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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