Circle Inversions with Software

Exercises

The PC inversion software uses these formulas to find the coordinates fo inverted points, and centers and radii of inverted circles.

In exercises 1 - 3 the inverting circle has center (2,1) and radius r = 4.

1. Find the inverse P' of P = (3,1) and then find the inverse of P'.

How are the inverses of P and P' related?

If a point is inside the circle, where is its inverse? If a point is outside a circle, where is its inverse? What does inversion do with respect to the insdie and outside of the inverting circle?

2. Find the inverse of P = (6,1).

Why is P its own inverse?

3. Find the inverses of these points

(3,1), (4,1), (5,1), (5.5,1) and (2.8,1), (2.5,1), (2.2,1), (2.1,1), (2.05,1)

As points inside the circle get closer to the circle, where are their inverses and how are they changing?

As points inside the circle get closer to the center, where are their inverses and how are they changing?

What is the inverse of the center of the circle?

4. For the circle with center (2,1) and radius 6, find the inverses U', V', and W' of the points U = (9,1), V = (11,1), and W = (13,1).

Find the distances UV, VW and U'V', V'W'.

Under inversion, are distances outside the circle expanded or contracted?

Under inversion, are distances inside the circle expanded or contracted?

Consider the distances UV and VW. Do these distances under inversion contract a uniform amount or does it depend on something? On what does it depend? Why is this? What can you say about the order of the points U, V, and W and their inverses?

5. Consider the circles C with center (2,1) and radius r = 10 and S with center (20,1) and radius s = 4. Find the center and radius of the circle of I_C(S).

Connect the centers of the two circles with a line and let U and V be the intersections of that line with S. Show U = (16,1) and V = (24,1). Find I_C(U) and I_C(V).

Find the average of I_C(U) and I_C(V).

Find the inverse of the center of S across C.

Are the inverse of the center of S and the center of the inverse of S the same?

Why is this?

6. If A has center (0,0) and radius 2, and B has center (4,0) and radius 3, find the center and radius of B inverted across A and sketch A, B, and AB = I_A(B).

How is the intersections of A and B and related to the intersection of A and AB?

7. If A has center (0,0) and radius 1, and B has center (1,1) and radius 1, find the center and radius of B inverted across A.

How are B and AB related?

What are the angles of intersection of A and B?

8. Suppose A has center (0,0) and radius 5, and B and C have centers (4,0) and (3.2,0). Note that points U = (4,3) and V = (4,-3) are on A. Find radii of B and C so they contain U and V.

Find the distance from the center of A to the closest points, P and Q, of B and C.

Find the center and radius of AB = I_A(B) and AC = I_A(C).

Verify the inverses of B and C contain U and V.

Compare the curvatures of AB and BA.

As a circle gets closer to the center of the inverting circle, what happens to the curvature of the inverted circle? When it passes through the center of the inverting circle, what is its inverse?

9. Suppose the circle B passes through the center of the circle A and is internally tangent to A at a point P. Describe I_A(B).

10. Suppose A has center (0,0) and radius 3, and B has center (1,0) and radius 1. Find I_A of the points on B with x-coordinates 2, 1, 1/2, and 1/4.

Show these inverse points all lie on the same line.

As the points on B get closer to the center of A, what happens to their inverses? Draw a picure illustrating this.

11. Make a problem investigting the inverse of a circle completely inside the inverting circle and passing through the center of the inverting circle.

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