  Persistence diagrams are one of the main tools in the field of Topological
Data Analysis (TDA). They contain fruitful information about the shape of data.
The use of machine learning algorithms on the space of persistence diagrams
proves to be challenging as the space lacks an inner product. For that reason,
transforming these diagrams in a way that is compatible with machine learning
is an important topic currently researched in TDA. In this paper, our main
contribution consists of three components. First, we develop a general and
unifying framework of vectorizing diagrams that we call the \textit{Persistence
Curves} (PCs), and show that several well-known summaries, such as Persistence
Landscapes, fall under the PC framework. Second, we propose several new
summaries based on PC framework and provide a theoretical foundation for their
stability analysis. Finally, we apply proposed PCs to two
applications---texture classification and determining the parameters of a
discrete dynamical system; their performances are competitive with other TDA
methods.
