Projection:

1. Imagine the stereographic projection of a sphere onto a plane, which projects any point on the sphere onto a plane by using the point where the line joining that point to a fixed point (called the North Pole) meets the plane. (Physically, you could think of projecting the Earth onto a plane, with the South Pole sitting on the plane and the North Pole directly above.) See STEREO for a dynamic 2D representation of this transformation in 3-space.

Now imagine a two-dimensional version of this projection: A circle sitting on a line. For any point (other than the North Pole) on the surface (circle), the projection is obtained by constructing the line through this point and the North Pole, and then constructing the point where this line meets the plane (line) on which the surface (circle) sits.

Construct a diagram to illustrate the projection, but starting with an arbitrary point on the fixed line and constructing the point on the circle that projects onto it. A static version looks like Diagram 1 below:

Inversion:

2. Surprisingly, the projection in #1 explains the funny mirror reflections of Alice on page 3 in Journey ... . Follow the construction process below to get the mirror effects.

a. Start with a circle 1 centered at point S (which corresponds to the South Pole), and a point P (which corresponds to any point on Alice) randomly chosen in the sketch plane. Drag P so that it is to the right of S and a little above S so that your diagrams will be more easily compared to the ones shown below.

Overview of the remaining instructions: We will now find a point that is P's mirror image by projecting P back onto a circle (as if we were finding the point on Earth that projects onto the point P on the map), reflecting that point onto the opposite hemisphere, and then projecting that point back onto the plane ... Well, you'll see how it goes! Check back here after going through the details.

b. Construct the ray from S through P. (S is the ray's endpoint.)

c. Construct the point of intersection, T, of circle 1 with ray SP.

d. Mark S as a center, and rotate T by 90 degrees about S, producing point N.

e. Construct the segment NS, its midpoint O, and a circle 2 centered at O and passing through N. So far, the diagram should look something like this:

f. Construct the segment NP and its point of intersection, Q, with circle 2. (This locates the point on earth that corresponds to P.

g. Now to reflect to the other hemisphere: Construct the line through O that is perpendicular to NS, mark it as a mirror, and construct the reflection of Q, Q', in this new line.

h. Finally, project the point on Earth, Q', onto the map: Construct the ray NQ', with endpoint N, and its point of intersection, P', with the ray SP. Now the diagram should look something like this:

Note that if your point P is inside circle 1, P' will be outside that circle, and Q and Q' will be in the southern and northern hemispheres, respectively.

The point P' is called the inversion of P with respect to the circle 1.

3. Go to TRIANGLE SEQUENCE to draw some conclusions and see why P' is called the inversion of P. It may be best to postpone this and go on to #4 unless you are just dying to know why the process is called inversion!

4. The final part of this lab is to investigate the mirror properties of inversion with the help of Sketchpad.

Start by hiding all those nice similar triangles and, actually, everything except the points S, P, P', and the control point for circle 1, which you may have already hidden. To find this control point if it is hidden, just hide everything but S, P, and P', then Show All Hidden. You'll see the control point, which is the only object not connected to anything, except circle 1. Being careful not to click outside the selections, deselect the control point, and then hide everything selected again. Label the control point C.

Create a new tool, as you did in Lab 3, the most important one for the remainder of Math 120 Sketchpad use: Select all four point and go to the Tool Bar and choose Create a New Tool by pressing the bottom and dragging it to choose Create New Tool. Click on Show Script View, title this script "Inversion," and press OK.

The script should appear on your monitor, with three objects as Given and about 13 Steps that construct other objects, eventually hiding them all, except P's inverse, P'.

Save your file (the sketch, that is), so that you can access this script; it will be available whenever you open this file.

Test how the script shows the funny mirror: Open a new sketch. You need to construct the givens, S, C, and P, but do it in the following way to utilize Sketchpad's dynamic capability. Construct a circle and about twenty points outside the circle, so that they outline some interesting asymmetric shape, like the letter P. Select those twenty points, in the "right" order, so that segments formed by successive points would outline the shape you want. (Actually, this doesn't matter - you'll be able to drag these points wherever you want later to produce the shapes you want.) Choose Polygon Interior from the Construct menu, and then with this interior still selected, choose Point on Interior again from the Construct menu. Make this random point a color different from the other points, say blue, so that it can be distinguished from the others and selected as one of the givens.

Play the script, with the circle's center as S, the circle's control point as C, and the random interior point of the polygon as P. P's inversion, P', with respect to the circle should appear. Drag P around to see where P' goes.

Now construct the locus of P' as P ranges around the polygon: Select P, the polygon inetrior, and P', and then choose Locus from the Construct menu. A lovely shape should appear: the image of Alice or the image of whatever shape you made.

Experiment by dragging points on the polygon around to see what images you can produce.

Try dragging all of the polygon to the interior of the circle.

As you read Journey ... try to keep this investigation in mind. Also, explore the dynamic sketch, INVERSION. A square exterior to a circle and the inverse of that square in the circle are shown. The points Q and Q' are inverses images. Point Q and one of the corners of the square can be dragged. More importantly, the size of the square and of the inverting circle can be adjusted.

Observe what happens to the inversion of the square as you drag the point called radius adj to the right to increase the size of the circle of inversion and drag side adj to the left to decrease the relative size of the square. (It will also help to drag the center of the circle all the way to the left border of the window just to get more viewing space of the critical objects.)

Note your observations on your experiments with INVERSION below: