Computer representation of surfaces

In technical applications of 3D computer graphics (CAx) such as computer-aided design and computer-aided manufacturing, 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.

If one considers a local parametrization of a surface:

then the curves obtained by varying u while keeping v fixed are sometimes called the u flow lines. The curves obtained by varying v while u is fixed are called the v flow lines. These are generalizations of the x and y lines in the plane and of the meridians and circles of latitude on a sphere.

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.

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.

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