To begin with, we recognize the physical foundation of the mirascope. To see an image coming out of the mirascope, there has to be some light from the environment that reaches the object inside and is somehow bounced back to the outside. Therefore, we need to understand and model how light works and further explain how the human eye could recognize the position of a virtual image. For our purpose, there are two types of mirrors that are interesting: a plane mirror and a parabolic mirror.
In the case of a plane mirror, there are two steps involved when the human eye sees a virtual image of an object in the mirror. First, there is some light that reaches the mirror and is reflected by its reflective surface to the eye, which allows the human observer to notice the image, not necessarily where it is in the mirror. Second, since there are multiple light rays that are reflected to the eye, the observer identifies the unique location of the image by the common intersection of these light rays in a process that can be called triangulation (Figures 2a, 2b). These two steps can be readily simulated using a dynamic tool like GeoGebra. Such a simulation allows us to understand how plane mirrors work, including the basic idea of geometric reflection. For simplicity, we assume, in all our examples, that the reflective surface of the mirror is coated on top of the glass, which is the case of a plastic mirascope toy. In other cases, the reflective surface is normally behind the glass and should be dealt with accordingly.
Figure 2a: How the human eye sees an object in a plane mirror.
A parabolic mirror is different than a plane mirror in that it has a curved surface which reflects incoming light to different directions, depending on the curvature at a specific point of the parabola. In a mathematical sense, when a light ray reaches a particular point of a parabolic mirror, it is reflected as if the light ray was applied to a plane mirror that is tangent to the curved mirror at that point. As shown in Figure 3, when a light ray leaves the object and reaches the parabolic mirror at point \(C\), it is reflected to the eye as if it was reflected by the tangent line \(b\). If so desired, the physical terms, such as point of incidence, the normal line, the angle of incidence, and the angle of reflection, could be discussed with students. It is worth noting that because of the lack of triangulation, the human eye in Figure 3 might not be able to see a clear image of the object, which is part of the reason that the mirascope requires two parabolic mirrors. However, when the object, the eye, and the focus of a parabolic mirror are aligned in some special positions, the eye may be able to see a clear and perhaps distorted image of the object or part of the object.
By its very nature, a parabolic mirror has a focus, toward which all incoming light rays parallel to its principal axis or line of symmetry are reflected (Figure 4a). Therefore, to locate the focus of a parabola, we can model at least two such light rays and find the intersection of the corresponding reflected light rays. This observation is useful when a parabola is defined algebraically and thus its focus is not immediately available.
It follows that if a parabola is defined in the form of \(f(x)=ax^2+c\), where \(a\neq 0\), its principal axis is the \(y\)-axis. We could use any two lines that are parallel to the \(y\)-axis to locate the focus of the parabola. An example is presented in Figure 4b, where the parabola is defined as \(p_u=ax^2\) with \(a>0\). We assume that two light rays parallel to the \(y\)-axis reach the upward parabola (Additional Figure 4c). Those two light rays are both reflected toward the focus, where they uniquely locate the focus of the parabola at their intersection (Figure 4b). Since the focus in our case is on the \(y\)-axis, there is, in fact, no need for a second light ray. Once the focus of the upward parabola is located, we may hide all the auxiliary objects in order to reduce the visual complexity of the construction, leaving visible only the focus \(F\) of the upward parabola (Additional Figure 4d). To hide any object in GeoGebra, we can right-click on it and toggle the "Show Object" property or toggle the radio button to the left of its definition.
Having analyzed the mirrors, we now have a better grasp of the mirage effect, which simply has to do with the way light travels and the properties of mirrors. However, we still do not have a starting point to build a model for it. In search of a starting point, we focus on the object itself. For the object at the bottom of the mirascope to show up at the top, there has to be some light rays that originate from the object and are somehow reflected to the eye. Although the light comes ultimately from another source such as the surrounding lighting or a small light bulb placed inside, we could start with an arbitrary point on the object to investigate the mathematical and physical processes behind a mirascope. Our assumption is that if we could see one point of the object, we could apply the same procedure all over the object to produce the whole image.
With such a starting point in place, we still need to model the two parabolic mirrors. Given our purpose and the complexity of 3-D modeling, we choose to model the mirascope using a two-dimensional approach or rather, a cross section of the 3-D mirascope. Accordingly, we need two parabolas. While a parabola can be constructed geometrically, an exclusively geometric approach is not well suited for our modeling task since we want to align the two parabolas with high accuracy. Furthermore, in building a dynamic model, we wish to leave room for open-ended exploration, which in our case may include changing the curvature of the parabolas and/or their vertical alignment. This analysis points to an algebraic way for constructing the two parabolas. Because the specific locations of the parabolas are not essential in this case, we could use a simple algebraic form: \(f(x)=ax^2+c\), where \(a\) controls the orientation and curvature of a parabola, and \(c\) moves it vertically for alignment. To facilitate subsequent open-ended explorations, we choose to use sliders for the initial parameters, \(a\) and \(c\), of the two parabolas. These sliders are simply visual representations of the two parameters and are shown as horizontal or vertical lines on the screen. Note that the symmetric shape of the mirascope dictates that if \(a\) is positive for the parabola that opens up, \(-a\) would be used for the one that opens down.
The above analysis is an essential phase of problem solving . Although it takes time and may take many iterative steps, it unveils the underlying structure of the mirascope, clarifies the interdependent relations, and ultimately helps us devise a plan of action. In my teaching experience, I found that this step could be challenging for some prospective and classroom teachers, who tend to take the problem at the surface level and thus create a mere visual replica of the given problem using a quadratic function \(f(x)=ax^2+bx+c\) without explicitly defining the mathematical relations among the integral components. Therefore, in spite of the technical utilities and representational resources of a dynamic environment such as GeoGebra, the problem solver still has to make sense of the problem before he/she could come up with a preliminary plan of action. With the analysis above, we now proceed to the construction stage, during which we may be able to learn more about the problem itself and revise our plan.