Math and Art: Math and Art by Sasho Kalajdzievski

Author:Sasho Kalajdzievski
Language: eng
Format: epub, pdf
Publisher: CRC Press
Published: 2021-09-08T00:00:00+00:00

Figure 4.2.1 Angles over diameters are right angles.

Here are the steps of the construction of , where D is an arbitrary point outside the inversion circle H.

STEP 1. The point D is now outside the circle H.

STEP 2. Find the midpoint M of the segment OD.

STEP 3. The circle centered at M and passing through O and D intersects H at B1 and B2. The image inv(D) of D is then on the intersection of OD and B1B2. □

We conclude that inversions with respect to a circle H shuffle points in the plane in a very special way: the points outside H move within H, and the points in H move out of H with one exception: the center of H is not within the scope of the inversion. The points on the circle H stay where they are.

Inversions in general do affect the shapes of planar objects. The following theorem is a partial description of how inversions change shapes. We continue to denote the inversion with respect to the circle H by inv.

Theorem.

If C is a circle that does not pass through the center of the inversion circle H, then its image inv(C) is a circle.

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