Arc vs. Interval — What's the Difference?

In geometry, an arc is a portion of the circumference of a circle or any other curve. It is defined by two endpoints and the path along the curve between them. In contrast, an interval in mathematics defines a set of numbers between two endpoints, which can include or exclude the endpoints themselves, depending on whether it is closed or open.

An arc is specifically related to curves and is measured by its angular distance in degrees or radians. On the other hand, an interval may represent a duration or span in various contexts like time, distance, or quantities in math, where no curvature is involved, just the linear span between two values.

In the context of a graph or a function, an arc can represent a part of a curve between two points, indicating a physical path. Whereas, an interval might be used to specify the domain or range of a function, focusing on the set of input or output values without implying a physical path.

In music, the concept of an arc can metaphorically describe the way a melody or a piece of music moves and flows, often in a figurative curved manner through its pitch and dynamics. Meanwhile, an interval in music strictly refers to the difference in pitch between two notes, which is a critical component of music theory.

From a narrative or artistic perspective, an arc can describe the developmental curve of a character or storyline in literature or film, emphasizing the progression and transformation over time. In comparison, intervals are not typically used in narrative arts but are central to film and music editing, describing segments or durations between specific scenes or musical measures.