transformation geometry 1 - t4 geometry... · transformation geometry 1 ... parabola abc is in the...

DESIGN & COMMUNICATION GRAPHICS

Transformation Geometry 1

NAME: ______________________________ DATE: _____________

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.

A

Key Principles • Under a horizontal translation all points move the _______ distance in a ____________ direction • Under a vertical translation all points move the ________ distance in a ____________ direction • The shape and size of the object are _________________

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

Trolley

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 screw in its initial position A. (a) The screw is to move along the surface of the wood to position A1.

Show the screw in the translated position. (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 wood. Show the screw in the new translated position.

A A1



NAME: ______________________________ DATE: _____________

The photograph shows a leisure boat. The diagram over shows the boat docked at pier A. (a) Complete the construction of the boat. (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 boat docks at point B. Show the translation of the boat into this position.

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

_________________ bisector P

P1

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.

X

Pier A

Pier B B



NAME: ______________________________ DATE: _____________

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

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.

The cone is translated horizontally along the Horizontal Plane.

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

Cone C



NAME: ______________________________ DATE: _____________

Key Principles

• Parallel lines always remain __________________ • Under a horizontal translation all points move the _______ distance in a ____________ direction • The shortest distance between two points is a ____________ ____________

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.

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.

Bus Stop B

Arrows are equally spaced

School

S

B



NAME: ______________________________ DATE: _____________

The photograph shows the Toastrack Hotel in Manchester. The building is generated by translating the parabola ABC in a vertical position along the horizontal line BD. Also shown is the incomplete elevation and plan of the structure. Complete the elevation and plan of the hotel

A

B

C A,C

B

A

C

B

D

x y

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 straight line BD. Also shown is the elevation and incomplete plan of the roof. Complete the plan of the roof showing all construction lines.

A

B

C A,C

B

D

B

A

C

x y

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.

Parabola ABC is in the vertical position with the line BD horizontal.

The parabola translates (pt B) along the line BD to form the hotel shape.



NAME: ______________________________ DATE: _____________

Identify all axes of symmetry in each of the following objects:

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.

Isosceles Triangle

Aeroplane Wing Alloy Wheel

Taj Mahal Axis of symmetry

Rotated

Square Star of David

Stop Sign

Equilateral Triangle Mercedes Logo

Reflection in nature

Maple Leaf



NAME: ______________________________ DATE: _____________

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 _______

A quilt pattern

A football jersey

Complete the drawing of the butterfly.

Complete the graph of the parabola

X

Y

Broom Pizza

Complete each of the designs below using axial symmetry.



NAME: ______________________________ DATE: _____________

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 join-

ing a point and its _______

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

G

F1 F2

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

Law of reflection of light

C



NAME: ______________________________ DATE: _____________

Key Principles • The set of points equidistant from the focus and the ____________ is the parabola • Every point on the perpendicular bisector of a line segment is equidistant from the ___________ of the line

When a ball is hit, the path the ball takes is in the shape of a parabola. 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

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.

D

Q

F

D

A



NAME: ______________________________ DATE: _____________

Key Principles • A rotation is a transformation that rotates a figure about a fixed ________ • The measure of the sides and angles in an equilateral triangle are all

__________ • 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 _________

The photograph shows a clock which is hexagonal in shape. The diagram below shows an equilateral triangle ABC. (a) Rotate the triangle about point B to create the remainder of the clock. (b) Identify the measure of the angle at B.

B

C

The photograph shows a football in which all the black areas are pentagonal in shape. The diagram below shows an isosceles triangle LMN. The side MN forms one side of the pentagon. (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.

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. (a) Identify the measures of angles A, B and C. (b) Reproduce a similar triangle ABC in the given

circle. (c) Rotate triangle ABC about A to complete the

octagon

M

L

N

A

B

C



NAME: ______________________________ DATE: _____________

Key Principles • Under a rotation a figure is turned about a fixed point through an angle. The fixed point is called the ________ of rotation and the angle is called the angle of __________. • The rotated figure is the same _____ and _____ as the original figure. • Every surface in a development represents the _____ _______ of that

surface. • Lines along which folds occur in a development are represented by

__________ lines.

Many home improvement projects require disposal of building waste and one of the best ways to do this is to hire a skip. The 3D graphic across shows a skip lorry.

The lifting arms are used to load and unload the skip without hitting the loading platform. The drawing below shows a representation of part of the skip and the lifting arm. The lifting arm rotates about the point O in a clockwise direction to unload the skip. (a) Draw the skip in the rotated position on the ground. (b) Write down the measure of the angle of rotation of the lifting arm. (c) Complete the surface development of the skip shown across.

O

lifting arm

X Y



NAME: ______________________________ DATE: _____________

The image shows a centrally pivoted directional sign post. The diagram below shows the plan and elevation of this sign post. The post is to be rotated 45o clockwise, as indicated in plan. Project the new plan and elevation of the sign post in this rotated position.

X Y

The photograph shows the Brazilian national flag flying in the wind. The diagram below shows the plan and elevation of the flag flying in an easterly direction. 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.

curves of rotation

Flag in rotated position

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

_______ _________ in elevation.



NAME: ______________________________ DATE: _____________

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.

Most modern wind turbines are three-bladed designs like the ones shown in the graphic below. 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.

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°. Draw the top part in its rotated position.

O

X Y

O

O A

B

C



NAME: ______________________________ DATE: _____________

Key Principles • A rotation is a movement of an object around a fixed point

(centre) or axis (line). • An object will rotate about a line called an ___________ or

___________ line.

The photograph shows the ancient Egyptian pyramids at Giza. Shown below is a plan and elevation of a square-based pyramid with is base ABCD on the horizontal plane. Rotate the pyramid about the edge AD until point O is on the horizontal plane. Construct the plan and elevation of the pyramid in this new position.

The photographs show a display unit and the hydrocarbon molecule, both based on a regular tetrahedron. The elevation and plan of a regular tetrahedron ABCO, which has its base ABC resting on the horizontal plane, is shown below. Rotate the tetrahedron about the edge BC until the surface BOC is in a vertical position. Construct the plan and elevation of the tetrahedron in this new position.

Pyramid in its rotated position on the Horizontal Plane.

Elevation of rotation

center of rotation / hinge line

Tetrahedron inclined to HP

Point O on HP

X Y

O

O

A B

C D

A,D B,C B,C A



NAME: ______________________________ DATE: _____________

Key Principle • In a 3 dimensional rotation, an object is rotated in a view which shows

the axis of rotation as a _______

The drawing below shows the plan and elevation of two spheres A and B which are in contact as shown. The elevation of a third sphere C, which is in mutual contact with both solids is also shown. Draw the plan of sphere C.

The 3D graphic shows a handcrafted wooden salt cellar, with a lid that rotates to reveal the salt. The projections of the salt cellar are shown in the drawing below. (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.

O

A B

C



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.

Rigel

Tellar Andoria

Risa

Drummin

Initial Position

Beta Wormhole

Starship Enterprise

Spacedock Space Station



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°.

earth

Romulan homeworld

Delta Vega

Jupiter 2

Jupiter

Klingon ship

Asteroid belt

shuttlecraft

P

transformation geometry 1 - t4 geometry... · transformation geometry 1 ... parabola abc is in the...

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