shall merely stretch it in the Y direction; the result
is still an ellipse. We can measure its major and minor axes, which
correspond to k.h/Q in the Y direction and h/√Q in the X direction,
although the former shall have been stretched (or squashed) by
unshearing.
We get the same cone if we replace k with −k, so
we can presume that k is positive. For v.y +w.z = h to specify a plane, at
least one of v and w must be non-zero; and applying a common non-zero scaling
to all of v, w and h makes no difference to which plane is specified, so we
can re-scale to make v = Sin(c), w = Cos(c) for some angle c. We still have
the freedom to reverse the directions of our co-ordinates, which we can use to
ensure that c lies between zero and a quarter turn (inclusive), with h
positive. The co-ordinate y.Cos(c) −z.Sin(c) will then, give or take a
suitable constant offset, constitute a suitable co-ordinate in our plane to,
along with x, provide us with an orthonormal system. The final step in our
preliminaries is then to find a suitable constant offset, by choosing one that
makes the intersection symmetric about the origin in our plane. 
