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Peaucellier's Linkage Homework

Here are some questions from the lab on Peaucellier's linkage, followed by some related questions using Sketchpad to study inversion.

    Peaucellier's linkage questions

  1. Create a Sketchpad sketch of Peaucellier's linkage.
  2. Create a Sketchpad script to construct the inverse of a point in a circle.
  3. Answer questions in Part 2B. Give brief justifications of your answers using the definition.
  4. Fill out the table in Part 2C.
  5. Part 2D: Explain how to construct the inverse of a circle c which goes through the center of the circle of inversion and turn in a Sketchpad script for this construction. Make sure that it works with a circle of any radius; in particular, it should not rely on intersection between the circle c and the circle of inversion.
  6. Part 2D: Explain how to construct the inverse of a line not through the center of the circle of inversion and create a script which does so.
  7. Part 3A: Using the definition of inversion, how can you verify that P and Q are related by inversion? (P and Q are shown in the picture from the lab shown above) What is the center of the circle of inversion?
  8. Part 3B: What is the radius of the circle of inversion in terms of j, k, and m? Explain how you can use segments of lengths j, k, and m to construct a segment of the length of the radius. (Again, j, k, and m are labelled above.) Create a Sketchpad sketch of Peaucellier's linkage which includes this circle of inversion.
  9. Create a Sketchpad sketch in which the inverse of the circular path of P is constructed using inversion (i.e. construct the shape traced by Q).

  10. General questions on inversion.

    It is assumed that you are familiar with Sketchpad. Thus, the following questions do not give step by step help on the technique of creating a sketch.

  11. Investigate what happens to angles under inversion as follows:
  12. Start with an angle determined by three points A, B, and C, and two segments AB and BC, shown in the above figure. Construct a circle of inversion with center not collinear with AB or BC. Invert the points and segments through this circle. Remember that the line segments will invert to curved arcs.

    The angle between two arcs meeting at a point is the angle formed by their tangent lines at that point. This is illustrated in the above figure. Using this, measure the original angle and the new angle, and compare your answers. Your answer to this problem should reference a Sketchpad sketch.

    In this problem you create the first four circles in the picture shown above. Starting with three tangent circles, two inside of the third, here is a method of constructing a circle tangent to all three circles. See the two figures below. Repeating this process gives the beautiful picture of arbelos, known to the ancient Greeks.

    Referring to the labels in the figure, create a circle c5 with center A and arbitrary radius, as shown in the figure below. Invert circles c1,c2, and c3 through circle c5. What do you get? Relate this to the chart you made for the lab.

    Create circle c6 tangent to the three inverses. Notice that by inverting, you made it possible to do this construction.

    Using your answer from the previous problem justify this statement: inversion preserves tangency. Using this information, create circle c4, tangent to c1, c2, and c3. This is the first step in constructing the arbelos picture. For extra credit, create two more circles in the arbelos picture.

    See Dan Pedoe, Geometry, A Comprehensive Course page 89 for more information.

  13. In standard Euclidean geometry, any pair of lines meets in one or no points. The shortest distance between two points is a straight line. Given a line and a point not on the line, there is exactly one line through that point which does not intersect the first line. (This is the famous "parallel postulate".)

    Inversion in the circle changes the objects we are familiar with in the Euclidean plane, and changes the relations between them. For instance, the inverse of a pair of parallel lines is a pair of circles that meet at the center of the circle of inversion. Other relations between objects stay the same: exercise 10 shows that inversion preserves angles if the "angle between two circles" is correctly defined.

    Choose some familiar definition, axiom, or theorem from Euclidean geometry and explore its manifestation in an inverted Euclidean geometry. Discuss the consequences of any changes you discover.

    For instance, the definition of "line" might be changed to mean "infinite straight objects and circles through the origin". This would have consequences for all axioms and theorems about lines. "Triangles" would no longer have straight line segments for sides, but the sum of their angles would still be 180 degrees. In an inverted geometry, statements about similarity and congruence of triangles become much more complicated.

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Author: Evelyn Sander
Comments to: webmaster@geom.umn.edu
Created: Jun 09 1996 --- Last modified: Jun 11 1996