Post on 21-Apr-2015
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Mathematicians often study theoretical constructs that seem utterly useless - then centuries later their studies
turn out to have enormous scientific value.
An example of this than the work done by the ancient Greeks on the curves known as the Conics: the ellipse, the
parabola, and the hyperbola. No important scientific applications were found for them until the 17th century,
when Kepler discovered that planets move in ellipses and Galileo proved that projectiles travel in parabolas.
A conic section is formed when a plane intersects a double-napped cone.
fig 1: Ice cream cone.
(Chocolate)
To grasp the concept of the double-napped cone, let us first consider a
cone of much greater familiarity to us, the ice cream cone. (fig 1)
fig 2: Ice cream cone.
(Cone only)
Now since we will be dealing with the cone part
mathematically, we prefer a cone devoid of such
frivolous contents as the actual ice cream. That is,
we will focus on the cone part only. (fig 2)
Our ice-cream-less cone of math is no ordinary cone. It is infinitely tall. It
stretches in height forever with no "top" per say, so in fact the removal of
the ice cream was moot. (fig 3)
fig 3: Infinitely tall cone of math.
(Imagination required)
Finally, we consider our infinitely-tall math cone, with yet another
infinitely-tall cone, this time facing downward.
Yes, our cone image has taken a turn for the bizarre. We have two
cones, joined at a single point we will now call the vertex.
As such, we have created what mathematicians call a double-napped
cone. (fig 4)
fig 4: The double-napped cone. Each
cone is a "nappe", infinite in height. An
imaginary vertex goes right through the
middle of each cone at the vertex.
Now that we have a basic idea of a double napped cone, lets apply a
more rigorous analysis on how it's constructed in the highly abstract
3-dimensional math world.
Consider this line, which by definition is infinite in length. It has no
beginning, no end. It does, however, have a "middle", which we will
call the vertex. (fig 5)
We will now rotate this line about the vertex, at some constant
arbitrary angle so that a three-dimensional shape generated. In
fact, that would be a good name for this line we're rotating - the
generator.
Can you imagine the shape of this three-dimensional object we
generate?
It's the double-napped cone! Crazy!
One more thing.....
We're not done with our double-napped cone yet.
Remember how the Ancients liked to consider the shapes formed
when a plane intersected these double-napped cones.
A plane, recall, can be thought of as a sheet of paper, except that it
has no "width", and is infinite in dimension. (It goes off forever in
every direction)
Our first conic section is relatively simple. It's formed when a plane
intersects one of the nappes perpendicular to axis.
fig 5: A line, with a vertex in "the
middle". This line is being
rotated about a vertical axis.
fig 6: The shape formed upon a full
rotation in our double-napped cone.
fig 7: The upper-nappe of a double-
napped cone intersected by a plane.
(This three-dimensional image is
compressed into two-dimensions here, so
that the plane looks like a line.)
Along with the circle, there are three other primary conic sections. (4 total)
In Pure Math 30, we need to know not only these types of possible conic sections, but also how they would be
created relative to the generator and cutting plain angles.
Consider a double-napped cone, with a generator angle and a cutting plane angle .
Cutting
Plane
Cutting Plane
Cutting Plane
Cutting
Plane
Let's consider a special case of the hyperbola. For our study, we will only consider acute cutting plan angles.
The vertex angle of a double-napped cone is , as shown.
Cutting Plane
(a) State the measure of the generator angle, .
(b) Determine the value or range of values for the cutting plane angle, , so
that each conic section is formed:
Hyperbola Parabola Ellipse Circle
A double-napped cone is intersected by a plane at an angle of to the axis. If the
vertex angle is , describe the conic section formed.
The shape of particular conic such as a circle is dependent upon factors such as the width of the cone (given by
the vertex angle and the distance between the plane in and the vertex:
Let's focus on the distance of the cutting plane from the vertex. (Recall - the point where the two nappes, or cones, "meet")
As this distance decreases, the size of the circle gets smaller and smaller.
Until finally, when the cutting plane is intersecting the double-napped cone right at the vertex, the intersecting
shape (circle) has degenerated to a: ___________________.
We can see a similar progression with an ellipse:
Degenerate Case of the CIRCLE / ELIPSE:______________________
Whereas the parabola is different:
Degenerate Case of the PARABOLA:______________________
Finally we consider the hyperbola:
Degenerate Case of the HYPERBOLA:______________________
The Degenerate Cases Summarized:
Conic sections are not limited to the intersection of a double-napped cone and plane.
We also consider the intersection of a cylinder (infinite in height, of course) and a plane.
What possible conic sections are formed
when a plane intersects with a cylinder?
In fact, there are five possible options! Two primary conics, plus three degenerates.....
The two primary conics possible are the circle and ellipse. (Shown below) Note that there is no parabola or
hyperbola possible with the intersection of a plane and cylinder.
List each of the four primary double-napped cone / plane conic sections and the
corresponding degenerate case.
List possible case of a conic section involving a cylinder and plane.
Thousands of years after the ancient Greeks pondered the conic shapes, physicists, astronomers, engineers, and
others found many applications.
Circles, of course, are abundant everywhere in nature!
But what about the other Conics - Ellipses, Parabolas, and Hyperbolas?
In the 17th century Johannes Kepler first discovered that the
motion of planets around the sun was not a perfect circle, as
suggested by early astronomers and supported by the
church, but was in fact elliptical.
In the image on the left, the Earth's orbit is elliptical, but
with a low eccentricity so that it is nearly circular.
The Pluto's and the comet's orbit has a much higher
eccentricity.
On a far smaller scale, the electrons of an atom move in an approximately elliptical
orbit.
In fact, elliptical shapes are present in many everyday
scenarios!
The easiest way to visualize the path of a projectile is to observe a waterspout. Each molecule of water follows the
same path and, therefore, reveals a picture of the curve
Parabolic mirrors are used to collect light and radio waves.
If a right circular cone is intersected by a plane parallel to its axis, part of a
hyperbola is formed. Such an intersection can occur in the patterns formed
on a wall by a lamp shade.
One of nature's best known approximations to parabolas is the path taken by
a body projected upward and obliquely to the pull of gravity, as in the
parabolic trajectory of a golf ball. Note that the effect of air resistance would
very slightly alter the path from a true parabola.
A sonic boom shock wave has the shape of a cone, and it intersects the
ground in part of a hyperbola. It hits every point on this curve at the
same time, so that people in different places along the curve on the
ground hear it at the same time.
1. Multiple Choice A cutting plane intersects a double-napped cone parallel to the axis. Which primary Conic is
formed:
(a) Circle (b) Ellipse (c) Parabola (d) Hyperbola
2. Multiple Choice A cutting plane intersects a double-napped cone at an angle parallel to the generator. Which
primary Conic is formed:
(a) Circle (b) Ellipse (c) Parabola (d) Hyperbola
3. Multiple Choice A cutting plane intersects a double-napped cone at an angle perpendicular to the axis.
Which primary Conic is formed:
(a) Circle (b) Ellipse (c) Parabola (d) Hyperbola
4. Multiple Choice A cutting plane intersects a double-napped cone at an angle slightly less than to the axis,
but greater than the generator angle. Which primary Conic is formed:
(a) Circle (b) Ellipse (c) Parabola (d) Hyperbola
5. The vertex angle of a double-napped cone is , as shown.
6. Consider a double-napped cone shown below. State the conic section formed when....
(a) State the measure of the generator angle, .
(b) Determine the value or range of values for the cutting plane angle, , so
that each conic section is formed:
(i) Hyperbola (ii) Parabola (iii) Ellipse (iv) Circle
(a) A plane intersects the cone parallel to axis
(b) A plane intersects a cone parallel to the axis, and traverses through the vertex
(c) A plane intersects a cone parallel to the generator
(d) A plane intersects a cone parallel to the generator, and traverses through the vertex.
(e) A plane intersects a cone perpendicular to the axis
(f) A plane intersects a cone perpendicular to the axis, and traverses through the vertex.
7. The vertex angle of double-napped cone is . If a cutting plane intersects a cone at an angle of , the
conic section formed would be a(n):
(a) Circle (b) Ellipse (c) Parabola (d) Hyperbola
8. The vertex angle of double-napped cone is . If a cutting plane intersects a cone at an angle of , the
conic section formed would be a(n):
(a) Circle (b) Ellipse (c) Parabola (d) Hyperbola
9. The vertex angle of double-napped cone is . If a cutting plane intersects a cone at an angle of , the
conic section formed would be a(n):
(a) Circle (b) Ellipse (c) Parabola (d) Hyperbola
10. The vertex angle of double-napped cone is . If a cutting plane intersects a cone at an angle of , the
degenerate case would be:
(a) A single point (b) A single line (c) Intersecting lines (d) Parallel Lines
11. The vertex angle of double-napped cone is 6 . If a cutting plane intersects a cone at an angle of , the
degenerate case would be:
(a) A single point (b) A single line (c) Intersecting lines (d) Parallel Lines
12. A plane intersects a cylinder at a perpendicular angle. The conic section formed is a:
(a) Circle (b) Ellipse (c) Single Line (d) Parallel Lines
13. A plane, parallel to a cylinder intersects it. The conic section formed is a:
(a) Circle (b) Single Line (c) Parallel Lines (d) Either b or c
14. Explain how "no locus" can be obtained in a conic section.
1. d 2. c 3. a 4. b 5. (i) (ii) (iv) (v) 6. (a) Hyperbola (b) Intersecting Lines (c) Parabola (d) Single Line (e) Circle (f) Single Point 7. b 8. c 9. d 10. b 11. a 12. a 13. d 14. A plane, parallel to a cylinder, outside of a
cylinder. "No locus" is not possible with a double-napped cone.