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Unity via diversity 81 Unity via diversity Unity via diversity is a concept whose main idea is that every entity could be examined by different point of views. Many sources name this conception as interdisciplinarity . The mystery of interdisciplinarity is revealed when people realize that there is only one discipline! The division into multiple disciplines (or subjects) is just the human way to divide-and-understand Nature. Demonstrations of how informatics links real life and mathematics can be found everywhere. Let’s start with the bicycle – a well-know object liked by most students. Bicycles have light reflectors for safety reasons. Some of the reflectors are attached sideway on the wheels. Wheel reflector Reflectors notify approaching vehicles that there is a bicycle moving along or across the road. Even if the conditions prevent the driver from seeing the bicycle, the reflector is a sufficient indicator if the bicycle is moving or not, is close or far, is along the way or across it. Curve of the reflector of a rolling wheel When lit during the night, the wheel’s side reflectors create a beautiful luminous curve – a trochoid . 82 Appendix to Chapter 5 A trochoid curve is the locus of a fixed point as a circle rolls without slipping along a straight line. Depending on the position of the point a trochoid could be further classified as curtate cycloid (the point is internal to the circle), cycloid (the point is on the circle), and prolate cycloid (the point is outside the circle). Cycloid, curtate cycloid and prolate cycloid An interesting activity during the study of trochoids is to draw them using software tools. We are not interested in demonstration-only software where trochoids are pre- built. Although this type of software assists a quick start for the student, it does not always correspond to a successful and effective learning. That is why we prefer to use programming environments where students explore mathematical topics through programming. For our experiments we will use Elica [1] – a free implementation of the Logo language which supports graphics, animation, and Object Oriented Programming. Let us start with the simplest trochoid – the cycloid. There are various approaches to construct a cycloid in Elica . Depending on the circumstances one or another of the approaches could be more appropriate in terms of learning. Method 1 – a parametric approach The first implementation is to use the parametric equation of the cycloid: x = Rθ − Rsin θ Cycloid (R,θ) = , y = R − Rcos θ where R is the radius of the rolling circle and θ is the parameter for the angle of rotation. An Elica program which draws a cycloid uses the equation to calculate x and y values of a specific point of the cycloid. For each pair (x, y) a point will be drawn on the screen. The variable R is the radius of the circle and the locus is a set of points generated during the iteration. Unity via diversity 83 Drawing a cycloid with Elica To draw the other two kinds of cycloids we introduce scale factors as follows: 0.5 for the curtate cycloid (the tracing point is inside the circle) and 2 for the prolate cycloid (the tracing point is outside of the circle). Drawing a curtate cycloid in Elica Drawing a prolate cycloid in Elica 84 Appendix to Chapter 5 We can decompose the general equation of a trochoid into two separate parametric equations representing the linear and the circular movement, respectively x = Rθ − r sin θ Trochoid (R,r,θ ) = y = R − r cos θ x = Rθ x = r sin θ Translatio n(R,θ ) = Rotation (r,θ ) = y = R y = r cos θ The parametric approach (as we shall call it) is straightforward; it gives quick results and is highly applicable to various types of curves. Method 2 – a constructive approach Although the parametric approach is easy to use, it is based on the knowledge of a specific formula. Some curves have rather complex parametrical representations. Although it is not very hard to program curves by their formalae this approach does not necessarily involve a real understanding of the properties of the curve. Thus the parametric approach is good mainly for demonstration and initial introduction. This is the reason we shall extend the cycloid study with other methods. Another possible approach is to make geometric constructions by describing the relations between some objects after which the locus of the curve is obtained ‘automatically’. For this purpose we can use a library in Elica called Geomland (a successor of standalone educational software Geomland considered in Chapter 5). Descriptive cycloid construction with Elica Geomland The program which generates the cycloid as a set of points describes the construction of a specific point of the cycloid. R is the radius of circle K, a is the angle of rotation. Point O is the center of K. Segment s starts from O and ends on the circle. The final Unity via diversity 85 point of s is point T which will trace the cycloid. The length of the arc covered by T at a certain moment T equals the length of the segment covered by O during s its movement parallel to the straight line. Until this moment we have only declarations about objects O R relations. The actual construction happens when the parameter a is being changed (see the command K repeat ), and the consecutive positions of T are collected in a set and displayed. The pointon function gets a point on a circle at a specific location, defined by an angle. The segment s is used only to enhance the animation effect of the rolling. Heterogeneous approach For curtate and prolate cycloids it is easy to find where to insert the scaling factor. The constructive approach follows the path of constructionist’s thinking – by making the construction of the curve themselves the students could get a deeper understanding of its intrinsic properties. This method could be applied to many of the mathematical problems in school and in real life. Method 3 – a transformational approach One of the main criteria when choosing a software platform for mathematical explorations is the flexibility. Elica provides a broad choice of techniques which students and their teachers can use. Users can create and use their own graphical libraries with custom notions and methods. Elica by itself has no graphical support – it is the external Elica -based libraries which define the graphical microworld which will be used. Now we will demonstrate another approach which is based on transformations. A geometrical world could be seen as a theory in which the basic canonic geometrical 86 Appendix to Chapter 5 objects correspond to axioms. Any other object could be generated by applying transformations to the canonicals. There are 9 basic transformations in the space – 3 translations, 3 rotations and 3 dilations. For example, the canonical sphere of radius 1 and center at the origin (0,0,0) can be transformed into an ellipsoid somewhere in the space oriented towards some other points. This is the essence of the transformational approach. In our case we take a point at (0,0,0) (the canonical object) and apply several basic transformations until we ‘plant’ the point on the cycloid. Initial and target points The first step is to move our point from a center of a circle to the circle itself. We do this by applying translation into y direction. The next step is to roll the circle. The result will be that the point which was at the ‘top’ will move along the circle to the desired angle. Building and rolling a circle Then we move the circle up until it ‘stands’ on the horizontal axis. This is again translation along y direction at distance R. Now we are ready to move the circle to the correct position to be taken after rolling. Unity via diversity 87 Placing the circle on the ground and rolling it forward (the horizontal translation is exaggerated for clarity) What we finally got is a point, which is on a circle rolled along a line at a given distance. A question may arise. Why do we move the point up twice? Could we do it once doubling the translation distance? The short answer is ‘no’. The long answer is: Rotation is applied when the circle center is at (0,0,0) because the basic rotation is around point (0,0,0) and our point should be rotated around the circle center. Cycloid as a result of transformations The instructions we have used in the Elica program are movex (translation in x direction), movey (translation in y direction) and spinz (rotation around z direction). Note that these instructions are used in reversed order because the transformations follow the mathematical notation, viz. if T1, T2 and T3 are transformations, O is an object, then the application of T1 then T2 and T3 is denoted by T3(T 2(T 1(O))) or T3T2T1O where the transformations are written in reversed order. 88 Appendix to Chapter 5 The transformational power (part I) The transformational approach as used so far does not reveal its true power. So let us make the problem with the cycloid a little bit harder. What if the ground is not flat but curved? What if the bicycle with wheel reflector is used by the Little Prince ? The Little Prince biking on his small planet Antoine de Saint-Exupéry was an aviator. Probably he knew that curves made by points on a circle rolling on the outside of another circle are called epitrochoids . Probably he even knew the parametric equation: R + r x = (R + r) cos θ − h cos θ r Epitrochoi d(R, r, h,θ ) = R + r y = (R + r) sin θ − h sin θ r Following the parametric approach students can just type in the equation and see the curve.
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