In the arts, line is considered one of the formal elements--that is, one of the purely visual aspects of an artwork. While two artworks may have the same subject matter, the artist's treatment of the formal elements can radically change how viewers understand each artwork.
Artists can create curved lines in many different ways, which result in various meanings. A curved line can be geometric, like the arc of a perfect circle. Curved lines can also be "organic," creating irregular lines and shapes.
Outline refers to a type of line which separates shapes and colors from other elements in a visual composition. Outlines are generally thick and do not vary in width--thus they have a flattening effect. Post-impressionist artists like Gauguin often used outline. They were inspired by Japanese woodcuts selling in France, and wanted the flat and decorative look of these woodcuts in their own paintings. Art historians refer to this style as cloisonnism.
Contour line follows the three-dimensional curves of a form: it gets darker where there is shadow and lighter where there is less shadow. Painters looking to create a three-dimensional effect often use contour line in tandem with light and dark values and perspectivally accurate renderings. The French Romantic painter Ingres is considered a master of contour line (Ingres is sometimes also considered Neoclassical--but most consider that an inaccurate designation).
Evenly rendered curved lines (like arcs and sections of perfect circles) are geometric. Many consider balanced and geometric shapes to represent rationality and the intellectual mind. Neoclassical painters, who celebrated the ideas and aesthetics of the ancient Greeks, as well as knowledge and reason, often carefully composed their paintings according to geometric shapes. For example, Jacques-Louis David's treatment of the arches in the famous painting "The Oath of the Horatii."
Organic Curves in Abstract Painting
Many artists of the 20th century used organic curved lines to create ambiguous and abstract shapes. Joan Miró, for example, made paintings using circles, curved lines and flat planes of color. The thing about Miró's use of curved lines and abstract shapes, is that he wanted the shapes to be multivalent--they can represent many different things. They can be body parts, stars in the universe, or microscopic cells.
Implied Lines and Curves
Artists do not always explicitly draw or paint a curved line. Instead, artists imply lines, between arrangements of shapes or using figures to point toward other parts of the composition. Jacopo Pontormo, for example, in his painting of the Deposition from the Cross, arranged figures and limbs to create a centrifuge around the central image of Jesus' dead body and the Virgin Mary's outstretched hand. The implied curved line created around the void between Christ and Mary signified the stress and tumult of the moment that Pontormo depicted.