In this part of the lab we will take a blank plane and create a system of affine coordinates. Compare this with they way we crated Euclidean (cartesian) coordinates last week.

• Draw a line AB and another line AC through A.
• Relabel points A, B, C to be points B, A1, A2.
• Now draw a point P anywhere in the plane and construct parallels to the two lines through P.
• Construct the points of intersection of these lines with lines BA1 and BA2. Label the point on BA1 as P1 and the other as P2. You may wish to color these new lines or make them dashed to distinguish them.

Now we consider the point B as the origin and the line BA1 the x1-axis, with |BA1| as the unit length on this axis. Likewise BA2 is the x2-axis and |BA2| is the unit length. There is no relation between the two length measures. Then we can have Sketchpad compute the coordinates of P by computing ratios as follows.

• Select the 3 points (IN ORDER) B, A1, P1 and HOLD DOWN THE SHIFT KEY while you select Ratio from the Measure menu. This will measure the ratio BP1/BA1 (the order of points determines which ratio you compute). This is the x1-coordinate.

• In the same way, select the points B, A2, P2 and measure the ratio. This is the x2-coordinate.

Finally, we can perform a little trick and label the coordinates differently. If we want to keep the ratio labels on the screen, we can measure the ratios a second time to get two copies of each. Now for the second copy of each ratio, change the label . Here is how we write the first ratio with label x1.

• Now, double-click on the ratio BP1/BA1 WITH THE FINGER (TEXT) TOOL. Then click the radio button that says Text Format. Replace the text BP1/BA1 = by x1 = . Then click OK to close the dialog.
• Use the same method on the other ratio to get x2 = ….

Finally, Select All and choose Make Script from the Work menu. This script should have 3 givens. It can be used to construct affine coordinate systems in one step. Be sure to select the ratio measures as well as the geometric objects. You want it all. NOTE: You can change the order of the givens in your script to the order you prefer by holding down the shift key and dragging the Given to a new position.

Now start with a new sketch. This time we will use the built-in coordinates.

• In a new sketch, choose Create Axes from the Graph menu. This will create the origin A and the unit point B. Construct the circle with center A through B and let C be the point of intersection of the y-axis with the circle. Rename the points A, B, C as O, E1, E2.
• Draw a point P anywhere in the plane.
• Apply your script to points O, E1, E2 and P to get our do-it-yourself coordinates and points P1 and P2. Construct the interior of the rectangle OP1PP2 and shade it a light shade.
• Also Measure the built-in coordinates of P and see that you get the same result.
• Now draw 3 new points B, A1 and A2 anywhere in the plane. Draw lines BA1 and BA2.

• Select the measure x1 and choose Mark Scale Factor in the Transform menu. Select B and choose Mark Center. Now, select point A1 and choose Dilate by Marked Scale Factor to get A1'.
• Repeat this with x2. Mark x2 as the scale factor and dilate A2 by marked ratio to get A2'.
• Now we construct P' as a vector sum or as an intersection by assuming that P1' = A1' and P2' = A2'. Construct the line through A1' parallel to BA2 and the line through A2' parallel to BA1. Let the intersection be called P'.

• Construct the parallelogram BA1'P'A2' to see the image of the rectangle OP1PP2.
• This is the affine transformation f(x) = Ax + b, where x is the coordinate vector of P and A is the matrix with columns A1 and A2.

• In the same sketch as above, draw a circle. Then Animate P on the circle by selecting first P and then the circle and creating an Animate button. Trace P' and see the result.
• For a more dynamic version of the same figure, create a Locus by selecting, in order, P', P and the circle. HOLD DOWN THE SHIFT KEY and select Locus from the Construct menu.

• Apply your affine coordinate script from #1 above 3 times in the following way.
• Apply it to A, B, C and P. Then apply it to B, C, A and P. Then apply it to C, A, B and P.
• Compare the resulting coordinates. How many are different? How are the different ones related?

• To figure out why the relationship is true, add the following to the figure. The 3 small triangles are similar to ABC. What is the ratio in each case and how is is related to the coordinates?