Not every topic in my mathematics education was covered equally well. I don’t recall learning anything about geometric transformations when I was in school. Many years after I graduated from college, when I was teaching mathematics, I learned how to make a shape rotate around the origin:

For each original point (x, y),
(-y, x) maps it 90° counterclockwise around the origin,
(y, -x) maps it 90° clockwise around the origin, and
(-x, -y) maps it 180° around the origin.

This year, the 9th-graders I work with at school need to know how to rotate a shape around a point that is NOT the origin. This is a topic I had never thought about before. To patch up this hole in my math knowledge, I decided to play with rotations in Desmos.

I began with
the point (-1, 2) and
the endpoints (2, 3) and (4, 7), and
the polygon function in Desmos to connect the endpoints.

I noticed that the point and the line segment had the same relationship as
(0, 0) and
the line segment with endpoints (3, 1) and (5, 5).
I noted that the following coordinates worked beautifully to rotate the line segment:
(x-1, y+2) maps it onto the line segment’s original endpoints, (2, 3) and (4, 7).
(-y-1, x+2) maps it 90° counterclockwise around (-1, 2),
(y-1, -x+2) maps it 90° clockwise around (-1, 2), and
(-x-1, -y+2) maps it 180° around (-1, 2).

I was quite pleased with the symmetry of those relationships, so I decided to add a few more ordered pairs to my table in Desmos: (4,-3), (6, 3), and (-2, 4). Desmos’s polygon function automatically rotated the line segments produced by those ordered pairs, and this lovely symmetrical design was produced:

Next, I wondered what would happen if I changed the center of rotation. I made an ordered pair, (a, b) and used sliders to move the point around. Now the point reminded me of a ball, and I got the idea to make the rotated shape look like a cat playing with the ball. I made the original cat a little darker than the rotated ones. This was the result:

That cat took a lot of ordered pairs to make. I got to thinking about how rotations are often used in tessellations. I found a simpler-looking kitten that tessellates and recreated it in this Desmos graph. Again, the original cat is darker than the rotated ones.

No cats were harmed in the production of these graphs.

Which of the cats do you like the best, or would you have used a different animal to play with the ball?