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A parametric surface is described by a vector function of two parameters, u and v, defined over a region D, which is the set of allowed values of u and v in the uv-plane.

The component functions of r are expressed as functions of u and v, with domain D.

These expressions form the parametric equations of the surface. Together, they define a position vector that traces the surface as u and v vary.

This traced surface is called a parametric surface.

A practical example is modeling a curved glass canopy at a building entrance as a parametric surface. Since the canopy’s curvature varies across its surface, parametric equations can capture its 3D shape.

Here, u and v act like surface directions, helping define smooth curves between support points and describe the desired curvature.

Holding one parameter constant generates lines across the surface, called grid curves.

When u is constant, the resulting v-direction curves can represent the main support paths. When v is constant, the u-direction curves can trace the canopy’s contour between those supports.

This helps designers align the canopy’s geometry with its physical framework.