An Approach to Conic Sections
Jia. F. Weng
1998
1
Introduction.
The study of conic sections can be traced back to ancient Greek mathematicians, usually to Applonious (ca. 220-190 bc) [2]. The name ‘conic section’ comes from the fact that the principle types
of conic sections, known as ellipses, hyperbolas and parabolas, are generated by cutting a cone
with a plane. However, most modern textbooks on calculus depart from this geometric approach.
Instead, conic sections are defined as some types of loci and studied through analytic geometry. In
this paper we show a new approach to conic sections which are defined as the intersections of two
cones. Then the vertices of two cones become the inherent foci of the conic section and a directrix
exists associated with each of the inherent foci. All known properties of conic sections still hold
for the inherent foci and their associated directrixes in this new approach. Moreover, when a conic
section and its foci and directrixes in space are projected to a horizontal plane, they become the
ones discussed in planar analytic geometry. This new approach seems simpler and more natural
than the classical geometric and analytic approaches in defining conic sections and proving their
properties. In the last section we show an application of the new approach in the network design
in mining industry. In Appendix we derive the standard equation of a conic section with respect
to the foci, lying on the cutting plane and referred to as the coplanar foci of the conic section. The
author does not know any textbook that gives such a derivation.
2
Intersections of Two Right Circular Cones.
Let xP , yP , zP denote the Cartesian coordinates of a point P . Suppose P and Q are two distinct
points. The length of line segment P Q is denoted by l(P Q), and the gradient of P Q is denoted by
g(P Q) which is defined as
|zQ − zP |
g(P Q) = q
.
(xQ − xP )2 + (yQ − yP )2
(1)
Let C(P ; m) denote a (double-napped) right circular cone whose vertex is P and whose generating
lines have gradient m (m > 0). The angle β formed by the axis of C(P ; m) and the generating lines
is referred to as the generating angle of the cone. Clearly, tan β = 1/m.
Now suppose A and B are two distinct points in space. The intersection of cones C(A; m)
and C(B; m) is denoted by C(A, B; m). Assume the horizontal distance between A and B is 2u
while the vertical distance between A and B is 2h. Then, after a transformation we may assume
A = (u, 0, h), B = (−u, 0, −h), u ≥ 0, h ≥ 0 (Fig. 1). Hence the equations describing C(A; m) and
C(B; m) are
(z − h)2
(z + h)2
2
2
(x − u)2 + y 2 =
,
(x
+
u)
+
y
=
.
(2)
m2
m2
If h = 0, then the intersection C(A, B; m) lies trivially in the Y Z-plane. Assume h 6= 0. Subtracting
the first equation from the second, we have
m2
u
x,
z = m2 x =
h
k
(3)
Z
Y
~
P
A
β
Y
VA
R1
VA A
α
O
R2
Z
X
S
X
O
B
VB
VB
B
R1
VB
B
A
O
VA
B VB VA A
R2 S
(1)
(2)
Figure 1: Intersections of two cones.
where k = h/u = g(AB). It follows that C(A, B; m) lies on a plane P̃ which contains the Y -axis
and meets the XY -plane at an angle α, referred to as the intersecting angle of two cones. We
call this planar curve C(A, B; m) a conic section, or simply a conic. In particular, if g(AB) ≥ m,
C(A, B; m) is a closed curve and called an ellipse (Fig. 1(1)); if g(AB) ≤ m, C(A, B; m) has two
separate branches, called a hyperbola (Fig. 1(2)). Trivially, in their degenerate cases in which
g(AB) = m, an ellipse becomes a line segment AB, and a hyperbola becomes two half-lines that
are the extensions of AB. Moreover, there are two special cases:
1. If A, B lie in a vertical line, then g(AB) = ∞ and the ellipse is a circle lying in a horizontal
plane.
2. If A, B lie in a horizontal plane, then g(AB) = 0 and the hyperbola lies in a vertical plane.
Substituting z with the right side of (3), from (2) we obtain
p
y=±
(m2 u2 − h2 )(m2 x2 − h2 )
.
mh
(4)
Hence the parametric expression of C(A, B; m) is
Ã
!
p
x, ±
(m2 u2 − h2 )(m2 x2 − h2 ) m2 u
,
x .
mh
h
(5)
Let VA and VB be the intersections of C(A, B; m) with the XZ-plane that are close to A and B
respectively (Fig. 1). Then by Equations (3) and (4)
VA = (
h
h
, 0, mu), VB = (− , 0, −mu).
m
m
2
(6)
Clearly, C(A, B; m) is symmetric with respect to two orthogonal lines: VA VB and the Y -axis.
Therefore, O is the center of the conic section C(A, B; m). When C(A, B; m) is an ellipse, let R1
and R2 be the intersections of C(A, B; m) with the positive and negative Y -axis respectively. Then
again from Equation (4) we find
µ
R1 = 0,
¶
µ
¶
uq 2
−u q 2
|k − m2 |, 0 , R2 = 0,
|k − m2 |, 0 .
m
m
(7)
When C(A, B; m) is a hyperbola, (7) also defines two points on the Y -axis. In the case of an
ellipse, VA VB is called the major axis while R1 R2 is called the minor axis of the ellipse. In the case
of a hyperbola, VA VB is called the transverse axis while R1 R2 is called the conjugate axis of the
hyperbola.
From Equations (4) and (3), it is easy to derive that
yx0 = ±(m2 /k 2 − 1)x/y, zx0 = m2 /k.
Therefore, the tangent vector t of the conic section C(A, B; m) is
Ã
t=
m2
x m2
1, ±( 2 − 1) ,
k
y k
!
.
(8)
Remark 1 Note t is completely determined by m and k, independent from the coordinates of A
and B.
Now suppose C(A, B; m) is a hyperbola. When x goes to infinity, by (4) y/x goes to
and the tangent vector becomes

s
t∞ = 0, ±
p
m2 /k 2 − 1

m2
m2 
.
−
1,
k2
k
The two lines through O in the directions of t∞ are the asymptotes of the hyperbola, which lie in
the plane P̃ where the hyperbola lies.
Conic sections have two important properties: constant sum/difference property and reflective
property. First we prove a lemma [5].
Lemma 2 Suppose the endpoint S of a line SA is perturbed in direction v. Let the angle between
−→
SA and v be θ. Then the directional derivative of l(SA) with respect to v is (− cos θ).
Proof: Suppose S moves to S ∗ in direction v. Let l0 = l(SA), l = l(S ∗ A), ε = l(SS ∗ ). Then
l2 = l02 + ε2 − 2εl0 cos θ, and
2l · dl = 2(ε − l0 cos θ) · dε.
Note l → l0 when ² → 0. Therefore
dl
= − cos θ.
ε→0 dε
lv0 = lim
The lemma is proved.
Theorem 3 (constant sum/difference) For any point S on an ellipse C(A, B; m), the sum of
the distances from S to the vertices A and B is constant. For any point S on a hyperbola C(A, B; m),
the difference of the distances from S to the vertices A and B is constant.
3
Proof: There is no loss of generality that we assume A = (u, 0, h), B = (−u, 0, −h) as before.
Because g(AS) = g(BS) = m, when k ≥ m and C(A, B; m) is an ellipse,
l(AS) + l(SB) =
p
p
1 + m−2 (|ZA − ZS | + |ZS − ZB |) = 2h 1 + m−2 .
The argument is similar for the case of C(A, B; m) being a hyperbola.
This property completely characterizes ellipses and hyperbolas, therefore, we can redefine ellipses/hyperbolas to be planar curves that satisfy the constant sum/difference property. That is,
an ellipse (or a hyperbola) is a planar curve such that the sum (or difference respectively) of the
distances between any point on the curve and two fixed distinct points is constant. This property
implies another property which is important in applications of conic sections.
Corollary 4 (reflection) For any point S on an ellipse or a hyperbola, the tangent line at S
meets SA and SB at the same acute angle.
Proof: Let the acute angle between t and SA, SB be θ, φ respectively. By Lemma 2, lt0 (SA) =
− cos θ, lt0 (SB) = − cos(π − φ) = cos φ. For an ellipse, since l(SA) + l(SB) is constant, lt0 (SA) +
lt0 (SB) = − cos θ + cos φ = 0. Hence θ = φ. The argument is similar for hyperbolas.
Remark 5 This corollary can also be proved by Equation (8).
Because of the reflective property A and B are called the inherent foci of C(A, B; m). Let Ā, B̄
and C̄(Ā, B̄) be the projections of A, B and C(A, B; m) on a horizontal plane respectively. Then
any point S̄ on C̄(Ā, B̄) is the projection of certain point S on C(A, B; m). Because
l(ĀS̄) ± l(S̄ B̄) = (l(AS) ± l(SB)) sin β = const,
C̄(Ā, B̄) is also an ellipse or a hyperbola with foci Ā and B̄. Since Ā and B̄ lie on the same plane
where the curve C̄(Ā, B̄) lies, we call them coplanar foci of the curve. Moreover, let V̄A , V̄B , R̄1
and R̄2 be the projections of VA , VB , R1 and R2 on the same plane respectively. Then V̄A V̄B and
R̄1 R̄2 are the major and minor axes (or the transverse and conjugate axes) of the ellipse (or the
hyperbola respectively) C̄(Ā, B̄). The equation of C̄(Ā, B̄) can be easily derived from Equation (4)
as follows:
x2 y 2
+ 2 = 1,
ā2
b̄
where
ā2 =
h2
h2 − m2 u2
(k 2 − m2 )u2
2
,
b̄
=
=
.
m2
m2
m2
(9)
(10)
Equation (9) is called the standard form of an ellipse or a hyperbola, depending on b̄2 ≥ 0 or b̄2 ≤ 0,
i.e. on k ≥ m or k ≤ m. Note that 2ā = l(V̄A V̄B ), 2b̄ = l(R̄1 R̄2 ). Let 2c̄ = l(ĀB̄) = 2u. Then it is
easy to see that
c̄2 = ā2 − b̄2 , if C̄(Ā, B̄) is an ellipse,
c̄2 = b̄2 − ā2 , if C̄(Ā, B̄) is a hyperbola.
4
_
B
_
A
_
B
_
A
Figure 2: Confocal ellipses and hyperbolas.
3
Confocal and Similar Conics, Parabolas.
The shape of a conic section is completely determined by two parameters m and k. Now we
investigate how the curve varies as m or k changes.
(1) If A and B are fixed, i.e. k is fixed, but m changes, we obtain a family of conics that lie on
different planes but share A, B as their common foci. These conics are called confocal. As argued
above, when these curves are projected to the same horizontal plane, their projections are also
confocal with Ā, B̄ as their common foci (Fig. 2, based on a figure in [3]).
(2) Now we study how C(A, B; m) varies if m is fixed but k changes. Without loss of generality
we assume B is fixed and A moves to A∗ that lies in the same vertical plane. Let A∗ = (u + ε, 0, h +
δ), ε ≥ 0, δ ≥ 0. Then the equations of cones C(A∗ ; m) and C(B; m) become
(x − u − ε)2 + y 2 =
(z − h − δ)2
(z + h)2
2
2
,
(x
+
u)
+
y
=
.
m2
m2
(11)
Of all possible moves of A we discuss the moves in three special directions: along BA, horizontal
and along VB A.
(2.1) A move of A either preserves the gradient between two cone vertices or does not. If it
does, then δ/ε = h/u = k. Solving the system (11) with δ = kε we have
z=
(h2 − m2 u2 )ε
m2
x+
.
k
2hu
Therefore, the new conic section C(A∗ , B; m) lies in a plane P∗ that is parallel to the plane P̃ where
C(A, B; m) lies. Besides, since both curves lie on the same cone C(B; m), it follows that the two
conics C(A∗ , B; m) and C(A, B; m) are similar relative to their common focus B. When projected
on the same horizontal plane, their projections are also similar relative to B̄ (Fig. ??). On the
other hand, if the move of A does not preserve the gradient between the vertices, then P∗ is not
parallel to P̃, and C(A∗ , B; m) is not similar to C(A, B; m) by the definition of similarity.
5
~
P
Z
Y
A*
A
X
L
O
S
VB
B
D
Y
L
VB
D
X
B
S
Figure 3: A parabola as an extreme ellipse.
Remark 6 A curve C1 (x1 (t), y1 (t), z1 (t)) is said to be similar to another curve C2 (x2 (t), y2 (t), z2 (t))
relative to point (x0 , y0 , z0 ) if
x2 (t) − x0
y2 (t) − y0
z2 (t) − z0
=
=
.
x1 (t) − x0
y1 (t) − y0
z1 (t) − z0
(2.2) The vertex A moves in the direction of the positive X-axis, that is, δ = 0 and ε monotone
increases. Assume we start with g(AB) = k > m and the conic is an ellipse. In this move, the
ellipse C(A, B; m) becomes narrower and narrower, and finally becomes a line segment, a degenerate
ellipse with k ∗ = (h + δ)/(u + ε) = m. If the increase of ε continues, then as stated in Section 2, the
line segment first suddenly jumps into two half-lines as a degenerate hyperbola, and then becomes
a normal hyperbola with k ∗ < m.
(2.3) As we have discussed in (2.2), in the degenerate case, an ellipse or a hyperbola overlaps
the line through A and B. One may ask if there is a non-degenerate common extreme of ellipses
and hyperbolas. Look at the move of A along VB A. In such a move δ = mε. Solving the system
(11) with δ = mε and then let ε go to infinity, we have
p
2 (h − mu)(mx + h)
y=±
, z = m(x − u) + h.
m
(12)
Therefore the parametric expression of the resulting curve is
Ã
!
p
2 (h − mu)(mx + h)
, m(x − u) + h .
x, ±
m
6
(13)
If starting with a hyperbola (h/k < m), we obtain the same result. Therefore, the resulting planar
curve, called a parabola, is the non-degenerate common extreme of ellipses and hyperbolas. The
parabola has a vertex VB and is symmetric about the line VB A∗ , the intersection of P̃ and the
XZ-plane. Let
2h
+ u, y, −h)}
L = {(−
m
be the point set that is the intersection of the plane P̃ and the horizontal plane through B (Fig.
3). For any point S on the parabola, let D be the foot of the perpendicular from S to L. Then the
gradient of SD is the slope of the plane P̃, which is equal to the gradient of any generating line of
C(B; m). Therefore, similarly to the argument for ellipses and hyperbolas, we have l(SB) = l(SD).
Theorem 7 (equidistance) For any point S on the parabola, the distance from S to B equals the
distance from S to the line L.
For this reason, B is called the inherent focus of the hyperbola, and L is called the directrix
associated with the inherent focus B. As ellipses and hyperbolas, this equidistance property completely characterizes parabolas and we can redefine a parabola to be a planar curve that satisfies
the equidistance property. The ratio of the distance from S to the focus to the distance from S
to the directrix is called the eccentricity of the parabola. Thus, a parabola can also be defined as
a planar curve whose eccentricity is one. Again, the equidistance property implies the reflective
property:
Corollary 8 (reflection of parabolas) For any point S on a parabola, the tangent line t at S
meets two lines at the same acute angle: the line joining S and the focus and the perpendicular
from S to the directrix.
Now denote the parabola by C(B, L; m). Let C̄(B̄, L̄), V̄B , B̄, S̄, L̄ and D̄ be the projections of
C(B, L; m), VB , B, S, L and D on a horizontal plane respectively. Because SB and SD have the
same gradient, l(SB) = l(SD) implies l(S̄ B̄) = l(S̄ D̄). Therefore, the projection C̄(B̄, L̄) is also
a parabola, whose coplanar focus is B̄ and the directrix associated with B̄ is L̄. From (12) the
equation of C̄(B̄, L̄) is
y 2 = 4p̄(x + h/m),
where p̄ = h/m − u is the distance from the vertex V̄B to the coplanar focus B̄ or to the directrix
L̄. This equation is the standard form of a parabola with vertex at (−h/m, 0). If the origin moves
to the vertex V̄B , then the equation is further simplified to y 2 = 4px.
4
Discussions
(1) It is easy to see that any ellipse (or hyperbola) C obtained by cutting a cone with a plane can
be generated by two cones. Factually, we may assume that the cone C is C(A; m). Then make a
copy of C(A; m) and turn the copy 180◦ around the Y -axis, the copy becomes the required C(B; m).
(2) As parabolas, an ellipse or a hyperbolas has two directrixes associated with its inherent foci
A and B: They are the intersections of P̃ and the horizontal planes through A and B. Figure
4(1) shows the directrix LB of an ellipse C(A, B; m) associated with B. For any point S on the
ellipse, let D be the foot of the perpendicular from S to LB . The eccentricity of the conic section
C(A, B; m) is still defined to be the ratio of l(SB) to l(SD). Let H be the foot of the perpendicular
7
Z
Y
O
FB
LB
D
~
P
~
P
VA
Z
Y
A*
FA
X
A
S
X
F
~ VB
LB
α
α
A
β
β
CB
O
~
LL
B
C
H
B
(1)
(2)
Figure 4: Coplanar foci and their directrixes.
from S to the horizontal plane through B. Since 6 HSB equals the generating angle β, and 6 HDS
equals the intersecting angle α (Fig. 4(1)),
l(SH)/ cos β
sin α
l(SB)
=
=
= const < 1.
l(SD)
l(SH)/ sin α
cos β
When projected to a horizontal plane,
l(S̄ B̄)
l(BH)
l(SH)/ cot β
tan α
=
=
=
= const < 1.
l(DH)
l(SH)/ tan α
cot β
l(S̄ D̄)
Thus, the third definition of ellipses is that an ellipse is a planar curve whose eccentricity is a
constant less than one. These argument can be similarly apply to hyperbolas. Hence, the third
definition of hyperbolas is that a hyperbola is a planar curve whose eccentricity is a constant greater
than one.
(3) Finally we should point out that an ellipse or a hyperbola C(A, B; m) also has its own
coplanar foci lying on the plane P̃. Take an ellipse as an example. According to Quetelet-Dandelin’s
construction [2], suppose the sphere inscribing C(B; m) and P̃ touches P̃ at a point FB . Then FB
is a coplanar focus of C(A, B; m). Another focus FA can be found in a similar way. Associated
with these coplanar foci, there are two directrixes. For instance, suppose the sphere defined above
touches the cone C(B; m) at a horizontal circle CB (Fig. 4(1)). Then the intersection of P̃ and
the plane through CB is the directrix L̃B associated with the focus FB . In a similar way we can
find the coplanar focus F and its associated directrix L̃ for a parabola C(B, L; m) (Fig. 4(2)). The
coordinates of the coplanar foci and the standard equations of conic sections with respect to its
coplanar foci and directrixes are derived in Appendix.
5
An application.
Given a point set, the minimum network problem asks for a network of shortest total length
interconnecting all given points. To shorten the network, probably some points not in the given
8
ground
A
deposit
S
deposit
C
B
Figure 5: A mining network.
point set are added. In the literature this minimum network is called a Steiner minimal tree and
the additional points are called Steiner points [4]. The Steiner tree problem was first posed by
Fermat who asked how to find the point S in a triangle ABC so that the sum of the distances
from S to the vertices A, B, and C is minimal [4]. There are many generalizations of the Steiner
tree problem. A recently studied one is the gradient-constrained Steiner tree problem [1]. Figure
5 depicts a simple example of an underground mining network in which the ore in two deposits is
extracted through tunnels to a vertical shaft and then hauled up to the ground. In the figure A and
B are two prescribed access points in deposits. In practice, the gradient of any tunnel cannot be
very steep. The typical maximum gradient m is about 1:7. In this example we need to find a point
S so that with this gradient constraint the total length of the network f = l(SA) + l(SB) + l(SC) is
minimized, where C is the access point to be determined on the vertical shaft. Clearly, to minimize
f , SC should be perpendicular to the vertical shaft. Hence, SC is horizontal and automatically
satisfies the gradient constraint. It can be shown that if g(AB) > m, then SA and SB must have
the maximal gradient m in order to minimize f [6]. It follows that S lies on the ellipse C(A, B; m),
and that 6 ASC = 6 BSC because of the reflective property. These conditions completely determine
S and C.
References
[1] M.Brazil, D.A.Thomas and J.F.Weng, Gradient-constrained minimal Steiner trees, DIMACS
Series in Discrete Mathematics and Theoretical Computer Science, Vol. 40 (1998), pp. 23-38.
[2] J.L.Coolidge, A history of the conic sections and quadric surfaces, Oxford University Press,
London, 1945.
[3] R.Courant and H.Robbis, What is mathematics? Oxford University Press, London, 1941.
[4] F.K.Hwang, D.S.Richard and P.Winter, The Steiner tree problem, North-Holland, 1992.
[5] J.H.Rubinstein and D.A.Thomas, A variational approach to the Steiner network problem, Ann.
Oper. Res., Vol. 33 (1991), pp. 481-499.
[6] J.F.Weng, A note on constrained shortest connections of two points to a vertical line, preprint.
9
Appendix. If C(A, B; m) is an ellipse, then let FA and FB be a pair of points lying on VA VB and
symmetric to O such that 0 ≤ ρ = l(OFA )/l(OVA ) ≤ 1. Similarly, if C(A, B; m) is a hyperbola,
then define FA and FB be the points lying on the extension of VA VB and symmetric to O such that
ρ = l(OFA )/l(OVA ) > 1. By the definition
µ
FA =
¶
µ
¶
ρh
ρh
, 0, ρmu , FB = − , 0, −ρmu .
m
m
(14)
Let S = (x, y, z) be a point on C(A, B; m), then
s
µ
l1 = l(FA S) =
ρh
−x
m
s
µ
l2 = l(FB S) =
¶2
¶2
ρh
+x
m
+ y 2 + (ρmu − z)2 ,
+ y 2 + (ρmu + z)2 .
Let l1 ± l2 = 2ã, ã ≥ 0. Then l12 = (2ã ∓ l2 )2 , l12 − l22 = 4ã2 ∓ 4ãl2 ,
±ãl2 = ã2 −
l12 − l22
ρhx
= ã2 +
+ ρmuz.
4
m
Squaring both sides, after a simplification we obtain
(h2 ρ2 − m2 ã2 )x2 + 2m2 huρ2 xz + (m2 u2 ρ2 − ã2 )m2 z 2
− m2 ã2 y 2 − (m4 u2 + h2 )ã2 ρ2 + ã4 m2 = 0.
Substituting y and z with (4) and (3) respectively, finally we have an equation containing only one
variable x:
f1 x2 + f2 = 0,
(15)
where
f1 = m4 u2 (1 + m2 )ã2 − (h2 + m4 u2 )2 ρ2 ,
f2 = h2 (h2 − m2 u2 )ã2 − m2 h2 ã4 + h2 (h2 + m4 u2 )ã2 ρ2 .
Equation (15) will hold for any x if both f1 = 0 and f2 = 0. Solving this system we obtain
s
ã =
√
h2
u 1 + m2
2
2
.
+m u , ρ=
m2
ã
Thus, the constant sum/difference of distances and the coordinates of the coplanar foci FA , FB are
both determined.
By the definition of the intersecting angle α, we have cos α = h/(mã), sin α = mu/ã. After
an anticlockwise rotation of the coordinate system around the Y -axis by α, let the new axes be
OX̃, OỸ and OZ̃. Then the X̃ Ỹ -plane is the plane P̃ where the conic section lies. Since the curve
lies on the X̃ Ỹ -plane, we have z̃ = 0, and the the transformation is x = x̃ cos α, y = ỹ, z = x̃ sin α.
From (4) we can derive the standard form of the conic section on the plane P̃:
(x̃)2 (ỹ)2
+
= 1,
(ã)2
(b̃)2
10
(16)
where b̃2 = (h2 /m2 ) − u2 . Note b̃2 ≥ 0 when g(AB) ≥ m and the conic section is an ellipse, and
b̃2 ≤ 0 when g(AB) ≤ m and the conic section is a hyperbola. It is easy to check ã = l(OVA ), b̃ =
l(OR1 ). Let c̃ = l(OFA ). Then as in the projection of C(A, B; m), the following equations hold:
c̃2 = ã2 − b̃2 , if C(A, B; m) is an ellipse,
c̃2 = b̃2 − ã2 , if C(A, B; m) is a hyperbola.
In the same way we can determine the coplanar focus F and its directrix L̃ for a parabola. As
shown in Figure 4(2), let ρ be the horizontal distance between VB and F . Since the gradient of
VB F is m and since VB has the same distance to F and to L̃, we have
µ
¶
F = ρ−
µ
¶
h
h
, 0, m(ρ − u) , L̃ = −ρ − , y, −m(ρ + u) .
m
m
Suppose S = (x, y, z) is a point on the parabola. Then the square of l(SF ) equals the square of
the distance from S to L̃,
µ
h
x−ρ+
m
¶2
Ã
!2
ρm2
+y + z−
+ mu
h
2
µ
h
= x+ρ+
m
¶2
Ã
!2
ρm2
+ z+
+ mu
h
.
Because y and z satisfy (12), this equation always holds if
ρ=
h − mu
.
m(1 + m2 )
√
Note now cos α = 1/ 1 + m2 . As argued in the case of ellipses and hyperbolas, after an anticlockwise rotation of the coordinate system by α, from (12) we obtain the standard form of the parabola
on the plane P̃:
√
h 1 + m2
2
),
(17)
ỹ = 4p̃(x̃ +
m
√
√
where p̃ = 4(h − mu)/(m 1 + m2 ) and h 1 + m2 /m = l(VB F ). If the origin moves to VB , then it
can be further simplified to ỹ = 4p̃x̃.
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