PMath 360, Introduction to Geometry

Set 3a: Centre and Axis,
Visualizing pairs of perspective triangles.

Make a Desargues configuration, and label the ten points V, A, B, C, A', B', C', X, Y, and Z, as in the accompanying figure. Notice that V is highlighed and so are triangles ABC and A'B'C' and so is the axis, XYZ. Make five more copies of this labeled figure. Instead of V, highlight A in the first copy, B' in the second, C in the third, X in the fourth, and Z in the last.

In each picture,

  1. Consider the highlighted point to be the centre of perspectivity for the configuration.
  2. Find the two triangles that are in perspective from the centre. (Hint: The vertices of these two triangle will lie on the three lines through the highlighted centre.)
  3. Shade or otherwise highlight these two triangles.
  4. Find the axis (the line) from which these two triangles are in perspective.
  5. At the bottom of the sketch, write an appropriate caption which states the center and the axis and the pair of triangle that are in perspective to the highlighted point.
Submit your five pictures.

Comments:

Regarding part (5) of the assignment, you might use wording analogous to one of the following, but there are other wordsing that will do equally well.

The triangles A B C and A', B', C' are in perspective from the point (centre) V and from the line (axis) X Y Z.

From the centre Y, and from the axis V B B', the two triangles Z A A' and X C C' are in perspective.

Points to ponder: The Desargues Configuration has

In this exercise, you were asked to elaborate on only 5 of the 10 possible choices for the centre of perspectivity. With the practice you have had, you should be able to find, for each of the other 5 points, the two triangles that are in perpective from that point. There is an easy algorithm for finding the axis, given the centre. The same holds, of course, for the dual.

Would you say the figure illustrates 10-fold symmetry? There are more symmetries. Can you find them?