Ellipse & Parabola

•No talking when the teacher is talking

•Arrive on time for class

•If you have something to say, raise your hand

•Take out and put away equipment in an orderly fashion

•Finally...

Show respect

Try your best

Mr Phelan

Ellipse & Parabola

http://www.google.ie/url?sa=i&rct=j&q=ellipse+sign&source=images&cd=&cad=rja&docid=prbBRbnue6-ksM&tbnid=BVMs6H3rE5ZGxM:&ved=0CAUQjRw&url=http://commons.wikimedia.org/wiki/File:Ellipse_sign_165.svg&ei=11GSUf_8NMzZsgasuICYDQ&bvm=bv.46471029,d.ZG4&psig=AFQjCNGyy6-t7ap1VenRLKcvrfuKc17M7w&ust=1368630099639145

Ellipse & Parabola

http://www.google.ie/url?sa=i&rct=j&q=mcdonalds+logo&source=images&cd=&cad=rja&docid=_xntqB7ROjA3UM&tbnid=DfzZJfg2CTMk3M:&ved=0CAUQjRw&url=http://www.lucidity.ie/blog/164-logo-design-and-how-important-it-is-for-your-business&ei=LfqZUYrZJpKwhAey74DgBg&bvm=bv.46751780,d.ZGU&psig=AFQjCNHXhWROSwSGeWyOdaPFDVS9PzsbHg&ust=1369131947059402

http://www.google.ie/url?sa=i&rct=j&q=real+life+examples+of+parabola&source=images&cd=&cad=rja&docid=nWpJPBp6sih9BM&tbnid=bTHqn_Syh6f1uM:&ved=0CAUQjRw&url=http://keenermath8.wikispaces.com/Chapter+5&ei=kvqZUbKwGcXChAeUuYGQDQ&bvm=bv.46751780,d.ZGU&psig=AFQjCNEI5S2ruThEHAeSdXiqDPSI75ajmw&ust=1369131982340585

The Ellipse

The circle is a very common curve in the world around us but it is not the curve we most often see. When you look at a circle at an angle you see a squashed version of it. This is called an Ellipse. Can you think of any other words we might use to call a squashed circle?

If we look at the picture above we can see a lot of circular shapes which appear to be elliptical because of the angle of our viewpoint.

The Ellipse

When a cylinder or a cone are cut at an angle, the resulting section is a ellipse. Hands Up What shape section are we left with if we cut a cylinder or a cone horizontally??

The Ellipse

Parts of an Ellipse

The Major Axis: The major axis is the longest line that can be drawn across an ellipse The Minor Axis: The minor axis is the perpendicular bisector through the mid point of the major axis. It is the shortest line that can be drawn across an ellipse The Focal Points: These are two points that lie on the major axis. They are denoted by F1 and F2.

Minor Axis

Major Axis

F1 F2

Focal Points F1 and F2

Parts of an Ellipse

The Ellipse

The Focal Points

The distance from one end of the minor axis to a focal point is always equal to one half of the measure of the major axis

We will go through how to locate the focal points once we know how to draw an ellipse.

The Ellipse

Sheet 1 – Title: Ellipse (Concentric Circles Method)

Draw the Major and Minor axes. In this case the major axis is 120mm and the minor axis is 70mm. Then draw the Major and Minor Concentric Circles

The Ellipse


Next, draw a series of 30˚ lines through the centrepoint until both circles are divided into 12 equal parts.

The Ellipse


Where each division meets the major circle, draw a line parallel to the minor axis. Where each division meets the minor circle, draw a line parallel to the major axis. The intersections of each of these lines gives us points on our ellipse.

The Ellipse


We now have 8 points on our ellipse. We have 12 in total if we count in each end of the major axis and each end of the minor axis. Join all of these points freehand to form the Ellipse.

The Ellipse

Sheet 2 – Title: Whiskas Logo

The figure shows the logo for Whiskas cat food. The design is based on an ellipse with major axis 140mm and minor axis 90mm. The ears of the logo are found by drawing lines from points P and Q to the focal points of the curve.

The Ellipse

Homework 1 – Title: Gate

The figure shows an entrance gateway. The design is based on a semi-ellipse with major axis 120mm and minor axis 70mm. The spacings between the 10mm vertical members are equal. Draw the gate, showing clearly all construction lines.

The Ellipse

Sheet 3 – Title: Mirror

The design for a bedroom mirror is shown opposite. Draw full size the given design showing clearly how the centre for the arc A is found.

The Ellipse

Homework 2 – Title: Shampoo Bottle

A drawing for a shampoo bottle is shown opposite. The outline of the bottle is a portion of an ellipse with major axis 180mm and minor axis 90mm. Draw the bottle full size showing all construction clearly.

The Ellipse

Sheet 4 – Title: Ellipse (The Trammel Method)

•Draw the Major and Minor axes. •Cut a piece of paper with a straight edge to use as a trammel. •Mark off point P about 10mm from one end of the trammel along the straight edge. •Mark off half of the minor axis from P. This gives us point A. •Mark off half of the major axis from P. This give us point B. •Place the trammel so that A is on the major axis and B is on the minor axis and mark a point at P.

The Ellipse

Homework 3 – Title: B&B Sign

A drawing of a B&B sign is shown opposite. Using a trammel, draw the sign full size. Tape the trammel to your drawing when you have completed it.

The Ellipse

Sheet 5 – Title: Hamburger Inn Logo – possible to leave out

The design for the Hamburger Inn logo is shown opposite. Draw the logo full size using the concentric circles method. Show all construction clearly.

The Ellipse

Sheet 6 – Title: Tangent to an Ellipse through a point on the curve

A tangent to an ellipse is a line the touches the curve at one point. This point is called the point of contact A normal is a line drawn perpendicular to the tangent through the point of contact.

Draw an ellipse with major axis 140mm and minor axis 80mm (in this case) and locate the focal points. The focal points are very important for the construction of any tangent as you will see. Join both focal points to a random (in this case) point P

The Ellipse

Sheet 6 – Title: Tangent to an Ellipse through a point on the curve

Bisect the angle F1PF2. The bisector of this angle is called the Normal Draw a line perpendicular to the normal through point P. This is the required tangent

The Ellipse

Homework 4 – Title: Tangents to the Ellipse

Draw an ellipse with major axis 160mm and minor axis 90mm. Pick four random points on the ellipse and construct tangents to the ellipse through each of these points.

The Ellipse

Sheet 7 – Title: Fish

The figure shows a design based on an ellipse with major axis 140mm and minor axis 90mm. F is one of the focal points of the ellipse. AB is a tangent through the point A. Draw the figure.

The Ellipse

Sheet 8 – Title: Computer Steering Wheel

The figure shows a design based on an ellipse with major axis 120mm and minor axis 90mm. The line AB is a normal to the curve at point A. Draw the figure.

The Parabola

Whenever a ball is thrown in the air, it follows a curved path called a parabola. Parabolas are used extensively in mathematics and science. One scientific use for a parabola is the satellite dish. The body of the dish is parabolic. Any radio wave which hits the surface of the dish is redirected or focused at a point called the focal point.

The Parabola

The same number of equal divisions must be used for

each half

4 equal divisions

4 e

qu

al d

ivis

ion

s

vertex

Sheet 9 – The Parabola Draw a rectangle with base 120mm and height 160mm

The Parabola

Sheet 10 – Title: Gateway Arch

The figure shows the Gateway Arch in St Louis. It is based on two parabolic curves. Draw the arch given the dimensions over.

The Parabola

Homework 5 – Title: Lava Lamp

The figure shows a lava lamp. It is based on two parabolic curves ABC and ADC, with vertices at B and D, respectively. Draw the lamp given the dimensions over.

230

70

15

75

100

40

The Parabola

Sheet 11 – Title: McDonald’s Logo

The figure shows the McDonald’s Logo. It is based on two parabolic curves. Draw the logo given the dimensions over.

The Parabola

Homework 6 – Title: Motorola Logo

The figure shows the Motorola Logo. It is based on two parabolic curves with vertices at A and B. Draw the logo given the dimensions over.

The Ellipse & The Parabola

The ellipse and the parabola are often combined in the drawing of many designs. One such example is the Sydney Harbour Bridge shown. http://www.youtube.com/watch?v=IiS-et1mts4 The bottom arch is a portion of a semi ellipse and the top curve is a parabola.

Drawings involving combinations of ellipses and parabolas

http://www.youtube.com/watch?v=IiS-et1mts4




Sheet 12 – Title: Sydney Harbour Bridge

Shown over is a drawing of the Sydney Harbour Bridge. The main arch of the bridge is in the form of a portion ABC of an ellipse having a semi-major axis length of 150mm and focal point F. The second arch is a parabola DEF having its vertex at E. The vertical cables are equally spaced. Draw the bridge.


Sheet 14 – Title: Wine Glass

Shown over is a design based on a wine glass. The curve ABC is based on an ellipse with major axis 130mm and focal point F. The curve JKL is a parabola with vertex K Draw the given figure.


Sheet 13 – Title: Fish

Shown over is a design based on a fish. The curve ABCDE is based on an ellipse with major axis 130mm and focal point F. The line BP is tangential to the ellipse at point B. The curve QDR is a parabola with vertex D Draw the given figure.


The Ellipse (Special considerations) In order to draw an ellipse we need only two pieces of information from the following list: • The major axis • The minor axis • The position of the focal points • A known point on the curve.

We already know (or should do!) how to draw an ellipse given the major and minor axes. We also know how to find the major axis given the minor axis and a focal point and vice versa. We will learn over the next few sheet how to draw an ellipse given the major (or minor) axis and a point on the curve. Finally we will learn how to draw an ellipse given the focal points and a point on the curve.


Sheet 15 - The Ellipse (Given major axis and a point on the curve)


Draw a major axis of 150mm. Mark off a random point P as shown over. Draw the major circle.



From the point P draw a vertical line to intersect with the major circle. From this intersection point join to the centrepoint. From the point P draw a horizontal line to hit the line we have drawn previously. The intersection of these two lines gives us a point on the minor circle.



Draw the minor circle. Now we can draw the ellipse using the concentric circles method as usual.

Sheet 16 - The Ellipse (Given minor axis and a point on the curve)


Draw a minor axis of length 90 mm. Position a random point P as shown over



Draw the minor circle as shown over. Similar to before, draw a vertical line from point P to hit the minor circle. And join this point to the centrepoint and extend. Draw a horizontal line from P to hit the extended line.



Draw the major circle through the intersection point as shown.

Homework 7- The Ellipse (Special conditions)


a) Given the major axis and a point P on the curve, draw the ellipse.

b) Given the minor axis and a point P on the curve, draw the ellipse.

Sheet 17 - Star Trek Emblem


The figure across shows a drawing of the Star Trek emblem. The curve ABCDE is based on an ellipse of major axis 150mm. The curves PQR and PSR are parabolas with vertices at Q and S, respectively. Draw the figure

Sheet 18 – Kawasaki Logo – possible to leave out


The figure across shows a drawing of the Kawasaki Logo. The curves DEF and GHI are based on the same semi-ellipse of minor axis 50mm. The curves AB and AC are parabolas with vertices at B and C, respectively. Draw the figure

Sheet 19 - The Ellipse (Given focal points and a point on the curve)


Draw a random triangle. Label the base F1F2 and label the apex P. F1 and F2 are the focal points of an ellipse and point P is a point on the curve. First bisect the line F1F2 to find the centre point of the ellipse. This will allow us to draw the major and minor circles later



Extend the line F2P on past P as shown here in light blue. Put the pin of your compass on P and get the distance from P to F1 on it. Swing and arc about P until it hits the extended line we have drawn. The distance F1Q is the length of the major axis of the ellipse we wish to draw.



Bisect the line F1Q to find the radius of the major circle of our ellipse.



Using the radius we have found previously we draw the major circle of the ellipse about the centrepoint located halfway between the two focal points We find the minor axis by get half the major axis on the compass and swinging an arc from either focal point.

Distance F1PF2 is equal to the major axis

Sheet 20 – Mouse and Cable


The figure across shows a drawing of a mouse and cable. The curve ABCDE is based on an ellipse of major axis 130mm. The curve AE is based on the same ellipse The curves VB and VF are based on the same semi-parabola with both vertices at V. Draw the figure

Sheet 21 – Tangent to an ellipse from an external point


Draw an ellipse with major axis 130mm and minor axis 90mm. Also locate the focal points and a random point P. The tangent will be drawn from point P to touch off of the ellipse at two points. With radius PF1, swing an arc from point P as show opposite.



With the major axis as radius draw an arc from F2 to intersect the first arc at points M and N. Next join the points M and N to the focal points. This locates two points of contact on the ellipse.



Join the points of contact R and Q to P. These are our required tangents.

Sheet 22 – Tangent that forms a required angle with ellipse.


Construct a tangent to the given ellipse which forms a 30˚ with the major axis.

Draw an ellipse of major axis 130mm and minor axis 90mm. Determine the focal points and draw the major circle (if you haven’t already done so) Next draw an line L at 30˚ to the major axis. The distance from the ellipse is random.

Sheet 22 – Tangent that forms a required angle with ellipse.


Construct a tangent to the given ellipse which forms a 30˚ with the major axis.

Draw two perpendicular lines from the focal points to the angled line L. Where these perpendiculars intersect with the major circle (A & B) are two points on our required tangent. Draw the tangent. How do we find the point of contact??

Sheet 23 – Trophy


The figure shows a design of a sports cup. It consists of a semi-ellipse with major axis 190mm and minor axis 86mm. The lines AB and CD are tangents to the ellipse The curves PQ and RQ are semi-parabolas with vertices at P and R, respectively. Draw the figure

Homework 8 – Mask


The figure shows a design of a mask. It is based on an ellipse with major axis 120mm and minor axis 70mm. The lines PQ and RS are tangents to the ellipse. The curve ABC is parabolic with vertex at B. Draw the figure

Documents

Ellipse & Parabola