Conics 1 Cut Cone

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:


3. Multiple Choice A cutting plane intersects a double-napped cone at an angle perpendicular to the axis.

Which primary Conic is formed:


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:


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):









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.

Conics 1 Cut Cone

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