1. Geometric Constructions With the Compass Alone Abstract Introduction Tools Curves construction Applications Bibliography

2. Abstract The topic of the thesis is focused on the constructions with compass alone. These constructions contain: Curves construction Fermat point Tooth – wheel coupling between epicycloid and hypocycloid Ellipse sliding in deltoid and deltoid circumscribing an ellipse Paper HomePaper Home Next section-IntroductionNext section-Introduction

3. Introduction In Mohr-Mascheroni geometry of compass proved that every Euclidean constructions can be carried out with compass alone. Paper HomePaper Home Next section-ToolsNext section-Tools

4. Tools This section reviews the main tools. In Mohr-Mascheroni geometry of the compass a straight line is, naturally, regarded as given or determined if two its point are known. Paper HomePaper Home review tools review tools

5. Tools Lemma 1. Construct a point, symmetric to a given point with respect to the given straight line. Construction Paper HomePaper Home Next toolNext tool

6. Tools Lemma 2. Construct a perpendicular to the segment AB at point B. Construction Paper HomePaper Home Next toolNext tool

7. Tools Lemma 3. Construct a circle determined by radius and center. Construction Paper HomePaper Home Next toolNext tool

8. Tools Construction 4. Given three points A,B,D, to complete the parallelogram ABCD. Construction Paper HomePaper Home Next toolNext tool

9. Tools Lemma 5. Given a circle C with center O and point A, construct the inverse of A with respect to C. Construction Paper HomePaper Home Next toolNext tool

10. Tools Lemma 6. Construct a segment n times the length of a given segment, n=2,3,4, …. Construction Paper HomePaper Home Next toolNext tool

11. Tools Construction 7. Construct a segment x times the length of a given segment, n=2,3,4, …. (a). x=1/n (b). x=2/n (c). X=3/n Paper HomePaper Home Next toolNext tool

12. Tools Lemma 8. Construct the sum and difference of two given segments. Construction Paper HomePaper Home Next toolNext tool

13. Tools Consequence 8-1 Given a circle C and straight line AB. Find the intersection of the circle C with the straight line AB. Case 1. Assume center does not lie on AB. Case 2. Assume center lies on AB. Paper HomePaper Home Next toolNext tool

14. Tools Consequence 8-2 Let two point A,B belong to circle C. Bisect the two arcs of the circle defined by the points A and B. Construction Paper HomePaper Home Next toolNext tool

15. Tools Lemma 9 Let a,b,c be defined as the length of three given segments. Find x such that x/c=a/b. Construction Paper HomePaper Home Curves constructionCurves construction

16. In preceding we reviews the main tools. Now we used these tools to construct plane curves, and avoided to construct the intersection of two straight lines. Paper HomePaper Home CycloidsCycloids

17. Curves construction Construct cycloid and the osculating circle of the cycloid : Let r=radius of rolling circle, r 1 =radius of base circle,where r 1 =nr point O = center of the base circle Point C = a cusp on the axis of the reals at the point r 1 point B = the point of contact of base circle and rolling circle θ= the angle COB. Step 1. Construct the point B ’ by rotating B with nθ about the center O. Step 2. Construct the point A by dilating B with respect to B ’ with factor (1+1/n). Then point A describes an epicycloid or a hypocycloid according to n is positive or negative. Step 3. Construct point R by dilating B with respect to A with factor (1+n/(n+2)). Paper HomePaper Home ExamplesExamples

18. Curves construction Epi- and Hypocycloid (1). Cardioid and Osculating circle of the Cardioid.Cardioid and Osculating circle of the Cardioid. (2). Nephroid and Osculating circle of the Nephroid.Nephroid and Osculating circle of the Nephroid. (3). Deltoid and Osculating circle of the Deltoid.Deltoid and Osculating circle of the Deltoid. (4). Astroid and Osculating circle of the Astroid.Astroid and Osculating circle of the Astroid. Paper HomePaper Home LemniscateLemniscate

19. Curves construction Lemniscate Method 1Method 1: construction based on “ Kite ” linkage Method 2Method 2: Construction based on 3-bar linkage Paper HomePaper Home ConicsConics

20. Curves construction Conics Construct the inverse of lemniscate. Construction Paper HomePaper Home ParabolaParabola

21. Curves construction Parabola The center of inversion coincider with the cusp, the inversion of cardioid is a parabola with focus at the cusp. Construction Paper HomePaper Home EllipseEllipse

22. Curves construction Ellipse Construction following parameter coordinates of ellipse and trochoid. Method 1Method 1 Method 2Method 2 Paper HomePaper Home ApplicationsApplications

23. In this section, we used preceding sections to construct dynamic geometry with compass alone. (1). Gear wheel tooth profilesGear wheel tooth profiles (2). SlidingSliding (3). Fermat pointFermat point Paper Home

24. Applications Gear wheel tooth profiles Without lose of generality, construct “ Tooth-wheel coupling between epicycloid and hypocycloid ”, we may assume that hypocycloid is located on left and epicycloid on right. There are two part: Paper HomePaper Home Part 1Part 1

25. Applications Construction 15: Tooth-Wheel Coupling Between m-cusped hypocycloid and n-cusped epicycloid, m is odd. Example 1Example 1. Tooth- wheel coupling between deltoid and cardioid. Example 2Example 2. Tooth- wheel coupling between deltoid and Nephroid. Paper HomePaper Home Part 2Part 2

26. Applications Construction 16: Tooth-Wheel Coupling Between m-cupsed hypocycloid and n-cusped epicycloid, m is even. Example 1Example 1. Tooth-wheel coupling between astroid and cardioid. Example 2Example 2. Tooth-wheel coupling between astroid and Nephroid. Paper HomePaper Home SlidingSliding

27. Applications-sliding We will discussion the phenomena of “ ellipse sliding in deltoid ” and “ deltoid Circumscribing an ellipse ”. First, we discussion (m-1)-cusped hypocycloid sliding inside m-cusped hypocycloid. Here,when m=3, the construction leads to a segment sliding inside deltoid. Paper HomePaper Home Ellipse sliding in deltoidEllipse sliding in deltoid

28. Applications-sliding Now we use the ellipse instead of the segment and the ellipse still sliding in deltoid. Method 1Method 1 Method 2Method 2 Paper HomePaper Home NextNext

29. Applications-sliding Construct “ m-cusped hypocycloid sliding outside (m-1)-cusped hypocycloid ” Here, when m=3, the construction leads to a deltoid sliding outside segment. Paper HomePaper Home Deltoid circumscribing an ellipseDeltoid circumscribing an ellipse

30. Applications-sliding Now we also use the ellipse instead of the segment and the deltoid still circumscribes the ellipse. Method 1Method 1 Method 2Method 2 Paper HomePaper Home Fermat pointFermat point

31. Applications-Fermat point If equilateral triangles ABR,ACQ,BCP are described externally upon the sides AB, AC, BC of triangle ABC, then AP, BQ, CR are meet in a point F. In order to construct the Fermat point with compass alone, we used the property that AP, BQ, CR meet at 120 0. Construction Paper HomePaper Home BibliographyBibliography

32. [1] Zwikker, C. The Advanced Geometry of Plane Curves and Their Application, Dover Publications, Inc., New York, 1963. [2] Dorrie, Heinrich. 100 Great Problem of Elementary Mathematics, Dover Publications, New York, 1965. [3] Aleksandr, Kostovskii. Geometrical Constructions Using Compasses Only, Blaisdell Publications, Co., New York, 1961. [4] Lockwood, E.H. A book of Curves, Cambridge, England, Cambridge University Press, reprinted, 1963. [5] Yates, Robert C. Geometrical Tools, Saint Louis: Educational Publishers, Inc, reprinted, 1963 [6] Eves, Howard. A survey of Geometry, Boston, Allyn and Bacon, 1963. Paper homePaper home