Conic Sections

FOUR CONIC SECTIONS

1Sections of a Cone

All four curves are obtained by slicing a double cone at varying angles

Circle Ellipse Parabola Hyperbola

The circle, ellipse, parabola and hyperbola are known as conic sections

The ellipse and circle are closed curves

Circle Ellipse Parabola Hyperbola

When this plane is perpendicular to the axis, the section is a circle When a plane cuts all generators on one side of the apex, the section is an ellipse When a plane is parallel to one generator, the section is a parabola When a plane cuts both parts of the double cone, the section is a hyperbola CONICS AS SECTIONS OF A CONE

The parabola has only one branch and is unlimited The hyperbola has two branches extending indefinitely in opposite

directions

Generators are lines we put

on the surface of curved objects

The circle is the simplest of all curves

However, the ellipse is the curve most

often "seen" in everyday life

This is because every circle, viewed

obliquely, appears elliptical

The orbits of the planets and

satellites are ellipses

THE ELLIPSE AROUND US

The ellipse is without doubt one of the

most popular curves to be seen.

For example, every time you look at a

circular figure at an oblique angle (one

that is not a right angle), the curve you

see is an ellipse.

In the example shown, when you look in

perpendicular to the piece you can see

that it is circular but as it rotates you see

that it becomes elliptical, this is also

evident when looking in at the top of

your cup of tea or when looking at the

top of any cylindrical object

Orthographic Projection + Developments – Part 2: Handout 7

We have already drawn an

ellipse and we did not know it!!

There are a number of ways to

draw the ellipse and not the

rest of the object

This is not the True shape of the

Ellipse.

When we are drawing an ellipse

using different methods, we will

always be finding the True Shape of

the Ellipse

WHISPERING GALLERIES

In rooms where the ceilings are elliptical, a sound made at one focus

can be heard very clearly at the other focus.

Any sound made at the focus will rebound of the elliptical ceiling and pass

through the other focus.

ELLIPTICAL BILLIARDS TABLE

The elliptical billiards table uses the same

principle as the whispering gallery above.

The pocket is positioned at one of the focal

points. When a ball passes over the other

focal point, regardless of which point on the

ellipse it bounces off, it will always pass

over the other focal point. Shown below is a

sample of a path traced by a ball on the

table.

Ellipses can be seen extensively in

logos. Here is one example of a logo

used by ford motors which is in the

shape of an ellipse.

What make of car is this?

What sports game uses a 3-D Ellipse?

As can be seen, when a cylinder is

cut by a plane the true shape of the

section is an ellipse

To find the focal points of the curve

the focal spheres can be used as

were used with the cone.

THE ELLIPSE AND THE CYLINDER

METHODS OF CONSTRUCTING AN ELLIPSE

There are numerous methods of constructing ellipses, here we will

take a look at the more popular methods.

1. AUXILIARY CIRCLES METHOD

2. THE PIN AND STRING METHOD

3. THE TRAMMEL METHOD

4. AN ELLIPSE IN A RECTANGLE

1. AUXILIARY CIRCLES METHOD

PROCEDURE

1. Draw the major and minor axis.

2. Construct the major and minor auxiliary circles as shown.

3. Divide the circles into a number of different segments.(using 30 degree set-square as

shown)

4. Where the line intersects the minor auxiliary circle project a line across parallel to the

major axis and where the line intersects the major auxiliary circle project a line parallel to

the minor axis until where it meets the previous line drawn.

5. Complete the construction for the rest of the points and draw in the curve.

2. THE PIN AND STRING METHOD

This method of drawing an ellipse has been used for hundreds of years. You can try it by

sticking two thumbtacks in a sheet of paper and hooking a piece of string around them. Keep

the string stretched with the point of your pencil and move your pencil around to trace an

ellipse.

PRINCIPLE

The two thumbtacks represent the foci and since the length of the string does not change it

proves that the sum of the distances (PF) and (PF1) [P= point on curve, F= foci] is a constant

and this constant is the major axis.

3. TO CONSTRUCT AN ELLIPSE IN A RECTANGLE PROCEDURE

1. Divide the sections of the rectangle into a number of equal parts as shown.

2. Draw a line from the sub-sections on the sides of the rectangle to the top of the minor axis.

3. Project a line from the bottom of the minor axis to meet these lines to locate points on curve.

4. Axial symmetry can be used to locate the other half of the curve

ELLIPSE

5 equal divisions

5 e

qual div

isio

ns

min

or

axis

major axis

The same number of equal divisions must be used for each quadrant

4. THE TRAMMEL METHOD

This is also a very old method of constructing an ellipse, it was Archimedes who

first used this method. You can try it by cutting out a strip of paper/cardboard and

marking it as shown;

PROCEDURE

1. Position the trammel so that the point B lies on the major axis and the point C lies on

the minor axis, mark the location of point A - this is a point on the curve.

2. Rotate the trammel keeping B on the major axis and C on the minor axis, mark the

position of a all the time

3. Join the various points and the resulting curve is an ellipse.

THE TRAMMEL METHOD

When a sliothar or golf ball is hit into

the air, it follows a parabolic path

THE PARABOLA AROUND US

The centre of gravity of a leaping

salmon describes a parabola

The parabola is used by engineers in

designing some suspension bridges

If a light is placed at the focus of a parabolic mirror (a

curved surface formed by rotating a parabola about its

axis), the light will be reflected in rays parallel to the axis

This property is used in the design of flashlights and

headlights

The bulb is placed at the focus for the high beam and a

little above the focus for the low beam

The opposite principle is used in the giant mirrors

in reflecting telescopes used to collect light and

radio waves from outer space

The beam comes toward the parabolic surface and

is brought into focus at the focal point

THE PARABOLA AROUND US

axis

vertex

4 equal divisions

4 e

qua

l div

isio

ns

PARABOLA

The same number

of equal divisions

must be used

for each half

1

1. Divide the sides of the

rectangle into a number of

equal points as shown.

2. Join the points on the

side of the parabola to V

(vertex) as shown.

3. Project lines parallel

to the axis from the points on

the bottom of the curve.

4. Where the corresponding

lines intersect gives the

required curve.

PARABOLA

The electrons of an atom move in an approximately

elliptical orbit with the nucleus at one focus

In an ellipse, any signal (light or sound) that starts

at one focus is reflected to the other focus

This principle is used in lithotripsy, a medical

procedure for treating kidney stones

The patient is placed in a elliptical tank of water,

with the kidney stone at one focus

High-energy shock waves generated at the other

focus are concentrated on the stone, pulverizing it

THE ELLIPSE AROUND US