Math 445 Lab 9

Part 1. Quick Plot

This is a quick plot to show the plots we were working with in class this morning.

Graph of ratio

Open a new sketch and choose New Function from the Graph Menu.

Notice the key in the calculator window marked X.  Use this to click in the formula x/(1-x). A function expression will appear in the sketch.

Select this function expression and choose Plot Function from the Graph menu.

Graph of Log ratio

Repeat this process with the function log(x/(1-x))

Dynamic endpoints

Construct two points on the x-axis and call them A and B.  Measure the x-coordinates of each one and label the measures a and b.  Then graph the function log((x-a)/(b-x)).

Part 2. Cross Ratio Several ways

Cross ratio of lines

• Go to preferences and choose Directed Degrees as the angle measure.
• Make a sketch with points O, A, B, C, D and compute r_c = sin(AOC)/sin(BOC).  Make this a tool.
• Then compute (with the tool) r_d = sin(AOD)/sin(BOD). 
• Finally, compute the ratio of ratios r_c/r_d.
• Make this a tool.  You can make it neater if you first make the order of the points in the givens = O, A, B, C, D.  The take the measure produced in the last step and rename it (include the = sign) ={1}({2}{3}{4}{5}) and check the Use in Sketch box.  Now make a new page and try out your tool.

Cross Ratio on a Circle

• In a new page, draw a circle with center P through Q and then draw 4 points A, B, C, D on the circle. 
• Also, place a point O in the plane and compute the cross ratio O(ABCD) from your tool above.  If you drag O around the cross ratio should change.
• Now merge O with the circle.  Now drag O around.  The cross ratio should not change.
• Finally, compute the distances CA, CB, DA, DB and compute the ratio (CA/CB)/(DA/DB).  This should be the same (except perhaps for sign) as the other cross ratio.
• Explain why this is so, by writing down the formula for the length of a chord CA in terms of angle CPA and then in terms of angle COA.

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Cross Ratio of Points

• In a new page, draw a line XY and then construct points A, B, C, D on the line.  Measure ratio CA/CB and DA/DB and take the ratio of ratios = (CA/CB)/(DA/DB).
• Then draw another point O somewhere in the figure and use your line cross ratio tool to compute the cross ratio of the lines.  You should get the same answer.

Cross Ratios and Projection

• Continue with the same figure but add another line and its intersections E, F, G, H with lines OA, OB, OC, OD.

• Compute the cross ratio R(EFGH) and check that this is the same as R(ABCD).  Then drag the new line on the opposite side of O, so that it intersects the original line in various ways, etc.  and see that the ratios still stay the same.
  

Part 3: Cross Ratio and Hyperbolic Distance

This figure shows how to measure the distance between two points in the Poincare disk.  Measure the cross ratio of the 4 lines through A, B, C, D and take the absolute value of the logarithm.

Make such a measurement.  Then you can either make your own tool or use a tool from the web.

Checking some P-distances

Now use this tool to check the following constructions:

• Construct a P- reflection A' of a P-point A in a P-line m. Let B be any point on m.  Check that the P-distances are equal: AP = A'P.

• Find the P-center of a P-circle and check that all the P-radii are equal.
  

Part 4. Harmonic Sets

1. Check the cross ratios of the 4 points ABCD in the following figures.

2. Draw a figure with a circle and a point A and its inversion B.  Let C and D be the points where the line AB intersects the circle.
3. Draw a circle and a point A outside the circle and the line through the tangent points as in the figure.  Then let B be any point inside the circle and on this line.  Label C and D the intersections of the line AB with the circle.