How does an object undergo circular motion on a non-circular trajectory?

[medwatt]

Hello,
Forgive me if this question is stupid.
How an object undergoes circular motion when the trajectory is not circular. I mean if it is not moving in a circle then why is it called circular motion.
http://emuonline.emu.edu.tr/phys101/PhIch6pg3_files/image009.gif

Let's suppose I am running along a circular path and I have a leaking cup of paint with me. Of course as long as speed is not changing, the paint will trace out a nice circular path.
But if I start increasing my speed, won't I go slightly of the circular track because I would have traveled just a tiny bit than was necessary ??

 

[A.T.]

Hello,
Forgive me if this question is stupid.
How an object undergoes circular motion when the trajectory is not circular. I mean if it is not moving in a circle then why is it called circular motion.
http://emuonline.emu.edu.tr/phys101/PhIch6pg3_files/image009.gif

Here a circle is used to approximate an arbitrary path locally. Given the same tangential acceleration a_T, the instantaneous centripetal acceleration a_R at the point P of the path is the same as on the circle. The curvature of a path is defined via the radius of a circle.
http://en.wikipedia.org/wiki/Curvature#Curvature_of_plane_curves

Let's suppose I am running along a circular path and I have a leaking cup of paint with me. Of course as long as speed is not changing, the paint will trace out a nice circular path. But if I start increasing my speed, won't I go slightly of the circular track because I would have traveled just a tiny bit than was necessary ??

You can stay on a circular path and just increase your speed. You just need more centripetal force then.