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AUTHOR Chuan, Jen-chungTITLE Geometric Constructions with the Computer.PUB DATE 1995-12-00NOTE 23p.; Lecture given at the ATCM (1st, Singapore, December

18-21, 1995).PUB TYPE Guides - Classroom Teacher (052) -- Speeches/Meeting

Papers (150)EDRS PRICE MF01/PC01 Plus Postage.DESCRIPTORS *Computer Uses in Education; Elementary Secondary Education;

*Geometric Constructions; Mathematics Activities;*Mathematics Instruction

ABSTRACTThe computer can be used as a tool to represent and

communicate geometric knowledge. With the appropriate software, a geometricdiagram can be manipulated through a series of animation that offers morethan one particular snapshot as shown in a traditional mathematical text.Ge:.atc:tric constructions with the compater enable the leaner v., see andunderstand a diagram,,in different ways. Engaging in the construction of theanimation encourages the learner to go through the abstract process offormulation of conjectures, generalization, condition-simplification, andclassification. This paper offers examples of such constructions on topicssuch as enveloping curves, linkage, polynomial interpolation, inversion,hypocycloid, and epicycloid. (ASK)

Reproductions supplied by EDRS are the best that can be madefrom the original document.

Geometric Constructions with the ComputerLecture given at the First ATCM, December 18-21, 1995, Singapore

Jen-chung Chuan

Department of Mathematics

National Tsing Hua University

Hsinchu, Taiwan 300

jcchuan@math.nthu.edu.tw

1 IntroductionComputer can be used as a tool to represent and communicate geometricknowledge. With the appropriate software, the geometric diagram can bemanipulated through series of animation that offer more than one particularsnapshot as shown in a traditional mathematical text. Geometric constructionswith the computer enables the learner to see and understand a diagram indifferent ways, Engaging in constructing the animation encourages the learnerto go through the abstract process of formulation of conjectures, generalization,condition-simplification and classification. In what follows we shall offer thevisual experience of geometric animation by guiding the audience throughdemonstrations of geometric construction from topics in enveloping curves,linkage, polynomial interpolation, inversion, hypocycloid and epicycloid.

2 EnvelopWhen a system of lines or curves expressed in the form of an algebraicequation, it is not clear immediately if there is another curve touching everymember of the system. A precise drawing allows the learner to perceive thisnotion of enveloping curve visually. Interesting enveloping curves occur whenthe system obeys certain common geometric conditions. It is thus desirable togenerate the curve by means of the direct geometric notion. Consequently adynamic geometric environment is particularly suitable for such exploration.

An enveloping curve may appear when only straight line segments are drawn:

1

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In this case all segments have the same length and have endpoints lying on twomutually perpendicular lines. The same envelop may appear as a curvetouching a family of ellipses:

All ellipses have the same major axis, the same minor axis, and the same sumof distances from the foci. This astroid also appears as a part of envelopingcurve associated with a family of circles:

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3 LinkageAt MIT library one can retrieve a massive book that weighs over 3 kilograms andwas published in 1951 in fulfillment of an MS thesis. The title of this dinosaur iscalled "Analysis of the Four Bar Linkage".

The four-bar linkage is one of the simplest mechanism and may be regarded asa basic mechanism. The equations which describe the displacement, velocityand acceleration of points on the connecting rod are long and difficult to handle,making mathematical synthesis a very tedious process which few designers willundertake. The publication claims to includes over 7000 displacement pathswith the velocity given at 72 equal intervals of drive crank angle along each pathand therefore represents about 500,000 solutions of interest to engineers.

Geometric construction can be adopted to simulate the movement of a linkage.We now show how to construct the Fermat linkage, a particular four-bar linkageused to draw a lemniscate of Bernoulli.

1) Take an equilateral right triangle ABC:

A0

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C2) With B as center, draw a circle passing through A and then take a point Dconfined on the circle:

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3) Find the point P so that CBDP is a parallelogram.

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C

4) Construct the reflection Q of P with respect to the line CD:

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5) Connect the line segments CQ,QD,DB. These are the bars in the linkage:

P0

6) Make the point D travel along the circle and then hide the circle, the linesegment CD as well as the points A and P:

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Animate

CQ

7) This is the locus of the mid-point of the segment DQ, when the Animatebutton is pressed:

Animate

4 Polynomial Interpolation

Suppose that we are given points (x1,y1),(x2,y2),...,(xn,yn) in the coordinate

plane with the xi distinct. The problem of polynomial interpolation asks for a

polynomial function whose graph passes through these points.

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Of the many methods of solution to this problem, the ones suggested by Aitkenand Neville can be most readily implemented under the dynamical geometricenvironment. The principle behind constructing the graph of the interpolatingpolynomial is the following lemma of Aitken:

Lemma. Suppose that the functions f and g satisfy f(x) = g(x) for x E T. If a #then the function

(b-x)f(x)+(x-a)g(x)h(x)

b-a

satisfies f(x) = g(x) for xE T, h(a) = f(a), h(b) = g(b).

Let xi, x2,IR xn be distinct points. Let y1, y2, yn be given points. For each

nonempty subset Tof {1,2,...,4 let pT be the (unique) polynomial of degree

1-11-1 satisfying pT(xi) = yi for jE T. If 15171 = 1 = , Aitken's lemma shows ushow to produce pa)T from ps and pT. The required interpolating polynomial

P{1,2,42 n}=P12a.n can be computed from this table:

P1 P12 P123 P1234 P12345

P2 P13 P124 P1235

P3 P14 P125

P4 P15

P5

with the obvious initial data po1 j/1 for j= 1,2,a n.

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The preceding procedure may be altered by finding the interpolating polynomialP12 building through this table:

P1 p12 p123 p1234 p12345

P2 p23 p234 p2345

P3 p34 p345

P4 p45

P5

5 InversionTake two circles, one inside another:

1.`2

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It is not always the case that one can find a series of circles kissing each otherso precisely as thus:

Steiner's porism states that if such situation arises once, it will occur infinitelyoften:

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The principle behind the animation of Steiner porism is inversion. We begin withthe particular situation

and then invert the configuration with respect to another circle which is to behidden. As this symmetric figure rotates, the resulting inverted figure, obeyingthe law that tangent circles be inverted to tangent circles, achieves the desiredresult. The method of this demonstration actually guides us to the proof ofSteiner's porism! Just invert the two disjoint circles into concentric ones. Theconfiguration of kissing circles close up if and only the inverted figure issymmetric. QED.

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6 Epicycioid and Hypocycloid

An epicycloid is the curve traced by a point in the circumference of a circlerolling outside a fixed circle. A hypocycloid is the curve traced by a point in thecircumference of a circle rolling inside a fixed circle. These curves possessmany remarkable properties that fascinate both mathematicians andnonmathematicians alike. It is possible to explore the magic and beauty of thesecurves through purely geometric constructions. Computer makes it possible forthe learner to be exposed to the geometric approach before hooked up toanalytical and algebraic methods.

We now show how to generate the epicycloid pictorially:

Starting with a circle with "the point of contact" attached.

Next, locate the center of the rolling circle.

Draw the rolling circle.

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Construct the "moving point" by suitably rotating the point of contact w.r.t. thecenter of the moving circle.

This is the result of pressing the "Animate" button:

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The animation of a hypocycloid is created through a similar process:

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Animate

There is a whole host of computer activities related to epicycloids, hypocycloidsby drawing the series of line segments joining various points of the curve to thecorresponding center of curvature.

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7 Double Generation

Every hypocycloid and every epicycloid can be generated in two ways. Considerthe case the hypocycloid traced by a point on the circumference of a rollingcircle:

This rolling circle determines another circle rolling in the opposite directioncontaining the same fixed point on its circumference that traces the hypocycloid.It is difficult to perform a dynamic demonstrate the double generation by meansof mechanical devices. With the assistance of computer, however, the

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animation of such phenomenon can be vividly shown.

Not convinced just by looking at these pictures? Then you have got to see thedemonstration.

References

1. H.S.M. Coxeter, Introduction to Geometry, Wiley, New York, 1961.

2. J. A. Hrones and G. L. Nelson, Analysis of the Four Bar Linkage, Wiley, 1951.

3. E.H. Lockwood, A Book of Curves, Cambridge University Press, 1961.

4. K. Rektorys (Editor), Survey of Applicable Mathematics, Illiffe, London, 1969.

5. J. Todd, Survey of Numerical Analysis, McGraw-Hill, New York, 1962.

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