that is -times continuously differentiable (that is, the component functions of are continuously differentiable), where , , and be a non-empty interval of real numbers. The of the parametric curve is . The parametric curve and its image must be distinguished because a given subset of can be the image of several distinct parametric curves. The parameter in can be thought of as representing time, and the trajectory of a moving point in space. When is a closed interval , is called the starting point and is the endpoint of . If the starting and the end points coincide (that is, ), then is a "closed curve" or a "loop". For being a -loop, the function must be -times continuously differentiable and satisfy for .