1. Some Mechanical Devices Alfredo Rodriguez July 03, 2001

2. 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

3. 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

4. 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

5. 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

6. Scales of Line of Lines Each ruler has equally space marks starting from point A.

7. Application for the Sector Ruler using line of lines 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.

8. Scales for Line of Superficies

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 AE 1 and radius AD 1 Find intersection this circle and line AB and call it B 1 This will be the first mark on the side of each ruler. Continue with each point AD i, AE i and B i.

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

11. Compare Areas Place 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

12. Pantograph Description: ABCD is a parallelogram Point O is fixed Point O, A,and E are collinear

13. Move point A along a circle

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

15. 2 pivot points (A and B) directing rule and describing rule at each pivot point 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

16. Tracing the Curve move point D along directrix trace point E (describen)

17. 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 A=(0,0) B=(a,0) <DAE=  <DBE=  AE : y=m x BE: y=n (x-a) Conditions Algebraic demonstration

18. Find location of point D Point D is on a line: Ax ’ + By ’ + C= 0 Equation of Conic Section

19. Let A=(0,0), B=(3,0),  ≈63.34 o,  ≈ 75.96 o, and directrix: -1 x+1/2 y-3/2=0 Example GSPMathematica

20. 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

21. 130 x^2 + 65 x y + 130 y^2 – 520 = 0

22. 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

23. 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?