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Second Reflection Images

The images considered on the previous page are the common reflections seen in a single mirror. It is the presence of another mirror that produces the infinity of images that are the hallmark of the kaleidoscope.

Consider the two light rays in the sketch below. The light originates at the center object and moves toward the bottom mirror. As we observed previously, the light reflects at the mirror's surface. But, the light paths do not end with a single reflection.

        2nd reflection image of object A.  Ray hits mirror 1 and then mirror 2.            

After moving away from the bottom mirror, the light reflects off the top right mirror. This second reflection gives rise to a second virtual image.

The outgoing light rays diverge. We extend them behind the mirror to locate the virtual image, marked at the intersection of the ray extensions.

       

The bottom left hand corner object produces an image from a bottom to upper right mirror reflection pair. This image is located via the ray diagram to the immediate right.

            2nd reflection image of object B.  Ray hits mirror 1 and then mirror 2.

       
        2nd reflection image of object C.  Ray hits mirror 1 and then mirror 2.            

Restricting ourselves to light paths that begin at the object and subsequently reflect from the lower to the top right mirrors, we locate our third object's virtual image. The rays used for this are shown in the figure on the left-hand side.

We summarize the findings from the bottom to right reflection pairs in the following schematic. The original flower petals are in the left triangle with the heavy perimeter, while the virtual objects are within triangle to the far right.

            Summary of first image set from 2nd reflection, mirror 1 to 2.        
        2nd reflection image of object A.  Ray hits mirror 1 and then mirror 2.            

Our next step is to consider the bottom to upper left mirror reflections. As before we start with the center object.

By symmetry we anticipate that the image would be found to the left of the object, as demonstrated in the ray diagram.

The lower left hand object has an image that appears on the left as well. The precise location of the image is best found by drawing a picture.

           

2nd reflection image of object B.  Ray hits mirror 1 and then mirror 2.

       
        2nd reflection image of object C.  Ray hits mirror 1 and then mirror 2.            

Our second set of second reflection images is nearly complete. The image location of the circle object in the upper section of the triangle is determined in the ray diagram that follows.

       

A summary of the images produced by the bottom to upper left mirror reflections is provided at right.

            Summary of second image set from 2nd reflection, mirror 1 to 2.        

Summarizing the second reflection images from the bottom (red) mirror we draw the following sketch where the objects are in black and the images appear in red.

Color coded summary.  Second reflection images are in red.

It is instructive to combine all first and second reflection images from the bottom mirror in a single figure. The objects are in black, first reflection images are in gray, and the second are in red.

Color coded summary.  First reflection images are gray, second are in red.

One can apply the procedure presented above for the bottom mirror second reflection images two more times to obtain the full set of second reflection kaleidoscope images. We present the result of this exercise below.

2nd reflection images.  Shading matches images with the first mirror reflection.

Note that original objects and first reflection images appear in black and gray, respectively. The bottom mirror and corresponding second reflection images obtained above are shaded in red. Blue and yellow-green are used to designate the upper left and right mirrors with their second reflection image sets.

This object-image template is used to create the actual kaleidoscope picture. They are shown together in the next figure.

1st and 2nd reflection image orientations.

Finally, we eliminate the template to reveal the anticipated kaleidoscope image produced by the object and first and second reflection images.

Summary of images from 1st and 2nd reflections.

Third reflection images will complete a hexagon, the pattern that is familiar to those who have looked through a 60°-60°-60° kaleidoscope.

Go to third reflection images.

©2004-2012 4Physics®




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