construction of transition pieces part...

Construction of Transition Pieces

Extension

We are now going to explore transition pieces involving circles.

We need to be able to find the coordinates of points around circles.

We will look at two cases: dividing the circle at 8 points and at 12 points around the circumference.

We could work out the coordinates for any number of points by using trigonometry. Ask you teacher to explain x = r cosθ, y = r sinθ

Case 1 – What are the coordinates (in the 2 dimensional number plane) of the points which divide a circle of radius 40 mm into 8 equal parts?

The following diagram shows a circle of radius 40 divided at points A1, A2, …, A8

Each of the angles at the centre is 45o

Focus your attention on triangle A2OB

Triangle A2OB is a right angled isosceles triangle.

Let OB = d. So A2B = d also. (Why?)

Using Pythagoras,

symmetry. using )3.28,3.28( bemust then

. 28.3) (28.3,point theis So

pl.) dec1(3.28800

800

16002

40

4

2

2

2

222

A

A

d

d

d

dd

• Case 2 – What are the coordinates (in the 2 dimensional number plane) of the points which divide a circle of radius 40 mm into 12 equal parts?

• The following diagram shows a circle of radius 40 divided at points A1, A2, …, A12

• Each of the angles at the centre is 30o

• Focus your attention on triangle A2OA12

It is an equilateral triangle (why?)

the length A2A12 is 40 mm

the length A2B is 20 mm (why?)

Using Pythagoras in triangle A2OB, we get

20,6.34 is So

pl) dec (1 6.341200

1200

4001600

2040

2

2

222

A

OB

OB

OB

Now focus on triangle A3OC.

It is congruent to triangle A2OB. (Why?)

So A3 is (20, 34.6).

Now A5 is ( −20, 34.6) by symmetry.

The other points are found by symmetry.

Example

A square to round transition piece is required to connect a 70 mm square to a circle of radius 25 mm.

The square and circle are on horizontal surfaces.

Their centres are in a vertical line 60 mm apart.

Take 12 points around the circle.

Produce the development then make the transition piece from an A4 sheet of paper or cardboard.

The piece looks like this.

This diagram shows the view looking down on the top of the piece. Equal lengths are shown in the same colour. We only need to find the lengths of AE, AQ and EQ, or other equivalent lengths. (Remember that AB = 70 mm).

Carefully check the following: A is (35, -35, 0), E is (25, 0, 60), Q is (21.7, -12.5, 60). Then: AE = 70.2, AQ = 65.4, EQ = 12.9 using the formula:

221

2

21

2

21 zzyyxxd

AE = 70.2, AQ = 65.4, EQ = 12.9, AB = 70 • Triangulation and construction of the arc

Exercise 1 Produce the development to make an oblique cone, open at the base.

The base is to be a circle of radius 40 mm.

The height of the cone is 50 mm, with the apex of the cone directly above the circumference of the base.

Take 8 points around the circle. Construct the cone using a sheet of A4 paper or cardboard.

Exercise 2 Produce the development to make a square to round transition piece.

The base is to be a square of side 80 mm.

The top is to be a circle of radius 30 mm.

The top and base are to be parallel surfaces 50 mm apart, with the line of centres perpendicular to the base.

Take 12 points around the circle. Construct the transition piece using a sheet of A4 paper or cardboard.

Exercise 3 Produce the development to make a round to rectangle transition piece.

The base is to be a circle of radius 45 mm.

The top is to be a rectangle 50 mm by 30 mm.

The top and base are to be parallel surfaces 60 mm apart, with the line of centres perpendicular to the base.

Take 12 points around the circle. Construct the transition piece using a sheet of A4 paper or cardboard.

Exercise 4 Produce the development to make a transition piece shaped as a frustum of an oblique circular cone.

The base is to be a circle of radius 45 mm.

The top is to be a circle of radius 30 mm.

The top and base are to be parallel surfaces 60 mm apart, with the centre of the top circle directly above a point on the circumference of the base circle.

Take 8 points around each circle. Construct the transition piece using a sheet of A4 paper or cardboard.