Mathematical Experiment 1

  1. Given a fixed circle with center O and radius k, the inverse of any point P (distinct from O) is defined to be the point P¢ on the ray OP whose distance from O satisfies the equation
  2. OP×OP¢ = k2.
    1. Draw the inverse of the circles
      1. {z Î C:|z-2| = 1/2}
      2. {z Î C:|z-2| = 3/2}
      3. {z Î C:|z-1/2| = 1/4}

      4. from the above definition.
    2. Prove the following theorems in plane geometry:
      1. The inverse of a circle is a circle or a straight line).
      2. The inverse of a straight line is a circle (or a straight line).
    3. Write a subprogram that draws the inverse of a circle w.r.t. the unit circle with the corrdinates of the center and the radius as parameters.
    4. Draw the inverse (w.r.t. the unit circle) of circles each having radius 1, touching 6 other circles of the same size and situated within the box [-21,21]×[-21,21].
    5. Draw the inverse of 21 circles
    6. {z Î C:|z-(5+ki)| = 1},k = 0,±2,±4,...,±20
      and the inverse of the straight lines y = 4 and y = 6. What happens when the original circles move vertically ?
    7. Illustrate Steiner's porism: If we have two nonconcentric circles, one inside the other, and circles are drawn successively touching them and one another, it may happen that the ring of touching circles closes, i.e., the last touches the first. If this happens once, it will always happen whatever the position of the first circle of the ring.

    8. Reference: H.S.M. Coxeter, Introduction to Geometry.
  3. Let ABC be a triangle with lengths AB = 5, BC = 6, CA = 7. Assume that A slides along the x-axis while B slides along the y-axis. Draw the locus of C.

  4. Repeat the same problem for AB = 10.
    For this problem you have to find out first the coordinates of the third vertex whenever the lengths of three sides and the coordinates of two vertices are given.
  5. Draw the shape of solid formed by rotating a cube about a main diagonal.