Transcript Geometric Constructions With the Compass Alone

Geometric Constructions With
the Compass Alone
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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
 Tooth –wheel coupling between epicycloid and
hypocycloid
 Ellipse sliding in deltoid and deltoid
circumscribing an ellipse
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Next section-Introduction
Introduction
In Mohr-Mascheroni geometry of compass
proved that every Euclidean constructions
can be carried out with compass alone.
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Next section-Tools
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.
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review tools
Tools
Lemma 1.
Construct a point, symmetric to a given
point with respect to the given straight
line.
Construction
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Tools
Lemma 2.
Construct a perpendicular to the
segment 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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Tools
Construction 4.
Given three points A,B,D, to complete
the 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 with
respect to C.
Construction
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Tools
Lemma 6.
Construct a segment n times the length
of a given segment, n=2,3,4,… .
Construction
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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
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Tools
Lemma 8.
Construct the sum and difference of two
given segments.
Construction
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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.
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Next tool
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
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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
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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.
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Cycloids
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)).
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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.
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Lemniscate
Curves construction
Lemniscate
Method 1: construction based on “Kite”
linkage
Method 2: Construction based on 3-bar
linkage
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Conics
Curves construction
Conics
Construct the inverse of lemniscate.
Construction
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Parabola
Curves construction
Parabola
The center of inversion coincider with the
cusp, the inversion of cardioid is a
parabola with focus at the cusp.
Construction
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Ellipse
Curves construction
Ellipse
Construction following parameter
coordinates of ellipse and trochoid.
Method 1
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Method 2
Applications
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
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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:
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Part 1
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.
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Part 2
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.
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Sliding
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.
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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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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.
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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
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Fermat point
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
0
property that AP, BQ, CR meet at 120 .
Construction
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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.
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