Parametric surface
The curvature and arc length of curves on the surface, surface area, differential geometric invariants such as the first and second fundamental forms, Gaussian, mean, and principal curvatures can all be computed from a given parametrization.
The first partial derivatives with respect to the parameters are usually denoted
In vector calculus, the parameters are frequently denoted (s,t) and the partial derivatives are written out using the ∂-notation:
The parametrization is regular for the given values of the parameters if the vectors
The tangent plane at a regular point is the affine plane in R3 spanned by these vectors and passing through the point r(u, v) on the surface determined by the parameters.
Any tangent vector can be uniquely decomposed into a linear combination of
The surface area can be calculated by integrating the length of the normal vector
to the surface over the appropriate region D in the parametric uv plane:
Although this formula provides a closed expression for the surface area, for all but very special surfaces this results in a complicated double integral, which is typically evaluated using a computer algebra system or approximated numerically.
Fortunately, many common surfaces form exceptions, and their areas are explicitly known.
This is true for a circular cylinder, sphere, cone, torus, and a few other surfaces of revolution.
This can also be expressed as a surface integral over the scalar field 1:
on the tangent plane to the surface which is used to calculate distances and angles.
Arc length of parametrized curves on the surface S, the angle between curves on S, and the surface area all admit expressions in terms of the first fundamental form.
If (u(t), v(t)), a ≤ t ≤ b represents a parametrized curve on this surface then its arc length can be calculated as the integral:
The first fundamental form may be viewed as a family of positive definite symmetric bilinear forms on the tangent plane at each point of the surface depending smoothly on the point.
This perspective helps one calculate the angle between two curves on S intersecting at a given point.
The first fundamental form evaluated on this pair of vectors is their dot product, and the angle can be found from the standard formula
expressing the cosine of the angle via the dot product.
Surface area can be expressed in terms of the first fundamental form as follows:
By Lagrange's identity, the expression under the square root is precisely
In the special case when (u, v) = (x, y) and the tangent plane to the surface at the given point is horizontal, the second fundamental form is essentially the quadratic part of the Taylor expansion of z as a function of x and y.
For a general parametric surface, the definition is more complicated, but the second fundamental form depends only on the partial derivatives of order one and two.
Its coefficients are defined to be the projections of the second partial derivatives of
The principal curvatures are the invariants of the pair consisting of the second and first fundamental forms.
Up to a sign, these quantities are independent of the parametrization used, and hence form important tools for analysing the geometry of the surface.
The points at which the Gaussian curvature is zero are called parabolic.
The coefficients of the first fundamental form presented above may be organized in a symmetric matrix:
[1] Now, if v1 = (v11, v12) is the eigenvector of A corresponding to principal curvature κ1, the unit vector in the direction of
Accordingly, if v2 = (v21,v22) is the eigenvector of A corresponding to principal curvature κ2, the unit vector in the direction of
