Some Mechanical Devices

Alfredo Rodriguez

July 03, 2001

Outline• Sector compass

– Description of device– Building one– Different scales

• Lines to lines• Line to superficies

– Application

• Pantograph– Description of device– Building one– Application

• Organic Construction– Description of device– Simulation– Application

Sector Compass

• Introduction

– Invented by Guidi Ubaldo de Monte

– The device has different scales• Lines of line

• Lines of superficies

– The compass was used for about two centuries

Description of Sector Compass

Given that AB=AC. If AE=AD, then AC and AB are cut proportionally.

Draw in BC and DE

AB:AD=BC:DE AB and AC are legs of sectors

A

BC

DE

Building a Sector Compass• Buy a pair of hinged rulers

Or

• Cut a wooden or cardboard model– Hinge the pieces together

• Mark the rules with the correct marks

Scales of Line of Lines

• Each ruler has equally space marks starting from point A.

A

BC

11

2

3

4

2

3

4

Application for the Sector Ruler using line of lines

p

q

p A

6 6

1 1

2

3

4

5

2

3

4

5

Increase a given line segment p by proportion of 3:5.

Given a segment, p. Open rulers up until p fits into 3-3, then find 5-5 segment.

This will be the desired length.

Scales for Line of Superficies

AD

C

B

E 987654321

1

23

45

6 78

9

Creating the Divisions (Superficies)

• Point A: center of Circle

• AB perpendicular to AC

• Marks of AD are 1/100 of AD

• AE= ½ AC

• Marks of AE are 1/100 of AE

• Create circles with center AE1 and radius AD1

• Find intersection this circle and line AB and call it B1

• This will be the first mark on the side of each ruler.

• Continue with each point ADi , AEi and Bi.

Compare the Area of two Squares

Given two square with side a and b. Use the line of superficies scale for this calculation.

p2

p1

a b

Compare AreasPlace side a on the 10-10 line and find where side b meet the

rulers. In this example side b meets the ruler at 4-4 marks.

Therefore, area of p1: area p2 = 10:4 =2.5:1

p2p1

abA

1 2 3 4 5 6 7 8 9 10

10987654321

Pantograph

Description: ABCD is a parallelogram

Point O is fixed

Point O, A,and E are collinear

O A

C

B

E

D

Move point A along a circle

O A

C

B

E

D

Applications

Given a triangle inscribe a square such that the base of the square is along one side of the triangle.

BA

C

BA D

C

GF

EBA D

C

GF

E

F'G'

D' E'

• 2 pivot points (A and B)

• directing rule and describing rule at each pivot point

directrix

ß

aB

A

D

E

• directrix - a line that directs the motion

• describen - the curve that is being traced

• the angle between each pair of rules ( and )

Components for this device

Tracing the Curve

directrix

ß

a

AB

D

E

• move point D along directrix

• trace point E (describen)

•Find equation of directing rules

Rotate line AE about point A by angle : line AD

Rotate line BE about point A by angle : line BD

directrix

a ß

A B

D

E

A=(0,0)B=(a,0)

<DAE=<DBE=

AE : y=m xBE: y=n (x-a)

•Conditions

y' =Hy +Hx - aLtanHbLL

x - y tanHbL- aHx¢- aL

y' =Hy +x tanHaLLx - y tanHaL

x¢

Algebraic demonstration

•Find location of point D

x' =x tanHbLa2 - y tanHaLtanHbLa2 +y2 tanHaLa - x2 tanHbLa +x y HtanHaLtanHbL- 1La

Hx2 - a x +y2 LHtanHaL- tanHbLL- a y HtanHaLtanHbL+1L

y' =y2 a+x2 tanHaLtanHbLa+x yHtanHaL+tanHbLLa - a2 y tanHbL- a2 x tanHaLtanHbL

Hx2 - a x+y2LHtanHbL- tanHaLL+a yHtanHaLtanHbL+1L

•Point D is on a line: Ax’ + By’+ C= 0

Equation of Conic Section

x2 HC HtanHbL- tanHaLL+a A tanHbL+a B tanHaLtanHbLL+x H- a CHtanHbL- tanHaLLL- a2 A tanHbL- a2 B tanHaLtanHbLL+xy Ha BHtanHaL+tanHbLL+a AH1 - tanHaLtanHbLLL+y2Ha B+CHtanHbL- tanHaLL- a A tanHaLL+yHA tanHaLtanHbLa2 +HtanHaLtanHbL+1LC a - a2 B tanHbLL=0

-2 -1 1 2 3x

-2

2

4

6

y

- 33 x2 -27 y2

2- 24 x y +99 x +

171 y

2=0

Let A=(0,0), B=(3,0), ≈63.34o, ≈ 75.96o, and directrix: -1 x+1/2 y-3/2=0

Example

GSP Mathematica

• Given five points:{{-2,0},{2,0},{0,2},{0,-2},{-1,2}}

Shift over by 2 units in the x-direction, we get• {{0,0},{4,0},{2,2},{2,-2},{1,2}} =ArcTan(2) and =ArcTan (2/3)• Points E’ and F’:{{-6,2},{10,30}}• Directrix: 4 y –7x – 50=0• 130 x^2 + 65 x y + 130 y^2 – 520 x – 130 y = 0

130 x^2 + 65 x y + 130 y^2 – 520 = 0

Overview• Sector compass

– Description of device– Building one– Different scales

• Lines to lines• Line to superficies

– Application

• Pantograph– Description of device– Building one– Application

• Organic Construction– Description of device– Simulation– Application

Shkolenok, A. G. (1972). Geometrical constructions equivalent to non-linear algebraic transformations of the plane in Newton’s early papers. Archive for History of Exact Science 9-2. p.22-44.

Whiteside, D. T. (1961) Pattern of mathematical thought in the later Seventeenth Century. Archive for History of Exact Science 3. p.176-388.

Wood, F. (1954) Tangible arithmetic II: the sector compasses. The Mathematics Teacher. 12. p.535-541

Yates, C.R. (1945) Linkages. In Multi-sensory aids in the teaching of mathematics. New York. p.117-129.

References

Any Questions?