Types of interpolation used for reconstruction in DSP? What are the different types of interpolation used in DSP for reconstruction of analog signal from discrete/digital signal

I am able to somehow learn two types of interpolation 
1st is "zero order hold interpolation"and 2nd is "linear interpolation(first order interpolation)"

Reconstructed output looks like stair case or square wave for zero order hold interpolation while reconstructed output for linear interpolation(1st order interpolation) looks like a triangular wave

There are other types of interpolation besides above? 

What are those types?  (That types give a much better/smoother reconstructed signal as compared to two types i discussed above)

What is the order of polynomial in those types (just like "zero order hold" type had zero order and "linear interpolation" type had order one)

For those other types of interpolation, what/how are the appearances/looks of reconstructed output waves ? (Please kindly try to include those wave graphs in your answer)

By reconstruction, i mean conversion of digital into analog signal
Zero-order hold will result in a piecewise-constant waveform.



Linear interpolation will result in a piecewise-linear waveform.



If you want a piecewise-quadratic or piecewise-cubic or higher order polynomial interpolation, it will not appear much different from the original bandlimited waveform.