What is a bounding box, and how is it mathematically represented?

Four numbers fully describe an axis-aligned rectangle, and there are two conventions for which four:

• Corner / xyxy: top-left (x1, y1) and bottom-right (x2, y2). Easy to draw and to compute overlap with.
• Center / xywh: center (cx, cy) plus width and height. This is YOLO's native format, and it's normalized by the image size so a box is 0-1 regardless of resolution.

You'll constantly convert between the two. Here's the exact conversion:

def xywh_to_xyxy(cx, cy, w, h, img_w, img_h):
    """Center form (normalized 0-1) -> pixel corners."""
    x1 = (cx - w / 2) * img_w
    y1 = (cy - h / 2) * img_h
    x2 = (cx + w / 2) * img_w
    y2 = (cy + h / 2) * img_h
    return x1, y1, x2, y2


def xyxy_to_xywh(x1, y1, x2, y2, img_w, img_h):
    """Pixel corners -> center form, normalized 0-1 (YOLO label format)."""
    cx = ((x1 + x2) / 2) / img_w
    cy = ((y1 + y2) / 2) / img_h
    w = (x2 - x1) / img_w
    h = (y2 - y1) / img_h
    return cx, cy, w, h

A YOLO label file stores one line per object as 'class cx cy w h', all normalized. Getting this format wrong (pixels instead of 0-1, or corners instead of center) is a classic reason a custom training run produces nonsense.