Tower cranes are a common fixture at any major construction site. Trolley

The 3D graphic on the left shows a tower crane which contains a long horizontal jib (or working arm), which is the portion of the crane that carries the load. A trolley runs along the jib to move the load in and out.

A

The drawing on the right shows an elevation of the crane and skip which is to be attached to the trolley. The elevation of the trolley is incomplete.

(a) Complete the elevation of the trolley, showing all construction lines.

(b) The trolley and skip move along the jib until they reach point A. Show the trolley and skip in the new position after this horizontal movement.

(c) When the trolley reaches the end of the jib the skip must be lowered to the ground to be refilled. Translate the skip vertically until the base of the skip reaches the ground. Skip

The photo shows a wood screw and a piece of wood. Two pieces of this wood are to be jointed together using the screw. The diagram over (left) shows the Key Principles screw in its initial position A.

(a) The screw is to move along the surface of the wood to position A . • Under a horizontal translation all points move the ______A1 1 Show the screw in the translated position. distance in a ______direction A (b) Having positioned the screw, it is to be inserted into the wood vertically until the top of the screw is flush (level) with the top surface of the • Under a vertical translation all points move the ______wood. Show the screw in the new translated position. distance in a ______direction

• The shape and size of the object are ______

DESIGN & COMMUNICATION GRAPHICS

Transformation Geometry 1

NAME: ______DATE: ______Pier B B

The photograph shows a leisure boat. The diagram over shows the boat docked at pier A.

(a) Complete the construction of the boat. X (b) The boat travels from Pier A S64°W for a distance of 105mm. Show the translation of the boat into this po- sition at sea. (c) The boat then travels to Pier B where point X on the Pier A boat docks at point B. Show the translation of the boat into this position.

The photograph shows the door number for a house. The diagram shows this number 4 setup using a square grid.

(a) The figure is subject to a translation. Find the image of the door number when point P moves to P1 under the translation. (b) The image is translated N110°E through a distance of 100mm. Construct the image figure under this translation.

P1 Key Principles

• Under a translation, all points on an object are moved the ______distance in the ______direction

• Corresponding lines on the object and its image are always ______

• A line must be ______to locate its midpoint

P • A line that bisects another line at right angles is called a ______bisector

DESIGN & COMMUNICATION GRAPHICS

Transformation Geometry 2

NAME: ______DATE: ______The photograph shows a training cone in contact with a football.

The diagram below shows the elevation and plan of the football S and an eleva- tion of the cone C. The centre of the sphere and cone are located on a line par- allel to the vertical plane (XY line) in plan.

(a) Translate the elevation of the cone until it is in contact with the sphere. (b) Project a plan of the solids in contact. (c) Show the projections of the point of contact between the solids.

Cone C The cone is translated horizontally along the Horizontal Plane.

The extreme generator of the cone will be tangential to the sphere.

Key Principles

• Under a translation, a line is mapped onto a ______line

• The extreme generator of the cone will be a ______to the circle (sphere) in elevation

• The elevation of the point of contact is located by drawing a line from the centre of the sphere ______to the tangent

DESIGN & COMMUNICATION GRAPHICS

Transformation Geometry 3

NAME: ______DATE: ______The photograph shows a road sign which is commonly seen at roundabouts. The arrows indicate to drivers the direction to follow. These arrows are translated across the rectangular sign. The diagram on the right shows a portion of the rectangular outline and the initial position of the arrow.

(a) Complete the road sign by translating the arrow. (b) Shade the road sign appropriately.

Arrows are equally spaced

School

The diagram on the right shows the plan of school and the position of a bus stop on an adjacent road.

Determine the position of the centerline of the path and the zebra crossing so that the path from the main entrance S to the bus-stop B is of minimum distance for students walking to the school. S

B Bus Stop B

Key Principles

• Parallel lines always remain ______

• Under a horizontal translation all points move the ______distance in a ______direction DESIGN & COMMUNICATION GRAPHICS

• The shortest distance between two points is a ______Transformation Geometry 4

NAME: ______DATE: ______The photograph shows the Toastrack Hotel in Manchester. The building is The photograph shows the inside of a building in Barcelona. The roof structure is generated by translating the parabola ABC in a vertical position along the generated by translating the parabola ABC in a vertical position along the straight horizontal line BD. Also shown is the incomplete elevation and plan of line BD. Also shown is the elevation and incomplete plan of the roof. the structure. Complete the plan of the roof showing all construction lines. Complete the elevation and plan of the hotel D B B B B

D

A,C A C x y A,C A C x y

A A

B B

C C

As the parabola translates along the inclined line BD, the points A and C on the parabola move below the HP.

The intersection of the HP and translated surface creates the outline of the plan.

DESIGN & COMMUNICATION GRAPHICS

Transformation Geometry 5 Parabola ABC is in the vertical position with the The parabola translates (pt B) along the line BD line BD horizontal. to form the hotel shape. NAME: ______DATE: ______Identify all axes of symmetry in each of the following objects:

Rotated

Reflection in nature Taj Mahal Square Star of David Axis of symmetry Isosceles Triangle

Equilateral Triangle Mercedes Logo Maple Leaf Stop Sign

Key Principles

• Axial symmetry is symmetry about a ______

• All symmetrical objects have at least ______axis of symmetry

• An object can have more than ______axis of symmetry.

DESIGN & COMMUNICATION GRAPHICS

Transformation Geometry 6 Alloy Wheel Aeroplane Wing NAME: ______DATE: ______Complete each of the designs below using axial symmetry.

Complete the drawing of the butterfly. Y A quilt pattern A football jersey

Pizza Broom X

Complete the graph of the parabola

Key Principles

• An object and its ______are the same shape and ______

• A point and its image are the same distance from the ______of reflection

• The axis of reflection is the perpendicular bisector of a line joining a point and its ______

DESIGN & COMMUNICATION GRAPHICS

Transformation Geometry 7

NAME: ______DATE: ______Plot the locus of the white ball if it is to strike the yellow ball so that it will enter the corner pocket marked C. P and Q represent two houses which require connection to a gas main represented by the line G. Find the location of a common connection point on the main supply line which will give the shortest length of pipe required.

P

Q

C

G Show how the point of contact between the ellipse and the tangent is obtained.

Key Principles

• An object and its are the same shape and ______

• A point and its image are the same distance from the ______F1 F2 of reflection

• The axis of reflection is the perpendicular bisector of a line join- ing a point and its ______Law of reflection of light

DESIGN & COMMUNICATION GRAPHICS

Transformation Geometry 8

NAME: ______DATE: ______When a ball is hit, the path the ball takes is in the shape of a parabola. D

The construction of the parabola begins with a fixed point for a focus and a line for a directrix. The drawing below shows the directrix DD of a parabola with the focal point at F.

Draw the parabola showing all construction lines.

D D

F F

Q

D The drawing shows the directrix DD and the focus F of a parabola. A series of points (such as Q) are marked on the directrix.

The axis of reflection which maps Q onto F has been determined as shown. This axis of reflection is a tangent to the parabola.

Locate the axes of reflection for the points on the directrix below the axis to generate a view of the skateboard ramp shown on the left. Key Principles

• The set of points equidistant from the focus and the ______is the parabola DESIGN & COMMUNICATION GRAPHICS

Transformation Geometry 9 • Every point on the perpendicular bisector of a line segment is equidistant from the ______of the line

NAME: ______DATE: ______The photograph shows a clock which is hexagonal in shape. The diagram below The photograph shows a football in which all the black areas are shows an equilateral triangle ABC. pentagonal in shape. The diagram below shows an isosceles triangle LMN. The side MN forms one side of the pentagon. (a) Rotate the triangle about point B to create the remainder of the clock. (b) Identify the measure of the angle at B. (a) Rotate the isosceles triangle about point L to form the pentagonal section of the ball. (a) Identify the measure of the angle at L. (b) Add the hexagonal patch shown on the football to the pentagonal patch.

A

B

C

L

The photograph shows an alternative rubix cube which is based on an octagonal prism. The diagram on the right shows a circle in which the octagon must be inscribed. M N Key Principles (a) Identify the measures of angles A, B and C. (b) Reproduce a similar triangle ABC in the given • A rotation is a transformation that rotates a figure about a fixed ______circle. (c) Rotate triangle ABC about A to complete the • The measure of the sides and angles in an equilateral triangle are all octagon ______

• A hexagon can be constructed from 6 ______triangles

• The angle at the centre of any polygon can be found by dividing the ______of sides into ______B

A DESIGN & COMMUNICATION GRAPHICS

C Transformation Geometry 10

NAME: ______DATE: ______Many home improvement projects require disposal of building waste and one of the best Key Principles ways to do this is to hire a skip. The 3D graphic across shows a skip lorry. • Under a rotation a figure is turned about a fixed point through an The lifting arms are used to load and unload the skip without hitting the loading platform. angle. The fixed point is called the ______of rotation and the The drawing below shows a representation of part of the skip and the lifting arm. The angle is called the angle of ______. lifting arm rotates about the point O in a clockwise direction to unload the skip. • The rotated figure is the same _____ and _____ as the original figure. (a) Draw the skip in the rotated position on the ground. • Every surface in a development represents the ______of that (b) Write down the measure of the angle of rotation of the lifting arm. surface.

(c) Complete the surface development of the skip shown across. • Lines along which folds occur in a development are represented by ______lines.

lifting arm

O

DESIGN & COMMUNICATION GRAPHICS

Transformation Geometry 11

NAME: ______DATE: ______The photograph shows the Brazilian national flag flying in the wind. The diagram The image shows a centrally pivoted directional sign post. The diagram below shows the plan and elevation of the flag flying in an easterly direction. below shows the plan and elevation of this sign post. The post is to be rotated 45o clockwise, as indicated in plan. The wind direction changes and rotates the flag to a southwesterly direction in plan (as indicated). Project the elevation of the flag in the new position. Project the new plan and elevation of the sign post in this rotated position.

X Y X Y

Key Principles • If an object / shape is parallel to the XY line in plan, it will be seen as a ______in elevation.

Flag in rotated position DESIGN & COMMUNICATION GRAPHICS

Transformation Geometry 12 curves of rotation NAME: ______DATE: ______Most modern wind turbines are three-bladed designs like the ones shown in the graphic below. The drawing shows a view of a paper stapler. The top part (shaded) rotates about the point O in a clockwise direction through an angle of 100°. The elevation of a wind turbine is shown. Redraw the elevation when the blades have been rotated through an angle of 75° in an anti-clockwise direction. Draw the top part in its rotated position.

O

Key Principles

• The centre of rotation remains ______under a rotation.

• To rotate a line it is sufficient to rotate the perpendicular line drawn from the ______of rotation to the line.

DESIGN & COMMUNICATION GRAPHICS

Transformation Geometry 13

NAME: ______DATE: ______The photograph shows the ancient Egyptian pyramids at Giza. The photographs show a display unit and the hydrocarbon Shown below is a plan and elevation of a square-based pyramid molecule, both based on a regular tetrahedron. The elevation with is base ABCD on the horizontal plane. and plan of a regular tetrahedron ABCO, which has its base ABC resting on the horizontal plane, is shown below. Rotate the pyramid about the edge AD until point O is on the horizontal plane. Construct the plan and elevation of the pyramid Rotate the tetrahedron about the edge BC until the in this new position. surface BOC is in a vertical position. Construct the plan and elevation of the tetrahedron in this new position.

O O

A,D B,C X Y X B,C A Y

A B C

O

O A

D C

Elevation of rotation

B Tetrahedron inclined to HP

center of rotation / Key Principles hinge line

Point O on • A rotation is a movement of an object around a fixed point HP (centre) or axis (line).

DESIGN & COMMUNICATION GRAPHICS • An object will rotate about a line called an ______or Transformation Geometry 14 ______line.

Pyramid in its rotated position on the Horizontal Plane. NAME: ______DATE: ______The drawing below shows the plan and elevation of two spheres A and The 3D graphic shows a handcrafted wooden salt cellar, with a lid B which are in contact as shown. The elevation of a third sphere C, that rotates to reveal the salt. The projections of the salt cellar are which is in mutual contact with both solids is also shown. shown in the drawing below.

Draw the plan of sphere C. (a) Complete the plan of the salt cellar by constructing a common tangent to the two circles. Show all constructions clearly. (b) The lid is rotated about the point O through an angle of 135° in a clockwise direction. Identify the axis of rotation in elevation. (c) Draw the elevation and plan of the lid in the open position.

A B C

O

Key Principle

• In a 3 dimensional rotation, an object is rotated in a view which shows the axis of rotation as a ______

DESIGN & COMMUNICATION GRAPHICS

Transformation Geometry 15

NAME: ______DATE: ______The photograph shows the Starship Enterprise which is travelling through the Vulcan star system. The diagram below shows the initial position of the Starship Enterprise which then travels under the following transformations:

(a) N105°E for a distance of 110 light years (I light year = 1 mm) (b) Through the Beta Wormhole under a central symmetry (c) Under an axial symmetry about the line joining the planets Tellar and Rigel. (d) Rotates about the Andoria solar system through an angle of 150° in a clockwise direction.

Starship Enterprise

Risa

Initial Position

Beta Wormhole

Andoria Tellar

Spacedock Space Station Drummin

Rigel

DESIGN & COMMUNICATION GRAPHICS

Transformation Geometry 16

NAME: ______DATE: ______The year is 3015. Space….. The final frontier….. These are the voyages of the Starship Enterprise. Its continuing mission: To explore strange new worlds... To seek out new life; new civilisations... To boldly go where no one has gone before! Wait a minute… Where’s Captain Kirk? Kirk? Kirk? We’ve got to find Kirk!

Your assignment is to help the crew of the Enterprise find its captain. To do this, the Starship Enterprise (P) must travel to 6 locations on the accompanying star map as directed. Use the circled points that are part of each object.

1. Starting at P, warp out of orbit to find P1 midway between the Romulan homeworld and Earth. 2. To find P2, travel one-third of the distance from P1 to the frozen planet Delta Vega. 3. To find P3, determine the image of the earth under a central symmetry in P2. 4. To locate P4, proceed in a northerly direction until you intersect the galactic barrier. The galactic barrier is the axis of symmetry of the transformation that maps the spaceship Jupiter 2 onto the planet Jupiter. 5. Meteor shower! You veer off course to find P5. It is the centre of a clockwise rotation through an angle of 90° that maps the Klingon ship onto the asteroid belt. 6. You find Captain Kirk at P6, which is the image of the shuttlecraft under a rotation about P5 in an anticlockwise direction through an angle of 120°.

Klingon ship P

Jupiter

Jupiter 2

Romulan homeworld shuttlecraft

Delta Vega

Asteroid belt

earth DESIGN & COMMUNICATION GRAPHICS

Transformation Geometry 17

NAME: ______DATE: ______