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

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Introduction

In Mohr-Mascheroni geometry of compass

proved that every Euclidean constructions

can be carried out with compass alone.

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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.

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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

Lemma 3.Construct a circle determined by

radius and center.

Construction

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Construction 4.Given three points A,B,D, to

completethe parallelogram ABCD.

Construction

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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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Lemma 6.Construct a segment n times the

lengthof a given segment, n=2,3,4,… .

Construction

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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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Lemma 8.Construct the sum and difference of

twogiven segments.

Construction

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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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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

ConicsConstruct the inverse of lemniscate.

Construction

Paper Home Parabola

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

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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:

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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

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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