My class had a good time redeveloping the unit circle under the taxicab metric. Now some of them want to do it again with another similar metric. I want to give this to some of my "honors" high-school* calculus I students to work on independently. As such, it can't be too difficult or require too many advanced concepts.
What is another metric that I could introduce them to where one can describe a "unit circle" and try to make sense of the sine, cosine and tangent functions?
*I have a group of high school students enrolled in a college course. The course, as planned, is far too easy for them. (At the same time, it is quite hard for the college students enrolled. Go figure.) After the high school students do what has been assigned by the online learning system I have them work on other stuff-- which I allow to be pretty self-guided.