1.) Give a closed plane curve C with k (curvature) > 0 that is not convex.
can someone please explain these concepts to me. How can you have a closed plane curve like this? Do you used signed curvature?
2.) Draw closed plane curves with rotation indices 0, 2, -2, and 3, respectively.
I am unsure about the rotation indices, would you just plot the points 0,2,-2,3 and draw a curve that curves around those points? Could someone show me a plot of such and explain the concept to me?
Thank you very much.
The key point is that the curve will need to have self-intersections. (A simple closed curve of positive curvature must be convex). Here's an inspirational example:
The parametric equations of the above curve are $$ x=3\cos(-t)+\cos5 t,\qquad y = 3\sin(-t)+\sin 5t $$
Imagine a planet orbiting the origin clockwise, and its moon orbiting the planet counterclockwise. That's the trajectory of the moon.
No, not at all. You need four different curves, one for each index. One that goes around the origin $0$ times, one that goes $2$ times, one that goes $-2$ times (negative means clockwise direction, positive is counterclockwise)...
For example, you could make the planet rotate twice as fast around the sun, or three times as fast... Try playing with coefficients of $t$ in the above formula, using for exmaple Function Plotter.