Which type of line is defined as being one-dimensional and able to vary in direction and width?

The correct choice identifies a line that is one-dimensional and can vary in both direction and width. In the context of theatre and visual arts, an identifiable path refers to a conceptual or visual representation that may not strictly adhere to a predefined shape, making it dynamic in appearance and interpretation.

While curved, diagonal, and horizontal lines each have specific definitions and characteristics, they are more rigidly defined in terms of their geometry and are typically used within those parameters. For instance, a curved line can change direction but maintains a consistent width, while horizontal and diagonal lines have established orientations and do not embody the flexibility of varying both direction and width that an identifiable path can convey. Therefore, the option that speaks to a line's ability to change both its direction and width in a fluid manner best captures the concept referred to in the question.