12. Ellipse and Parabola

Class: Technical Graphics - Year 2

Learning Intentions and Success Criteria

Date Week beginning:

13/10/2015

Learning Intention

Success Criteria: You will know you can do this because you will be able to:

List the terms associated with the ellipse

Provide examples of how ellipse is used

Drawing an Ellipse using the Concentric Circles Method

Drawing an Ellipse using the Trammel Method

Drawing an Ellipse using the Rectangle Method

Draw a variety of logos involving ellipse

Draw tangents from points outside an ellipse and explain the geometry involved

At the end of these lessons you will:

Use the correct terminology when describing Ellipse

Be able to provide real life examples the use of the Ellipse

Be able to draw an ellipse using three different methods

Be able to draw ellipse and objects involving ellipse accurately

Be able to draw tangents and lines at 90 degrees to ellipse accurately and neatly

Textbook References:

JCTG: Junior Certificate Technical Graphics (O’Callaghan)

UTG: Understanding Technical Graphics (O’Sullivan)

WS: Worksheets

Specialist equipment required for class

• HB, H and 2H pencil

• Pencil sharpener, eraser

• Colouring pencils

• 45, 30 and 60 degree set-squares

• Compass

• Drawing clips or masking tape

• A3 Folder

• Textbook: Understanding Technical Graphics

• Piece of paper/card for a trammel (template)

• iPad

Equipment required at home

• T-square

• Drawing Board

• Internet connection and access to SharePoint

Keywords

Circumference
Arc
Chord
Diameter
Semicircle
Quadrant
Segment
Radius
Sector
Secant
Concentric
Eccentric
Diameter
Semicircle
Intersect
Volkswagen
Tangent
Contact
Equilateral
Triangle
Auctioneers
Trophy
Radius
Construction
Guaranteed
Ellipse
Cylinder
Major
Auxiliary
Umbrella
Entrance
Quadrant
Trammel
Method
Shampoo
Rectangular/Rectangle
Tangent
Normal
Significant
Initially
Bisect

There are three main methods to draw an ellipse; (1) the concentric circles method, (2) the trammel method and (3) the rectangular method. You need to be familiar with these methods, other geometry such as how to construct tangents as well as the properties of tangents such as the role of the focal points.

Scheme for this topic

Sheet 1 Ellipse (Concentric circles and Trammel methods)

Fill in your name, SHEET ?: ELLIPSE 1 and the date
Read the instruction on page 105 of your textbook
Measure 130mm vertically and draw a light construction line across the page with your T-square(This is your centre line)
Measure 95mm horizontally along this line; this is the centre point for two circles (known as auxiliary circles).
Draw a circle of radius 64mm and another of radius 40mm with the same centre point
Divide the larger circle into 12 equal segments (using your 30/60 setsquare
From the points where these lines intersect the larger circle, draw a series of lines, PERPENDICULARLY TOWARDS the MAJOUR AXIS.
From the points where these lines intersect the smaller circle, draw a series of lines, PERPENDICULARLY AWAY FROM the MINOR AXIS.
Draw a smooth curve through the points where the lines from the larger and smaller circle intersect.
Label the Major and Minor Auxiliary Circles
From the bottom of the large circle measure down 10mm, draw a light horizontal construction line
Measure down a further 5mm and draw another light horizontal construction line and write CONCENTRIC CIRCLES METHOD in between the guidelines
Measure 30mm horizontally along the centre line from the circumference of the large circle. Draw a vertical line from the title block to the margin to divide the page into two.
Measure 30mm along the centre line from the dividing line, then measure a further 130mm. These are the two ends of the major axis of a ellipse.
From the mid-point of the major axis draw a vertical line 40mm above, and 40mm below (80mm in total); this is the minor axis
Read the instructions for the Trammel method on page 109
Use a strip of paper/card with a straight edge (called a Trammel)
Mark half the length of the Major Axis on the trammel (mark P at the circumference of the ellipse and MINOR at the other end).
Move the trammel so it lies along the minor axis with point P at the circumference of the ellipse.
Mark half the length of the Minor Axis on the trammel (place P at the circumference of the ellipse and mark MAJOR at the point where the trammel intersects the major axis)
Rotate the trammel slightly so the Major point moves a little along the Major axis and the Minor point moves slightly along the Minor axis. Mark the new position of P
Rotate the trammel slightly further so the Major point moves further along the Major axis and the Minor point moves slightly along the Minor axis. Mark the new position of P
Repeat the last step again until the trammel coincides with the major axis. Then continue rotating with the major point moving along and back the major axis and the Minor point moving along and back the Minor axis.
Join the series of points generated from point P in a smooth curve. This should form an ellipse.
Mark the four quadrants of the ellipse, the major and minor axis
From the bottom of the large circle measure down 10mm, draw a light horizontal construction line
Measure down a further 5mm and draw another light horizontal construction line and write TRAMMEL METHOD in between the guidelines

Sheet 2 Ellipse (Whiskas Logo and Umbrella)

Fill in your name, SHEET ?: ELLIPSE 2 and the date
Read the instruction on page 105 of your textbook
Measure 125mm vertically and draw a light construction line across the page with your T-square(This is your centre line)
Measure 95mm horizontally along this line; this is the centre point for two circles (known as auxiliary circles).
Draw a circle of radius 70mm and another of radius 35mm with the same centre point
Divide the larger circle into 12 equal segments (using your 30/60 setsquare)
From the points where these lines intersect the larger circle, draw a series of lines, PERPENDICULARLY TOWARDS the MAJOUR AXIS.
From the points where these lines intersect the smaller circle, draw a series of lines, PERPENDICULARLY AWAY FROM the MINOR AXIS.
Draw a smooth curve through the points where the lines from the larger and smaller circle intersect to form your ellipse
Measure HALF THE LENGTH OF THE MAJOR AXIS on your compass. With the pin of the compass at the vertex of the minor axis, mark off this distance along the Major axis (as shown on page 104). This gives you the two FOCI.
Draw a line 55mm parallel to the major axis as shown in Question 1, page 105. Join the focal points to the points marked as P and Q
Heavy in the ellipse and the other outlines as shown in the diagram
From the bottom of the large circle measure down 10mm, draw a light horizontal construction line
Measure down a further 5mm and draw another light horizontal construction line and write WHISKAS in between the guidelines
From the left margin, measure 200mm horizontally and draw a vertical line from the top of the title box to the top margin. This divides the sheet into two equal parts
Read the instructions for question 2 on page 106 of your textbook
In the second part of the page, measure 50mm from the circumference of the Whiskas logo along the centre line. Then another 70mm, this will be the centre of the umbrella and the ellipse.
Draw two semi-circles radius 70mm and 40mm
Bisect these semi-circles into 6 equal segments using the 60/30 degree set-square
From the points where these lines intersect the larger circle, draw a series of lines, PERPENDICULARLY TOWARDS the MAJOUR AXIS.
From the points where these lines intersect the smaller circle, draw a series of lines, PERPENDICULARLY AWAY FROM the MINOR AXIS.
Draw a smooth curve through the points where the lines from the larger and smaller circle intersect to form your ellipse
Mark off 35mm increments along the bottom of the diagram (majour axis) to mark the bottom points of the umbrella
Using a radius of 35mm put the point of the compass on each of these marks, draw arcs of 35mm to find the centre of the curves. Using these centres complete the diagram of the umbrella as shown.
From the bottom of the large circle measure down 10mm, draw a light horizontal construction line
Measure down a further 5mm and draw another light horizontal construction line and write UMBRELLA in between the guidelines

Sheet 3 Ellipse (Mirror and Bottle)

Fill in your name, SHEET ?: ELLIPSE 3 and the date
Line out the sheet with a 10mm margin and Title bar at the bottom. Measure 200mm from the left margin and draw a vertical line from the Title Bar to the top margin. This should divide the sheet into two equal parts.
Read the instruction on page 107 of your textbook
Measure 125mm vertically and draw a light construction line across the page with your T-square(This is your centre line)
Measure 95mm horizontally along this line; this is the centre point for two circles (known as auxiliary circles).
Draw a circle of radius 65mm and another of radius 40mm with the same centre point
Divide the larger circle into 12 equal segments (using your 30/60 setsquare)
From the points where these lines intersect the larger circle, draw a series of lines, PERPENDICULARLY TOWARDS the MAJOUR AXIS.
From the points where these lines intersect the smaller circle, draw a series of lines, PERPENDICULARLY AWAY FROM the MINOR AXIS.
Draw a smooth curve through the points where the lines from the larger and smaller circle intersect to form your ellipse
Starting from the centre line, draw the stand for the mirror. Measure out 5mm from one vertex of the smaller circle then draw an outline of the stand without the curved edges.
Measure up 10mm and measure in 10mm to find the centres of the inside curves of the stand
Measure in 25mm from the bottom left and bottom right of the stand. These two points are on the arc. The third point (A) is along the centre line of the ellipse and 10mm from the bottom of the stand. Join the points 25mm from the bottom left/right to point A to from a triangle. Bisect these lines to find the centre of the curve. Look at the diagram on page 107.
From the bottom of the large circle measure down 10mm, draw a light horizontal construction line
Measure down a further 5mm and draw another light horizontal construction line and write MIRROR in between the guidelines
Read the instructions for exercise 1 on page 108 of your textbook
In the second part of the page, measure 95mm from the page divide along the centre line. This will be the centre of the bottle and the ellipse.
Draw two semi-circles radius 90mm and 45mm
Bisect these circles into 12 equal segments using the 60/30 degree set-square
From the points where these lines intersect the larger circle, draw a series of lines, PERPENDICULARLY TOWARDS the MAJOUR AXIS.
From the points where these lines intersect the smaller circle, draw a series of lines, PERPENDICULARLY AWAY FROM the MINOR AXIS.
Draw a smooth curve through the points where the lines from the larger and smaller circle intersect to form your ellipse
Examine the diagram on page 108. Draw a horizontal line 80mm above the centre line and another 70mm below the centre line. These are the top and bottom of the bottle.
Draw lines from the centre to the cap as shown in the diagram. Draw another line 20mm above the top horizontal line to form the top of the cap
From the bottom of the large circle measure down 10mm, draw a light horizontal construction line
Measure down a further 5mm and draw another light horizontal construction line and write GEL BOTTLE in between the guidelines

Sheet 4 Ellipse (Lamp and Arch)

Fill in your name, SHEET ?: ELLIPSE 4 and the date
Line out the sheet with a 10mm margin and Title bar at the bottom. Measure 200mm from the left margin and draw a vertical line from the Title Bar to the top margin. This should divide the sheet into two equal parts.
Read the instruction on page 108 of your textbook
Measure 145mm vertically and draw a light construction line across the page with your T-square(This is your centre line)
Measure 95mm horizontally along this line; this is the centre point for two circles (known as auxiliary circles).
Draw a semi-circle of radius 70mm and another of radius 40mm with the same centre point
Divide the larger circle into 6 equal segments (using your 30/60 setsquare)
From the points where these lines intersect the larger circle, draw a series of lines, PERPENDICULARLY TOWARDS the MAJOUR AXIS.
From the points where these lines intersect the smaller semi-circle, draw a series of lines, PERPENDICULARLY AWAY FROM the MINOR AXIS.
Draw a smooth curve through the points where the lines from the larger and smaller semi-circle intersect to form your ellipse
Along the bottom of the lamp shade (Minor axis), mark off 20mm increments. Through these increments mark off lines at 45 degrees as shown in the diagram.
From the minor axis measure 85mm downwards to draw a horizontal line to form the bottom of the lamp
Mark points A and C as shown in the diagram. Measure 15mm along the centre line to find point B
Join either A or C to B. Bisect this line and where the bisector line strikes the centre line from the ellipse will be the centre of the curve at the base of the lamp. Join this centre point to either A, B or C and join an arc between them.
From the bottom of the large circle measure down 10mm, draw a light horizontal construction line
Measure down a further 5mm and draw another light horizontal construction line and write LAMP in between the guidelines
Read the instructions for exercise 3 on page 106 of your textbook
In the second part of the page, measure 95mm horizontally from the page divide and measure 130mm vertically and draw a horizontal line. This will be the centre of the arch and the ellipse.
Draw two semi-circles radius 60mm and 35mm
Bisect these semi-circles into 6 equal segments using the 60/30 degree set-square
From the points where these lines intersect the larger semi-circle, draw a series of lines, PERPENDICULARLY TOWARDS the MAJOUR AXIS.
From the points where these lines intersect the smaller semi-circle, draw a series of lines, PERPENDICULARLY AWAY FROM the MINOR AXIS.
Draw a smooth curve through the points where the lines from the larger and smaller circle intersect to form your ellipse
Examine the diagram on page 106. Draw the outline of the arch first. Measure 55mm vertically from the centre of the ellipse, this is the top point of the arch.
Draw a horizontal line from the vertex of the minor circle. Measure 70mm along either side of the centre line. This gives the sloping part of the top of the diagram.
Add the width of the planks (10mm) together to get ? Subtract this number from 140mm and divide the answer to give you the distances between the planks. Mark these distances along the bottom of the diagram and draw vertical lines through these points (to the majour axis). Complete the diagram as shown.
From the bottom of the large circle measure down 10mm, draw a light horizontal construction line
Measure down a further 5mm and draw another light horizontal construction line and write ARCH in between the guidelines

Homework

Watch the following video on how to draw an ellipse using the concentric circles method: https://www.youtube.com/watch?v=UszZlFoa2jc

Watch the following video on how to draw an ellipse using the concentric circles method: https://www.youtube.com/watch?v=LEkfwttiWzw

Watch the following video on how to draw an ellipse using the concentric circles method: https://www.youtube.com/watch?v=g68ioPKMZI4

Parabola

Video 1 on the parabola: https://www.youtube.com/watch?v=IXuO76DrfYI

Video 2 on the parabola: https://www.youtube.com/watch?v=Ct-AoSbvPQY

Video 3 on the parabola: https://www.youtube.com/watch?v=E9ofGxtMu5Y

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