Fly-eye Sensors and Circles Description: Part of the research conducted in the Electrical and Computer Engineering Department at the University of Wyoming involves a sensor platform based on the function of a fly's vision system. The physical layout of the sensors involves extensive use of circle geometry. This lesson will use the sensor layout to facilitate numerous exercises and connection to geometry. This applies to high school geometry and/or general mathematics. Background: This lesson ideally relies on physically inspecting actual sensor prototypes. If the prototypes are not available, actual dimensions can be contrived. These sensors were constructed by electrical engineering students at the University of Wyoming in Laramie, WY. They mimic certain behaviors present in the vision system of the common housefly. Knowledge of electrical engineering is not necessary. This lesson is more concerned with using the way the sensor is built to lead a discussion about geometry. The worksheet leads students through an in depth analysis that uses a model of the sensor as a basis. A small collection of photos of the sensor are included below in the event that actual prototypes are not available. These images are property of the author and should not be redistributed outside the classroom or without this lesson plan. The rounded, yellowish components are the diodes, and the black lens housing is the cartridge. These are referred to in the included worksheet.
Concepts: The following concepts apply to this lesson: circle geometry (including inscribed circles and angles, central angles, concentric circles, tangent and secant lines, and arc length), caliper use, limitations of human measuring, model development, and potential model flaws. Vocabulary: These words are interesting and important words found throughout this lesson. Students may benefit from learning about these prior to the lesson, perhaps as a homework assignment. Arc Length the distance of a portion of the circumference of a circle. Calipers tools for precisely measuring distance. Central having to do with the midpoint (center) or a shape. In this case, angles that have a vertex at the center of a circle are referred to as central angles. Circumference the distance around a circle. Concentric sharing a center or central point. Diameter maximum distance from one point on a circle to another point on the same circle. These two points are 180 (π radians) apart. Diode any semiconductive electrical component. In this case, photodiodes are used. These measure the quantity of light present. Inscribed when one shape is completely containing within another, the smaller shape is said to be inscribed into the larger. Model an approximation of a system, device, or process that allows for a simplified analysis to occur. Models differ from the actual system, device, or process in varying degrees. Parallel Pair two lines that are parallel constitute one parallel pair. A single line may be part of more than one parallel pair. Radius one half of the diameter of a circle. This is the distance from the center to any point on the edge of the circle. Ratio a way of comparing one value to another. In this lesson, the challenge problem is asking students to develop a ratio. Secant Line a line that passes through a circle, touching that circle s edge at two points. The line continues one beyond the edge of the circle (whereas a chord terminates at the edges of the circle). Tangent Line a line that intersects a circle at only one point and continues on indefinitely. Vision sensor an electrical and/or optical device that provides data about the surrounds. This data can then be used to make decisions. Take Away Message: Circle geometry is applicable in many, many areas. Knowing how to compute and apply knowledge and information about circles is useful to advanced, cutting-edge research. Lesson: How is circle geometry applicable to advanced topics such as machine vision research? This lesson covers the following Wyoming State Math Standards and requires approximately 55 minutes. MA 12.2.1 12.2.2, and 12.2.3 Materials: Single-cartridge sensor prototypes, calipers (both are suggested but not required, the photos may be substituted for the sensor).
About the Author: Rob Streeter is a graduate student at the University of Wyoming. He is currently pursuing a MS of Electrical Engineering. A Wyoming native, Streeter appreciates the need for Wyoming high school graduates to be aware of the extensive opportunities available to them. He genuinely hopes this lesson inspires interest in engineering and the University of Wyoming. Any feedback on this lesson would be appreciated and should be directed to Mr. Streeter (rstreete@uwyo.edu), NSF GK-12 EE- Nanotechnology Fellow, University of Wyoming, Laramie, Wyoming. The following pages include a suggested worksheet and a solution to that worksheet. Values are provided for the student measurements. These values are approximate (measured by hand by the author), and when the worksheet is given to students (if the sensors are not available), values within a couple (1-2) millimeters can be given to each student (or each group of students) to provide a simulation of human measuring error. The calipers report in mm, but the worksheet deals with cm. The conversion between the two is an optional aside.
Name: The sensor models before you are one version of the sensor. Just by looking at it, can you see how circles may be important to the sensor? HINT: look at the layout of the photodiodes and the "cartridge." Sketch what you see and how the diodes are laid out. What is the relationship between the larger circle of the cartridge and the 7 smaller circles of the diodes?
Check your sketch on the previous page and make sure it looks similar to the figure below. If you know the diameter of the cartridge (the large outer circle) is about cm, what does that tell you about its circumference? Additionally, what does that diameter tell you about the diameters of the diodes (smaller circles)? What would the circumference of the diodes be? Use the digital calipers to check your diode diameter measurement. Does it match up? If not, why might it be different?
Now, consider the same circle layout from the last page and answer the follow questions. Is it possible to draw a tangent line of the larger circle than is also a tangent line of one or more of the smaller circles? If so, sketch it above, if not, why not? Is it possible to come up with an angle that is central to both the larger circle and one of the smaller circles? If so, sketch it above, if not, why not? Is it possible to come up with an angle that is inscribed to both the larger circle and one of the smaller circles? If so, sketch it above, if not, why not?
Now use the image below to sketch on. If you draw lines along the diameters of adjacent smaller circles, how many parallel pairs (any two parallel lines make up one pair) of lines can you develop? Draw them below. Of the lines you drew, how many are along the diameter of the larger circle? Those that weren't have a special classification or name, what is it?
Finally, consider the figure below. A C b B Assume that Point B is at the center of both the large and the center small circles. Using what you know about the diameter of the circles and the fact that angle b is 75, compute the arc length of the inner (smaller) and outer (larger) circles, between lines BA and BC.
As a challenge problem, find an expression that relates the difference in arc length of your two answers to the difference in diameters of the two circles. Why would this be useful?
Name: The sensor models before you are one version of the sensor. Just by looking at it, can you see how circles may be important to the sensor? HINT: look at the layout of the photodiodes and the "cartridge." The smaller diodes are within the larger cartridge. All are circular. The layout of the diodes is such that the 7 diodes fill the circular base of the cartridge nearly completely. Sketch what you see and how the diodes are laid out. Refer to the diagrams on the next few pages. Encourage students to use those as their working model. There are differences between the diagrams and the actual layout if the sensor, these are brought to light when the students are asked to explain whether or not their calculations match their measurements. They should differ because the model used has the diodes right next to one another, while in actuality they are separated slightly. What is the relationship between the larger circle of the cartridge and the 7 smaller circles of the diodes? The diodes are inscribed inside the cartridge.
Check your sketch on the previous page and make sure it looks similar to the figure below. If you know the diameter of the cartridge (the large outer circle) is about 1.417 cm, what does that tell you about its circumference? Circumference = 2 * π * radius where the radius is one-half (1/2) the diameter. Thus... C = 2 * π * 1.417 / 2 = 4.452 cm Additionally, what does that diameter tell you about the diameters of the diodes (smaller circles)? According to the model, the diameter of the diodes is one-third (1/3) the diameter of the cartridge. Thus... D diode = D cartridge / 3 = 0.4723 cm What would the circumference of the diodes be? Similar to above, C = 2 * π * r Thus... C diode = 2 * π * r diode = 1.484 cm Use the digital calipers to check your diode diameter measurement. Does it match up? If not, why might it be different? The measurement should not match. This is due to differences in the model that cause the computed answers above to be approximations. If the gaps between the diodes are considered, the answers are very close to correct. The diodes are actually 0.32 cm in diameter.
Now, consider the same circle layout from the last page and answer the follow questions. 1 2 3 Is it possible to draw a tangent line of the larger circle than is also a tangent line of one or more of the smaller circles? If so, sketch it above, if not, why not? See the line indicated as "1" above. There are 6 such lines that can be drawn (each time one of the diodes intersects with the cartridge). Is it possible to come up with an angle that is central to both the larger circle and one of the smaller circles? If so, sketch it above, if not, why not? See the angle indicated as "2" above. This is possible with any central angle of the larger cartridge, since it is concentric with the central diode. Is it possible to come up with an angle that is inscribed to both the larger circle and one of the smaller circles? If so, sketch it above, if not, why not? See the angle indicated as "3" above. This is possible at all the same points a tangent line can be drawn as in the first question on this page.
Now use the image below to sketch on. If you draw lines along the diameters of adjacent smaller circles, how many parallel pairs (any two parallel lines make up one pair) of lines can you develop? There are 9 parallel pairs of lines. Draw them below. Of the lines you drew, how many are along the diameter of the larger circle? Of the lines drawn, 3 are along the diameter of the cartridge. Those that weren't have a special classification or name, what is it? Depending upon whether or not the students' lines have arrows on the ends, the lines are either called chords (no arrows, and terminating at the edge of the circle) or secant lines (arrows).
Finally, consider the figure below. A C b B Assume that Point B is at the center of both the large and the center small circles. Using what you know about the diameter of the circles and the fact that angle b is 75, compute the arc length of the inner (smaller) and outer (larger) circles, between lines BA and BC. Use the values computed a few pages ago. This is really two problems "hidden" in one. Arc length = angle / 360 * circumference = angle / 360 * 2 * π * r The inner circle (the diode) has an arc length of The outer circle (the cartridge) has an arc length of 75 / 360 * 1.484 = 0.309 cm 75 / 360 * 4.452 = 0.928 cm Note that the relationship is positive and linear. Since the diameter of the larger circle is three times the diameter of the smaller circle, the arc length is also three times as large. Realizing this helps with the next challenge problem.
As a challenge problem, find an expression that relates the difference in arc length of your two answers to the difference in diameters of the two circles. Why would this be useful? This is a potentially difficult problem to set up, and it involves some algebraic manipulation to solve. Let the arc length of the larger circle be denoted as L L and the arc length of the smaller circle be L S. Similarly, let the circumference of the larger circle to be C L and the smaller be C S. Any other variables used will follow the same subscript convention. Since the problems asks for a difference, use subtraction to set up the problem... L L - L S = (75 / 360 * C L ) - (75/360 * C S ) Make another substitution for the circumferences... = (75 / 360 * 2 * π * r L ) - (75/360 * 2 * π * r S ) Now change the radius to diameter... = (75 / 360 * π * D L ) - (75/360 * π * D S ) Factor out the common terms... L L - L S = (75 / 360 * π) * (D L - D S ) This is an acceptable solution, however, it can be made more general (and thus more useful) by making the angle a variable. L L - L S = (Θ / 360 * π) * (D L - D S ) This is the most powerful form of the ratio. It allows any arc length to be computed by knowing three things: 1) The diameters of the two circles 2) One of the arc lengths 3) The angle being used Depending on what is known about the circles and what is desired, the ratio can be rearranged appropriately. Such manipulation is very common in engineering, and engineers of any type use this technique often.