15TH INTERNATIONAL CONFERENCE ON GEOMETRY AND GRAPHICS ©2012 ISGG1-5 AUGUST, 2012, MONTREAL, CANADA

THEOREMS AND CONSTRUCTIONS CREATED USING DYNAMIC GEOMETRY SYSTEM GINMA

Vladimir SHELOMOVSKIIMurmansk State University, Russia

ABSTRACT: Geometry is a powerful tool for the development of creative thinking and ability to see the total in the seemingly unrelated phenomena. The most powerful methods of geometry, such as inversion, projective transformations of space are beyond the knowledge of most of today's students. Creation of new dynamic geometry systems such as GInMA allows us to return the wealth of methods in the area of available knowledge. The paper assumes the demonstration of the classical problems of geometry and new constructions and theorems, creation of which is associated with the capabilities of visualization. This became possible due to capabilities of GInMA program which allow us to create complicated surfaces and curves in space and on a plane, to perform such transformations as the construction of the caustics and orthotomic surfaces, to perform the projective transformations associated with the transition from a plane into space, to execute transitions to the limit and the construction of the solids of intersection of the surfaces in space. It contributes to the implementation of such complicated calculations, as the calculation of the volumes and the surface areas of the solids of intersection of quadrics. GinMA illustrations provide an opportunity to enter into an interactive image directly from a typical text file of the electronic textbook or scientific paper. This feature helps to make it more convenient attraction of computer technology to the studying, contributes to a deeper understanding of geometry. All the figures from this paper have the appropriate interactive files in GInMA Library, accessed if your computer has installed GinMA; you can download demo from www.deoma-cmd.ru

Keywords: Dynamic geometry system, computer algebra system, inversion, projective transformations, orthotomic, caustic.

1. INTRODUCTIONGeometry is a powerful tool for the development of creative thinking and ability to see the total in the seemingly unrelated phenomena. Geometric methods are often stronger than algebraic methods. The most powerful methods of geometry, such as inversion, projective transformations of space, affine transformations are beyond the knowledge of most of today's students. Creation of new dynamic geometry systems such as GInMA allows you to return the wealth of methods in the area of available knowledge. Experience of teaching math with GinMA is described at [3]. GInMA demo version is accessed at www.deoma-cmd.ru

GInMA provides complicated geometric constructions and theorems, creation of which is associated with the capabilities of visualization.

2. CONSTRUCTIONS WITH GINMADeep study of the geometry requires imagining the objects with which it deals. Dynamic Geometry System GInMA allows creating and exploring geometric constructions including the most complicated ones.

2.1 Lively curvesTo study the curves and their properties it is important to see the dynamics of the curve modifying when the parameters change and to

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15TH INTERNATIONAL CONFERENCE ON GEOMETRY AND GRAPHICS ©2012 ISGG1-5 AUGUST, 2012, MONTREAL, CANADA

check the geometrical phenomena that describe the curve. It is important to see the connection of the geometrical phenomena and the algebraic relations. GinMA provides us with this opportunity thanks to the easy change of the algebraic expression parameters defining the curve.

Figure 1: Algebraic curve

Figure 2: Lissajous curve

Power of GInMA graphics apparatus shows Hyrotaka Ebisui curve, for which construction forty thousands points are required.

Figure 3: Hyrotaka Ebisui curve

Constructing of evolute, evolvent, pedal curve and orthotomics develop the imagination.

Figure 4: Evolute for curves

Figure 5: Evolvent for curves

Figure 6: Pedal curve and orthotomics

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15TH INTERNATIONAL CONFERENCE ON GEOMETRY AND GRAPHICS ©2012 ISGG1-5 AUGUST, 2012, MONTREAL, CANADA

The most complicated in this group of the curves is the caustic curve.

Figure 7: Caustic curve

Even seeing the image it isn’t easy to understand how the circle evolvent transforms into the spatial helicoid.

Figure 8: Helicoidal surface

It is useful to imagine not only the plane curves but also the curves lying on different surfaces, for example on the torus.

Figure 9: Torus curves

The formation of the curves in space is interesting in the cases of limit transformations. As an example we can show the construction of

the stereographic projection for the curve of intersection of the shrinking sphere and the ellipsoid.

Figure 10: Stereographic projection for the curve of intersection of the shrinking sphere and the ellipsoid

2.2 Lively surfaces

Sometimes it is difficult even just imagine three-dimensional surfaces and complicated solids in geometric problems. Visualization using the computer is very helpful to understand the complicated problems, but the computer program should have a powerful computational tool to make the most vivid construction and at the same time to provide precise and correct solution of the problem.

Figure 11: Icosahedron and a sphere

Let us consider an icosahedron and an inscribed sphere. Figure 11 demonstrates the accuracy of the GinMA CAS calculations, which provides synchronization of changes in all intersection curves when the sphere size changes.

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15TH INTERNATIONAL CONFERENCE ON GEOMETRY AND GRAPHICS ©2012 ISGG1-5 AUGUST, 2012, MONTREAL, CANADA

Without good graphics it is difficult to understand, for example, the theorem about the equivalent replacement of the rotating region.

Let the plane region F be placed on the plane perpendicular to yOz plane in Cartesian coordinate system, the plane region F’ is the orthogonal projection of F on the plane xOz. Then the volumes of the solids of revolution obtained by the rotation of the regions F and F’ about the line Oz are equal.

Figure 12: Replacement of the rotating region

The same is true for the theorem about the replacement of the rotational axis.

Let the plane region F be placed on the plane having a slope to the axis of rotation .ϕ Then the volume of the solid of revolution obtained by the rotation of the region F about the rotational axis is equal to the volume of the solid of revolution obtained by the rotating of the same region F about the orthogonal projection of the rotational axis onto the plane containing the region F, multiplied by .cosϕ

Figure 13: Replacement of the rotational axis

These theorems are used for the solutions of such problems as, for example, the search of the volume of the solid of revolution of the cube using different methods.

Figure 14: Solid of revolution of the cube

GInMA graphics helps to present the complicated solids of intersection, for example, a sphere intersecting an ellipsoid.

Figure 15: Solid of intersection of a sphere and

an ellipsoid

It is quite hard to imagine the solid, which is a result of cutting in a cube a hole, such that the second cube equal to the given one can pass through this hole.

Figure 18: Cube with a hole through which the identical cube passes

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GInMA comes to help when the students try to imagine eight spheres touching the planes of the tetrahedron faces or eight tetrahedra touching in pairs by the pieces of its surfaces with non-zero areas.

Figure 16: Tetrahedron’s spheres

Figure 17: Eight contiguous tetrahedra

2.3 Extreme curvesIt is known that the bifurcation of solutions is an important phenomenon which can make a good decision as unfit, which accompanies the investigation of aircraft accidents and less critical of engineering errors. It is convenient to study bifurcation with the help of the models.

For example, we can use the models of the shortest polygonal path inside the triangle or a quadrangle, a tetrahedron or a cube, that contain points located by one on each side of the polygon or on each face of the polyhedron.

Figure 19: The shortest polygonal path inside the quadrangle

Figure 20: The shortest polygonal path inside the tetrahedron

Figure 21: The shortest polygonal path on the surface of the cube

Figure 22: The peach in the beaker

GInMA is an indispensable tool for the

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construction of the Meissner solid.

Figure 23: Meissner solid

GInMA is also a tool for the construction of any solid of constant width.

Figure 24: Solid of constant width construction

Figure 25: Solid of constant width (another type)

3. SOLUTIONS USING GINMA

Geometers appreciate the beautiful geometric relations. Classic computational methods don’t always contribute to take notice such relations. The observation of the dynamics of the geometric configuration allows us to notice some regularity, which after the proof could become the statement and sometimes the theorem. After the conference ATCM 2011 when the program GInMA became widely known to mathematicians, the author have had the contacts in which mathematicians suggested to explore some geometric constructions using GInMA. Let us consider some of these problems.

3.1 Finding of the Rotational homothety

Zilacci Lajos, Hungary have found the solution of the problem concerning the construction of the square if its vertices are placed on the given lines. But he needed in solution of the following related problem.

Problem 1 Let points GED ,, are placed on the lines ,,, BCABAC respectively and

DEFG be a square. Prove that the locus of point F is a straight line.

Solution: The author performs the construction using GInMA, move the lines, the square, and find that the following statement is true.

Let −H point lie on AB and CBHC ⊥ . Let −K point lie on BC and .AKBA ⊥ Let −'K point lie on ВC and .' CHCK = The

circle ω is tangent to AK at point K and contains −'K point. Let O be a point of the intersection of the circle ω and the circle

ACKH . HK is the diameter of this circle. Then .' ACOOCKAKOAHO ∠=∠=∠=∠

Let .'OGKCODHOE ∠=∠=∠ So,, CODHOE ∆∆ and OGK '∆ are similar.

Hence, the rotational homothety transforms H into ,C C into ,'K E into D and D into .G CKHC ⊥ and .CKHC = So

DGED ⊥ , and .DGED = The fourth vertex of the square is obtained from the three given

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vertices by using the linear transformation and also lies on the straight line.

Figure 26: Solution to Problem 1

3.2 Transformations of the space

There are several methods for performing the constructions and proofs based on the transformation of plane or space. These methods use the transformation of the geometrical configuration into the trivial configuration, the performance of the solution for the trivial configuration and the return to the original configuration. Widespread adoption of these methods is hampered by the lack of dynamic models, such as converting an arbitrary triangle into an equilateral triangle. GInMA program has three such models, see Figures 27-29 below.

Figure 27:Transformation of a plane converting an arbitrary triangle into an equilateral triangle

Figure 28: Projection of the plane converting an arbitrary triangle into an equilateral triangle

Figure 29: Central collineation converting an arbitrary triangle into an equilateral triangle

M. Sejfried, Poland had found the solution of some problem concerning the construction where three segments intersect on the incircle [3]. But he had the difficulties with the building of this construction. So the problem aroused:

Problem 2 Let triangle ABC be given, point 0A lies on 0, BBC lies on 0,CAC lies on

,AB 0AA intersect 0BB at 0, AAF intersect 0CC at 0, BBH intersect 0CC at .K Find

point 0A such that points ,, HF and K lie on the incircle of .ABC

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15TH INTERNATIONAL CONFERENCE ON GEOMETRY AND GRAPHICS ©2012 ISGG1-5 AUGUST, 2012, MONTREAL, CANADA

Solution: The solution have been performed using standard transformations which are easy fulfilled by GinMA, see Figure 30.

Figure 30: Solution to Problem 2

3.3 Geometric interpretation

Here we present an example of the solution where the algebraic solution of the geometric problem concealed the geometric interpretation of the solution. Let us consider the problem [4].

Figure 31: The smallest sum of the squares of the distances

Problem 3 Given a set of smooth surfaces that

do not have common points. Point A is located on one surface; points ,...,CB are located on the other surfaces (by one on each surface). Find such a position of points that the sum of the squares of the distances from the point A to the other surfaces was the smallest.

Solution: It is obvious that ,...,CB are the bases of the perpendiculars dropped from A to the other surfaces. But how can we find A ?

The analysis of the construction performed using GInMA allows us to see that the perpendicular to the surface containing A always passes through the center of gravity of points ,...,CB See Figure 31.

The analysis of [4] reveals that the authors have deduced all the necessary formulas for proving this statement, but this simple ratio remained outside of the review.

3.4 Dynamic visualization for the proofLet us discuss the use of dynamic visualization for searching the proof. In the beautiful geometric work of Maeda [1] we find construction of a circle perpendicular to two given circles, using the properties of the hyperboloid.

Figure 32: Common perpendicular to AB and CD lines in the Poincare model

We have built and explored construction from this problem using GInMA and noticed that the

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proof is drastically simplified if we use the following statement.

Problem 4 Let O be A projection onto the plane ', DDBCO is a common perpendicular to the lines AB and .CO The angle between

AB and CO is equal to .sin

',ϕ

ϕ DDDE =

Prove that °=∠ 90AEB .

Figure 33: Solution to Problem 4

Solution: Let us consider the view along .CO 'D and C coincide with ,O ,'' BDAD ⊥

'.DDAB ⊥ Hence, the product of AD and BD projections is equal to .'2DD The

segment projections are ϕsin times shorter than the segments themselves. Hence

,2DEBDAD =⋅ therefore °=∠ 90AEB .

3.5 Mirroring exploration

A situation similar to the previous one arose in finding the shortest path inside a regular tetrahedron. We performed mirror reflections of the tetrahedron with GInMA, rotated configuration and saw something that turned into a statement.

Problem 5 Prove that the regular tetrahedron and the sequence of its mirroring images are inscribed in a cylinder with the axis crossing the medians of the faces and dividing them at a ratio 3 : 2 from the vertex.

Solution: Distances from any vertex to the axis are equal to ,33.0 a where a is the edge of the tetrahedron.

Figure 34: Solution to Problem 5

3.6 Limits for integration in finding the volume of a solid of intersectionLet us consider the problems of finding the volume or the surface area of a solid having a complicated surface, for example, the solid of intersection of some surfaces. The solution requires integration with certain boundary conditions. The boundaries are the equations of the curves of intersection of the surfaces. Experience with calculation programs shows that such programs have serious difficulties and they can’t find the equation of the boundary in many cases. This equation can be found "by hand", but the error is likely to occur in this case. In addition, the solutions often change substantially when the parameters change in the condition. For example, an ellipsoid turns into a hyperboloid.

The issue of verification of the numerical results arises. As a rule, it is not easy to test the calculated values of the volume and the surface area by independent methods. Therefore, verification of the boundary conditions is important for the calculation of the integrals, or spatial curves. The author's practice shows that GInMA features for such control can hardly be overestimated

The human eye can easily note the falsity of the geometric construction, the disparity of the solid elements, the overlapping of the surfaces. However, this is easily done only in the dynamic mode, as the incorrect construction at first is often masked by bad view angle or the coincidence of control points. And only the

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15TH INTERNATIONAL CONFERENCE ON GEOMETRY AND GRAPHICS ©2012 ISGG1-5 AUGUST, 2012, MONTREAL, CANADA

ease of configuration changes quickly reveals the slips in the calculations.

Figure 34: Solid of intersection of an arbitrary quadric and a plane

4. CONCLUSIONSExperience in conducting research with the help of GInMA program show its effectiveness in the following cases.a. For students understanding the complex geometrical concepts (caustic, ortotomic, ...);b. For performing the transformations which allow us to make a variety of constructions or proofs;c. To create a text document in pdf, doc, odt, html file formats intended for teaching geometry and algebra. GInMA program is convenient to use in conjunction with calculation programs such as Maple in calculations for the searching the volume or the surface area of intersection solids with complicated boundary lines as well as for teaching students the methods of performing such calculations.

ACKNOWLEDGMENTSThe author is grateful to Svetlana Nosulya and Dmitry Lyzhin, without whose large and generous assistance the research and this paper could not be created.

REFERENCES[1] Y. Maeda. Dynamic Construction of the Common Perpendiculars in the Three-sphere. Proceedings of the ATCM 2011, pp151 - 160 2011

[2] M. Sejfried. Amicable triangles and perfect circles. Abstract ATCM 2011 for 18956. http://atcmmathandtech.org/EP2011/pages/abstracts.html#18956.[3] V. Shelomovskii, and S. Nosulya. Development of Creativity Using 3D Dynamic Geometry System. The Electronic Journal of Mathematics and Technology, Vol. 6, Num. 1, ISSN 1933-2823. [4] W.-C. Yang, Some Geometric Interpretations of The Total Distances Among Curves and Surfaces., Electronic Journal of Mathematics and Technology (eJMT), ISSN 1933-2823, Issue 1, Vol.3, June, 2009.

ABOUT THE AUTHORS

1. Vladimir Shelomovskii, Russia. www.deoma-cmd.ru

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