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Geometric Constructions With the Compass Alone

Geometric Constructions With the Compass Alone. Abstract Introduction Tools Curves construction Applications Bibliography. Abstract. The topic of the thesis is focused on the constructions with compass alone. These constructions contain: Curves construction Fermat point

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Geometric Constructions With the Compass Alone

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  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 HomeNext section-Introduction

  3. Introduction In Mohr-Mascheroni geometry of compass proved that every Euclidean constructions can be carried out with compass alone. Paper HomeNext 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 Home review tools

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

  6. Tools Lemma 2. Construct a perpendicular to the segment AB at point B. Construction Paper HomeNext tool

  7. Tools Lemma 3. Construct a circle determined by radius and center. Construction Paper HomeNext tool

  8. Tools Construction 4. Given three points A,B,D, to complete the parallelogram ABCD. Construction Paper HomeNext 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 HomeNext tool

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

  12. Tools Lemma 8. Construct the sum and difference of two given segments. Construction Paper HomeNext 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 HomeNext 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 HomeNext 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 HomeCurves construction

  16. Curves construction 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 HomeCycloids

  17. Curves construction Construct cycloid and the osculating circle of the cycloid: Let r=radius of rolling circle, r1=radius of base circle ,where r1=nr point O = center of the base circle Point C = a cusp on the axis of the reals at the point r1 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 HomeExamples

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

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

  20. Curves construction Conics Construct the inverse of lemniscate. Construction Paper HomeParabola

  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 HomeEllipse

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

  23. Applications In this section, we used preceding sections to construct dynamic geometry with compass alone. (1). Gear wheel tooth profiles (2). Sliding (3). Fermat 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 HomePart 1

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

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

  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 HomeEllipse sliding in deltoid

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

  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 HomeDeltoid 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 2 Paper HomeFermat 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 1200. Construction Paper HomeBibliography

  32. Bibliography [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 home

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