Computer representation of surfaces

An open surface with u- and v-flow lines and Z-contours shown.

In computer aided design, surfaces are one way of representing objects. The other ways are wireframe (lines and curves) and solids. Point clouds are also sometimes used as temporary ways to represent an object, with the goal of using the points to create one or more of the three permanent representations.

Open and closed surfaces

Surfaces are said to have two directions, commonly called u and v. That is, any point on a surface can be described by a u and v coordinate pair.

Open surfaces are not closed in either direction. This means moving in any direction along the surface will cause an observer to hit the edge of the surface. The top of a car hood is an example of a surface open in both directions.

Surfaces closed in one direction include a cylinder, cone, and hemisphere. Depending on the direction of travel, an observer on the surface may hit a boundary on such a surface or travel forever.

Surfaces closed in both directions include a sphere and a torus. Moving in any direction on such surfaces will cause the observer to travel forever without hitting an edge.

Places where two boundaries overlap (except at a point) are called a seam. For example, if one imagines a cylinder made from a sheet of paper rolled up and taped together at the edges, the boundaries where it is taped together are called the seam.

Flattening a surface

Some open surfaces and surfaces closed in one direction may be flattened into a plane without deformation of the surface. For example, a cylinder can be flattened into a rectangular area without distorting the surface distance between surface features (except for those distances across the split created by opening up the cylinder). A cone may also be so flattened. Such surfaces are linear in one direction and curved in the other (surfaces linear in both directions were flat to begin with). Sheet metal surfaces which have flat patterns can be manufactured by stamping a flat version, then bending them into the proper shape, such as with rollers. This is a relatively inexpensive process.

Other open surfaces and surfaces closed in one direction, and all surfaces closed in both directions, can't be flattened without deformation. A hemisphere or sphere, for example, can't. Such surfaces are curved in both directions. This is why maps of the Earth are distorted. The larger the area the map represents, the greater the distortion. Sheet metal surfaces which lack a flat pattern must be manufactured by stamping using 3D dies (sometimes requiring multiple dies with different draw depths and/or draw directions), which tend to be more expensive.