I am trying to implement a tool into my application. the tool would allow the user to plot a triangle meshes using the mouse. I have looked everywhere for a way to do this,
I have read your question till the end :-) In contrast to many people here you are at least providing a clear description of what you want to achieve. According to my own experience I would say that what you want to do is easily possible with JavaFX and the MeshView is the way to go here. You can use your image as the texture for this mesh and you can distort the image by manipulating the vertices of this mesh. I have implemented part of your functionality myself for a project so I know that it works.