DirectX or OpenGl are included with Windows, and should not be considered 3rd party.
Mapping an image to a triangle in 3d space for rendering on a 2d device. Not something I can give you a Quick Answer to.
If you don't mind wading trough a lot of code - http://away3d.com/
] provides one solution to yur problem - look at the pre flash 10 version.
] also comes with a "software" based renderer.
The irrlicht software renderer is pretty impressive in terms of speed.