I see many posts on both this forum and elsewhere the discuss the use of a wide angle on a crop sensor. For example, they discuss how at 14mm lens is 14mm on a 35mm film or full frame, but with a 1.6x crop sensor it is an effective 28mm. My question is, is a 14mm on a crop sensor the same image as a 28mm on a full frame. Or does "effective" include some other conotation. I read that a 14mm can create distortion on a full frame and some vignette, so people seem to say using it on a crop takes out some of those edge problems. My confusion is how the lens is actually taking in the extra image. Reading kenrockwells blog I learned that wide angle keeps lines straight, while fish eye actually skews the lines.
So with this as my understanding, I am still a bit confused how a lens that is curved in order to get a greater angle of view, will be the same as a cropped version of a less curved lens. Or does "effective" only refer to the objects to the sides but not to the distortion of the image. Or, if it is different, is it so small that for photography it does not matter? A technically description would be welcome as long as a basic understanding of the situation.
Edit: To be more clear after the comments,
I am specifically talking about the end resulting 2d projection from 3d space. I have read both those previous answers. the first one is closer to what I am talking about. However, it is still confusing for me. I have experience with 3d modeling and projection matricies, which might be confusing me with this. But for example, i dont understand the images having a 50 mm lens being drawn Starting from the sensor, so it shows a different fov for crop and full frame . If the lens is the same, it takes the same amount of information from the world, which makes a ray from a certain degree off the last lens projected into a smaller space, there is no projection from the sensor, so drawing the line from the sensor doesnt make sense to me. The distance of the sensor to the rear of the lens must have some impact on the light projected but is not represented in the images. Further, again from that first answer it says cropping is the same as zoom, but from what i understand from how perspective works, it is different, since a wide angle will project lines differently than a small fov to the same size, so cropping the center of a wide angle and zooming into something is very different. This is easily recreated in an opengl application with varying fov projection matrix, i imagine in painting class people learn this as well. A fisheye would be a good example, since if cropping was the same as zooming, a fish eye lens would keep the center exactly like a normal lens and then a gradient weighted towards the outside would rapidly create a warped perspective, but from what I see, it is even. To me those images just look like its comparing cropped with full frame in regards to orthogonal projections.