When is it OK to place the subject in the middle of a picture?
I took this picture and feel very compelled to put the subject in the centre rather than on the sides.
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Q: When is it OK to place the subject in the middle of a picture?
A: Whenever you feel that it works best!
The general rule of not centering your subject is time-honored, and comes from one basic idea: the center of an image is a stable, straightforward place. When you put something there visually, it stays there visually, usually resulting in a static composition.
When you have your subject off-center, you can use tension and dynamic balance, which tend to make a more engaging composition.
Other factors can contribute to this: the lines from the subject's eyes and the way the subject is facing; color weight; other objects and motion in the composition and their balance. Overall, these can add dynamic interest even if your main subject is static.
You may, though, want the simple, straightforward, and more-static image. That's okay. Think about the flow of interest as you are observing the photo, and decide if a centered or dynamically-balanced composition fits your intent better.
In your particular example, the dog's face (and particularly eyes) aren't actually centered at all: they're quite towards the top of the frame. The overall subject is centered, but the face has considerable off-center visual weight. The leaves on the right side contrasting with the bright yellow flowers on the center-right provide some reason to keep the horizontal as it is; a tighter crop either cuts out the context of the plants or leaves the frame feeling cluttered.
@mattdm has given a nice overview. I'll add some specific situations where middle might turn out to give the best composition:
When you have a subject looking straight at camera and no directional lighting, placing the subject to a side would often feel synthetic. Especially so with a tighter crop. Your photo is an example of this category.
Photos with nothing else than subject on it, such as product photos. If there is no background to look at, there's also no need to leave much room for it.
When shooting with a fisheye lens, middle is the area with least barrel distortion, so it's often a good idea to place your subject there and use the rest of frame to convey context.
Middle is a good place when you want to stress that your subject is isolated, surrounded by emptyness.
On a square format (such as 6x6 cm medium format), middle is actually suggested by diagonal method.
When talking about the alignment of the subject in the frame, I'll take this as the prime example of the kind of composition where not centering your subject will result in a more pleasing picture.
The major factor in composing this image is the balance. When I see your photo, I feel like I'm teetering on a point in the middle, and could fall either way (left or right). I don't feel balance when I look at it and it makes me receive the intent of the photo in an unpleasant way.
When you place the subject to the right or left of a frame like this, you're balancing the background with the foreground. You might think of it as a line that goes from the subject to the background on the opposite side of the frame, but intersecting the middle and balancing things out. Kind of like a see saw. On one end is your subject, on the other, your background or environment.
Now, if you were to crop the photo in portrait orientation so that the dog filled up most of the frame, you wouldn't need to worry about this. The dog is falling on either side of the middle and is balancing itself out.
Hope I made sense.