coordinate curve

For other coordinate systems the coordinates curves may be general curves.

Then the area A of any quadrilateral formed by the coordinate curves is smaller than .

This procedure does not always make sense, for example there are no coordinate curves in a homogeneous coordinate system.

In three-dimensional space the intersection of two coordinate surfaces is a coordinate curve.

In the Cartesian coordinate system the coordinate curves are, in fact, straight lines.

For example, the coordinate curves in polar coordinates obtained by holding r constant are the circles with center at the origin.

The vector field b is tangent to the q coordinate curve and forms a natural basis at each point on the curve.

The coordinate axes are determined by the tangents to the coordinate curves at the intersection of three surfaces.

The coordinate curve q represents a curve on which q, q are constant.

Lemma 3: Let be a coordinate neighborhood of such that the coordinate curves are asymptotic curves in .

Lemma 2: For each exists a parametrization , such that the coordinate curves of are asymptotic curves of and form a Tchebyshef net.

It therefore generalizes the notion of a partial derivative, in which the rate of change is taken along one of the coordinate curves, all other coordinates being constant.

Moreover, is an isometric immersion and Lemmas 5,6, and 8 show the existence of a parametrization of the whole , such that the coordinate curves of are the asymptotic curves of .

In two dimensions, if one of the coordinates in a point coordinate system is held constant and the other coordinate is allowed to vary, then the resulting curve is called a coordinate curve (some authors use the phrase "coordinate line").