We define a parameterized curve to be any function Z→[a,b]→C where c belongs to the set of  complex number.
A parameterized curve is said to be a smooth curve if Z : [a,b]→C is a function such that z(t) is differentiable, and z'(t)  is continuous on [a,b] and z't≠0 at any point t.
A piecewise(parameterized) smooth curve is any function Z : [a,b]→C for which there exists a finite number of real numbers a=t0<t₁<....<tn=b such that Z is restricted to each closed interval [ti, ti+1] is parameterized smooth Curves..
A parameterized curve Z : [a,b]→C is said to be closed if z(a)=z(b). It is also said to be closed if Z is not self-intersecting except possibly at its endpoints that is if z(t1)=z(t2),then t1=t2 or t1,t2=a,b.

Methods of parameterizing Curves.
(1) let C be a circle of radius r at a point z0 then C can be parameterized by

Since 
It's derivative is given by 
Hence the parametrization is smooth since z'(t) is continuous and never equals to zero(0) anywhere.  
(2)line segments connecting two distinct complex numbers z0 to z1, if we want to parameterized the line segments joining z0,z1 with the function z(t) within the closed interval [0,1] where z(0)=z0, and z(1)=z1 then we can let z(t)=z0+t(z1-z0).one quickly checks z'(t)=z1-z0≠0 is continuous and hence the parametrization is smooth. 
(3)parametrization the arcs of a circle when integrating along a full circle or an arc of a circle in the anti-clockwise direction,  we use the following parametrization. 
Where z0 is the Centre of the circle. R is the radius of the circle and α and and are the angles from z0 to z1 and z2. So for full circle you use θε[0,2π]. Another important thing to consider is the direction of the contour as θ increases in the above equation, the contour moves round anti-clockwise to make it move in the clockwise direction.