Applet shows a small particle's motion on a smooth disc rotating with a constant angular velocity. The particle is initially located at some distance from the center of the disc and is riding with the disc. At some moment the string holding it to the center breaks and the the particle starts sliding on the surface. The position of the particle on the disc at a given moment is marked as it travels across the disc. This is the path of the moving particle in a reference frame attached to the disc. The yellow vector defines the position of the particle.
Along side the motion as it appears to a ground observer is shown. The green vector defines the position of the particle here too. Notice that the length of the yellow vector is same on both the discs.
To illustrate the idea, consider this example. An insect is flying northward at a constant speed. If you mark it's shadow on a horizontal surface and connect the positions of the marks obtained you will get a northward line. If instead the shadow was cast on a horizontal surface moving eastward at constant speed, the line obtained connecting the marks will be west of north. This is the path relative to the surface. Or this is how the insect appears to move, to say, another insect riding the surface.
This should bring out the need for using the right reference frame for analyzing motion. What is a very simple straight line motion in a reference frame attached to the ground, is motion along a complicated curve in a reference frame attached to the rotating disc.