Conics around us
Some properties of conics
(curves of the second degree)
Johannes Kepler, working with data painstakingly
collected by Tycho Brahe without the aid of a telescope,
developed three laws which described the motion of the
planets across the sky.
1. The Law of Orbits: All planets move in elliptical
orbits, with the sun at one focus.
2. The Law of Areas: A line that connects a planet to the
sun sweeps out equal areas in equal times.
3. The Law of Periods: The square of the period of any
planet is proportional to the cube of the semimajor axis
of its orbit.
All planets move in elliptical orbits, with the sun at one
focus.
The eccentricity of an ellipse can be
defined as the ratio of the distance
between the foci to the major axis of
the ellipse. The eccentricity is zero
for a circle. Of the planetary orbits,
only Pluto has a large eccentricity.
Mercury
.206
Venus
.0068
Earth
.0167
Mars
.0934
Jupiter
.0485
Saturn
.0556
Uranus
.0472
Neptune
.0086
Pluto
.25
–
–
–
–
v1 = 7.9 km/s = first
cosmic velocity: circular
orbit around the Earth
v2 = 11.2 km/s = second
cosmic velocity: leaves
Earth (parabolic orbit)
v3 = 16.7 km/s = third
cosmic velocity: leaves
Sun if launched from
Earth (parabolic orbit)
v4 = 31.8 km/s = falls
on the Sun
The largest "dish" antenna in the world, the radio telescope at Arecibo
Observatory, Puerto Rico.
It is 1000 feet (305 meters) in diameter; has a gain of about 10 million.


Temperature
up to 3,000 C
(5,400 F)
1MegaWatt
power
delivered
onto 0.5 m
(1.5 feet)
diameter
spot


Temperature
up to 3,000 C
(5,400 F)
1MegaWatt
power
delivered
onto 0.5 m
(1.5 feet)
diameter
spot
Temperature up to 3,500 C (6,300 F)
Temperature up to 3,500 C (6,300 F)
Smaller is relative – I’ve seen it, it is not “small”!
Solar furnace – Google Search
1MW & 3,500 C (6,300 F) Four Solaire a Odeillo, France
The 2nd century AD
author Lucian wrote
that during the Siege of
Syracuse (c. 214–212
BC), Archimedes
destroyed enemy ships
with fire.
In 2005 a group of MIT students carried out an experiment with 127 one-foot (30 cm)
square mirror tiles, focused on a mock-up wooden ship at a range of around 100 feet (30
m). Flames broke out on a patch of the ship, but only after the sky had been cloudless
and the ship had remained stationary for around ten minutes.
Definition. Ellipse is the
locus of points P(x,y) such
that the sum of the distances
to two given points F1(f,0)
and F2(-f,0) is the same for
all points P(x,y).
What does the sum of these two
distances equal to?
What dimension of the ellipse
does it define?
Y
P (x,y)
X
F2(-f,0)
F1 (f,0)
The sum of the distances from
any point P on the ellipse to its
two foci is constant and equal
to the major axis,
|PF1 | + |PF2 | = 2a .
In the method of coordinates,
Upon squaring this equation
twice we obtain the equation of
an ellipse,
Y
P (x,y)
X
F2(-f,0)
F1 (f,0)
Excersize: The sum of the
distances from any point inside
the ellipse to the foci is less —
and from any point outside the
ellipse is greater — than the
length of the major axis.
Proof: Use the triangle
inequality.
B
|AB|+|AC| ≥ |BC|
|AB|-|AC| ≤ |BC|
A
C
Fermat principle states that the
light always travels along the
shortest path. It is clear from
the Figure that of all reflection
points P on the line l (mirror)
the shortest path between
points F1 and F2 on the same
side of it is such that points F1,
P, and the reflection of F2 in l,
F’2, lie on a straight line.
If a ray of light is reflected in a mirror, then the reflection
angle equals the incidence angle.
Suppose a line l is tangent
to an ellipse at a point P.
Then l is the bisector of
the exterior angle F1PF2
(and its perpendicular at
point P is the bisector of
F1PF2).
A light ray passing
through one focus of an
elliptical mirror will
pass through another
focus upon reflection.
Proof. Let X be an
arbitrary point of l
different from P. Since X
is outside the ellipse, we
have XF1 +XF2 > PF1 +PF2,
i.e., of all the points of l
the point P has the
smallest sum of the
distances to F1 and F2.
This means that the
angles formed by the
lines PF1 and PF2 with l
are equal.
Definition. Parabola is the
locus of points equidistant
from a given point F (a,0)
and a given line l, (x = -a)
which does not contain this
point.
In method of coordinates,
Horizontal parabola
Parabolas can open up,
down, left, right, or in
some other arbitrary
direction.
Any parabola can be
repositioned and rescaled
to fit exactly on any other
parabola — that is,
all parabolas are similar.
Not true for ellipses!
Vertical parabola Parabola at an angle
Y
Y
y = kx 2
( x F ,y F )
(0 ,a )
y=
ax
+b
(0,b)
X
O
(0 ,-a )
y = -a
O (-B /a,0)
X
Excersize: For the points
inside a parabola the
distance to the focus is less
than the distance to the
directrix, and for the points
outside the parabola
the opposite is true.
Suppose a line l is tangent
to a parabola at point P.
Let P’ be the projection of
P to the directrix. Then l is
the bisector of angle FPP’.
If a point light source,
such as a small light bulb,
is placed in the focus of a
parabolic mirror, the
reflected light forms
plane-parallel beam.
Light rays incident along the axis of a parabolic mirror upon
reflection pass through its focus (the beam is focused to F).
Proof. Let point P belong
to a parabola and l’ be a
bisector of the angle FPP’,
where |PP’| is the
distance to the directrix l.
Then, for any point Q on
l’, |FQ| = |QP’| ≥
|QQ’|. Hence, all points
Q on l, except for Q = P,
are outside the parabola,
so l’ is tangent to the
parabola at point P.
Definition. Hyperbola is
the locus of points where
the absolute value of the
difference of the distances to
the two foci is a constant
equal to 2a, the distance
between its two vertices.
In method of coordinates,
or,
where
This definition accounts for many of
the hyperbola's applications, such as
trilateration; this is the problem of
determining position from the
difference in arrival times of
synchronized signals, as in GPS.
Excersize: Let d be the
difference of the distances
from any point on the
hyperbola to the
foci F1 and F2 and let Γ be the
branch of the hyperbola
inside which F1
lies. Then for the points X
outside (inside) Γ the
quantity |XF2| − |XF1| is
less (greater) than d.
Suppose a line l is tangent to
a hyperbola at point P. Then
l is the bisector of the angle
F1PF2, where F1 and F2 are
the foci of the hyperbola.
If a point light source,
such as a small light bulb,
is placed in the focus F1 of
a hyperbolic mirror, the
reflected beam looks like
it emerges from its image
at the other focus F2.
The Hubble Telescope has used two
of these giant hyperbolic Mirrors
Ellipse is the locus of centers of all
circles tangent to two given nested
circles (F1,R) and (F2,r). Its foci are the
centers of these given circles, F1 and
F2, and the major axis equals the sum
of the radii of the two circles, 2a = R+r
(if circles are externally tangential to
both given circles, as shown in the
figure), or the difference of their radii
(if circles contain smaller circle (F2,r)).
Hyperbola is the locus of the centers
of circles tangent to two given nonnested circles. Its foci are the centers
of these given circles, and the vertex
distance 2a equals the difference in
radii of the two circles.
Indeed, consider circles (F1,R) and
(F2,r) that are not nested. Then the
loci of the centers O of circles
externally tangent to these two satisfy
|OF1| - |O F2| = R - r.
As a special case, one given circle may be a point located at
one focus; since a point may be considered as a circle of zero
radius, the other given circle—which is centered on the other
focus—must have radius 2a. This provides a simple technique
for constructing a hyperbola.
Parabola is the locus of the centers of circles passing through
a given point and tangent to a given line. The point is the
focus of the parabola, and the line is the directrix.
Indeed, If the radius of one of the given circles is zero, then it
shrinks to a point, and if the radius of the other given circle
becomes infinitely large, then the “circle” becomes just a
straight line.
Parabola is the locus of points such that the ratio of the
distance to a given point (focus) and a given line (directrix)
equals 1 (by its definition).
Ellipse can be defined as the locus
of points P for which the distance
to a given point (focus F2) is a
constant fraction of the
perpendicular distance to a given
line, called the directrix,
|PF2|/|PD| = e < 1.
Hyperbola can also be defined as the locus of points for
which the ratio of the distances to one focus and to a line
(called the directrix) is a constant e. However, for a hyperbola
it is larger than 1, |PF2|/|PD| = e > 1.
This constant is the eccentricity of the hyperbola. By
symmetry a hyperbola has two directrices, which are parallel
to the conjugate axis and are between it and the tangent to
the hyperbola at a vertex.
Using the method of coordinates, we can write the equation
which describes ellipse, hyperbola and parabola alike,
or,
This can be transformed to a canonical form
A curve of second degree is a set of points whose coordinates
in some (and therefore in any) Cartesian coordinate system
satisfy a second order equation:
In stereometry, an ellipse is
defined as a plane curve that
results from the intersection of
a cone by a plane in a way that
produces a closed curve.
Hyperbola is the curve of
intersection between a right
circular conical surface and a
plane that cuts through both
halves of the cone.
Parabola is a unique conic section, created from the
intersection of a right circular conical surface and a plane
parallel to a generating straight line of that surface.