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Geometric Constructions With the Compass Alone
Abstract Introduction Tools Curves construction Applications Bibliography
Abstract The topic of the thesis is focused on theconstructions 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 Home Next section-Introduction
Introduction
In Mohr-Mascheroni geometry of compass
proved that every Euclidean constructions
can be carried out with compass alone.
Paper Home Next section-Tools
Tools
This section reviews the main tools. In
Mohr-Mascheroni geometry of thecompass a straight line is, naturally,regarded as given or determined if
two itspoint are known.
Paper Home review tools
Tools
Lemma 1.Construct a point, symmetric to a givenpoint with respect to the given straightline.
Construction
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Tools
Lemma 2.Construct a perpendicular to thesegment AB at point B.
Construction
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Tools
Construction 4.Given three points A,B,D, to
completethe parallelogram ABCD.
Construction
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Tools
Lemma 5.Given a circle C with center O and
point A,construct the inverse of A withrespect to C.
Construction
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Tools
Lemma 6.Construct a segment n times the
lengthof a given segment, n=2,3,4,… .
Construction
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Tools
Construction 7.Construct a segment x times the
length ofa given segment, n=2,3,4,… .(a). x=1/n(b). x=2/n (c). X=3/n
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Tools
Lemma 8.Construct the sum and difference of
twogiven segments.
Construction
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Tools
Consequence 8-1Given a circle C and straight line AB.
Findthe intersection of the circle C with thestraight line AB.Case 1. Assume center does not lie on AB.Case 2. Assume center lies on AB.
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Tools
Consequence 8-2Let two point A,B belong to circle C.Bisect the two arcs of the circle
definedby the points A and B.
Construction
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Tools
Lemma 9Let a,b,c be defined as the length ofthree given segments. Find x such
thatx/c=a/b.
Construction
Paper Home Curves construction
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 Home Cycloids
Curves constructionConstruct 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 Home Examples
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 Home Lemniscate
Curves constructionLemniscate
Method 1: construction based on “Kite” linkageMethod 2: Construction based on 3-bar linkage
Paper Home Conics
Curves construction
ParabolaThe center of inversion coincider with
the cusp, the inversion of cardioid is aparabola with focus at the cusp.
Construction
Paper Home Ellipse
Curves construction
EllipseConstruction following parametercoordinates of ellipse and trochoid.
Method 1 Method 2
Paper Home Applications
Applications
In this section, we used preceding sections to
construct dynamic geometry with compassalone.(1). Gear wheel tooth profiles(2). Sliding(3). Fermat point
Paper Home
Applications
Gear wheel tooth profilesWithout lose of generality, construct“Tooth-wheel coupling between epicycloidand hypocycloid”, we may assume thathypocycloid is located on left and epicycloidon right. There are two part:
Paper Home Part 1
ApplicationsConstruction 15:
Tooth-Wheel Coupling Between m-cuspedhypocycloid 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 Home Part 2
ApplicationsConstruction 16:
Tooth-Wheel Coupling Between m-cupsedhypocycloid 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 Home Sliding
Applications-slidingWe will discussion the phenomena of“ ellipse sliding in deltoid” and “ deltoid Circumscribing an ellipse”. First, wediscussion (m-1)-cusped hypocycloidsliding inside m-cusped hypocycloid.Here ,when m=3, the construction leads toa segment sliding inside deltoid.
Paper Home Ellipse sliding in deltoid
Applications-sliding
Now we use the ellipse instead of the
segment and the ellipse still sliding in
deltoid.Method 1 Method 2
Paper Home Next
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 Home Deltoid circumscribing an ellipse
Applications-sliding
Now we also use the ellipse instead of the
segment and the deltoid still circumscribes
the ellipse.Method 1
Method 2
Paper Home Fermat point
Applications-Fermat pointIf equilateral triangles ABR,ACQ,BCP aredescribed 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 Fermatpoint with compass alone, we used the
property that AP, BQ, CR meet at 1200.Construction
Paper Home Bibliography
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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