19.1 Parametric Surfaces and Arc Element

A surface may be given by an implicit form f (x, y, z) = 0 or, more useful for CAGD, by its parametric form

(19.1)

where the cartesian coordinates x, y, z of a surface point are differentiable functions of the parameters u and v and [a, b] denotes a rectangle in the u, v-plane; see Figure 19.1 (sometimes other domains are used, for example, triangles). To avoid potential problems with undefined normal vectors, we will assume

that is, that both families of isoparametric lines are regular (see Section 10.1) and are nowhere tangent to each other. Such a parametrization is called regular.¹

Any change r = r(u) of the parameters will not change the shape of the surface; the new parametrization is regular if det[r_u, r_v] ≠ 0 for u ∈ [a, b], that is, if one can find the inverse u = u(r) of r.

A regular curve u = u(t) in the u, v-plane defines a regular curve x[u(t)] on the surface. One can easily compute the (squared) arc element (see Section 10.1) of this curve: from we immediately obtain

which will be written as

(19.2)

where

The squared arc element (19.2) is called the first fundamental form in classical differential geometry. It is of great importance for the further development of our material. Note that the arc element ds, being a geometric invariant of the curve through the point x, does not depend on the particular parametrization chosen for the representation (19.1) of the surface.

For the arc length of the surface curve defined by u = u(t), we obtain

19.2 The Local Frame

The partials x_u and x_v at a point x span the tangent plane to the surface at x. Let y be any point on this plane. Then

is the implicit equation of the tangent plane. The parametric equation is

The normal x_u ∧ x_v of the tangent plane coincides with the normal to the surface at x. The normalized normal

together with the unnormalized vectors x_u, x_v form a local coordinate system, a frame, at x (see Figure 19.3). This frame plays the same important role for surfaces as does the Frenet frame (see Section 10.2) for curves. The normal is of unit length and is perpendicular to x_u and x_v, that is, n² = 1 and nx_u = nx_v, = 0. In general, the local coordinate system with origin x and axes x_u, x_v forms only an affine system; it is also (unlike the Frenet frame) dependent on the parametrization (19.1).

19.3 The Curvature of a Surface Curve

Let u(t) define a curve on the surface x(u). From curve theory we know that its curvature is defined by t′ = κm; the prime denotes differentiation with respect to the arc length of the curve. We will now reformulate this expression in surface terms. Since t = x′ and u′ = du/ds, v′ = dv/ds, we have

Let ϕ be the angle between the main normal m of the curve and the surface normal n at the point x under consideration, as illustrated in Figure 19.4. Then

Inserting t′ and keeping in mind that nx_u = nx_v = 0, we have

(19.4)

Furthermore, nx_u = 0 implies n_ux_u + nx_uu = 0, and so on. Thus, using the abbreviations

(19.5)

Eq. (19.4) can be written as

(19.6)

This expression is called the second fundamental form in classical differential geometry. For any given direction du/dv in the u, v-plane and any given angle ϕ, the second fundamental form, together with the first fundamental form (19.2), allows us to compute the curvature κ of a surface curve having that tangent direction.

19.4 Meusnier’s Theorem

The right-hand side of (19.4) does not contain terms involving ϕ. For ϕ = 0, that is, cos ϕ = 1, we have that m = n: the osculating plane of the curve is perpendicular to the surface tangent plane at x. The curvature κ₀ of such a curve is called the normal curvature of the surface at x in the direction of t (defined by du/dv). The normal curvature is given by

(19.7)

Now (19.6) takes the very short form

(19.8)

This simple formula has an interesting and important interpretation, known as Meusnier’s theorem. It is illustrated in Figure 19.5: the osculating circles of all surface curves through x having the same tangent t there form a sphere. This sphere and the surface have a common tangent plane at x; the radius of the sphere is ρ₀.

As a consequence of Meusnier’s theorem, it is sufficient to study curves at x with m = n; moreover, these curves may be planar. Such curves, called normal sections, can be thought of as the intersection of the surface with a plane through x and containing n, as illustrated in Figure 19.6.

19.5 Lines of Curvature

For Meusnier’s theorem, we considered (osculating) planes that contained a fixed tangent at a point on a surface; we will now look at (osculating) planes containing the normal vector at a fixed point x. We will drop the subscript of κ₀ to simplify the notation.

Setting λ = dv/du = tan α (see Figure 19.6), we can rewrite (19.7) as

In the special case where L : M : N = E : F : G, the normal curvature κ is independent of λ. Points x with that property are called umbilical points.

In the general case, where κ changes as λ changes, κ = κ(λ) is a rational quadratic function, as illustrated in Figure 19.7. The extreme values κ₁ and κ₂ of κ(λ) occur at the roots λ₁ and λ₂ of

(19.9)

It can be shown that λ₁ and λ₂ are always real. The extreme values κ₁ and κ₂ are the roots of

(19.10)

The quantities λ₁ and λ₂ define directions in the u, v-plane; the corresponding directions in the tangent plane are called principal directions. The net of lines that have these directions at all of their points is called the net of lines of curvature. If necessary, it may be constructed by integrating (19.9).

Therefore, this net of lines of curvature can be used as a parametrization of the surface; then (19.9) must be satisfied by du = 0 and by dv = 0. This implies, excluding umbilical points, that

The first equation, F ≡ 0, states that lines of curvature are orthogonal to each other; the second equation states that they are conjugate to each other as defined in Section 19.9.

At an umbilical point, the principal directions are undefined; see also Remark 9.

19.6 Gaussian and Mean Curvature

The extreme values κ₁ and κ₂ of κ = κ(λ) are called principal curvatures of the surface at x. A comparison of (19.10) with κ² – (κ₁ + κ₂)κ + κ₁κ₂ = 0 yields

(19.11)

and

(19.12)

The term K = κ₁κ₂ is called Gaussian curvature, whereas is called mean curvature. Note that both κ₁ and κ₂ change sign if the normal n is reversed, but K is not affected by such a reversal.

If κ₁ and κ₂ are of the same sign, that is, if K > 0, the point x under consideration is called elliptic. For example, all points of an ellipsoid are elliptic points. If κ₁ and κ₂ have different signs, that is, K < 0, the point x under consideration is called hyperbolic. For example, all points of a hyperboloid are hyperbolic points. Finally, if either κ₁ = 0 or κ₂ = 0, K vanishes, the point x under consideration is called parabolic. For example, all points of a cylinder are parabolic points. In the special case where both K and H vanish, one has a flat point.

The Gaussian curvature K depends on the coefficients of the first and second fundamental forms. It is a very important result, due to Gauss, that K can also be expressed only in terms of E, F, and G and their derivatives. This is known as the Theorema Egregium, which states that K depends only on the intrinsic geometry of the surface. This means it does not change if the surface is deformed in a way that does not change length measurement within it.

19.7 Euler’s Theorem

The normal curvatures in different directions t at a point x are not independent of each other. For simplicity, let the isoparametric curves of a surface be lines of curvature; then we have F ≡ M ≡ 0 (see Section 19.5). As a consequence, we have

and κ(λ) may be written as

(19.13)

The coefficients of κ₁ and κ₂ have a nice geometric meaning: let Ψ denote the angle between x_u and the tangent vector of the curve under consideration, as illustrated in Figure 19.9. We obtain

where as before. Hence κ(λ) may be written as

This important result was found by L. Euler.

19.8 Dupin’s Indicatrix

Euler’s theorem has the following geometric implication. If we introduce polar coordinates and Ψ for a point y of the tangent plane at x by setting , then setting Euler’s theorem can be written

This is the equation of a conic section, the Dupin’s indicatrix (see Figure 19.10). Its points y in the direction given by Ψ have distance from x. Taking into account that a reversal of the direction of n will effect a change in the sign of ρ, this conic section is an ellipse if K > 0, a pair of hyperbolas if K < 0 (corresponding to and ), and a pair of parallel lines if K = 0 (but H ≠ 0).

Dupin’s indicatrix has a nice geometric interpretation: we may approximate our surface at x by a paraboloid, that is, a Taylor expansion with terms up to second order. Then Remark 7 of Chapter 10 leads to a very simple interpretation of Dupin’s indicatrix: the indicatrix, scaled down by a factor of 1 : m, can be viewed as the intersection of the surface with a plane parallel to the tangent plane at x in the distance . This is illustrated in Figure 19.11.

19.9 Asymptotic Lines and Conjugate Directions

The asymptotic directions of Dupin’s indicatrix have a simple geometric meaning: surface curves passing through x and having a tangent in an asymptotic direction there have zero curvature at x; in other words, these directions are defined by

(19.14)

They are real and different if K < 0, real but coalescing if K = 0, and complex if K > 0.

The net of lines having these directions in all of their points is called the net of asymptotic lines. If necessary, it may be calculated by integrating (19.14). In a hyperbolic region of the surface, it is real and regular and can be used for a real parametrization.

For this parametrization, one has

and vice versa.

As earlier, let y be a point on Dupin’s indicatrix at a point x. Let denote its tangent direction at y. The direction is called conjugate to the direction from x to y. Consider two surface curves u₁(t₁) and u₂(t₂) that have tangent directions and at x. Some elementary calculations yield that is conjugate to if

Note that this expression is symmetric in , . By definition asymptotic directions are self-conjugate.

19.10 Ruled Surfaces and Developables

If a surface contains a family of straight lines, it is called a ruled surface. It is convenient to use these straight lines as one family of isoparametric lines. Then the ruled surface may be written

(19.15)

where p is a point and q is a vector, both depending on t. The isoparametric lines t = const are called the generatrices of the surface; see Figure 19.12.

The partials of a ruled surface are given by and x_v = q. The normal n at x is given by

A point y on the tangent plane at x satisfies

in other words, the tangent planes along a generatrix form a pencil of planes. However, if and q are linearly dependent, that is, if

(19.16)

the tangent plane does not vary with v.

If (19.16) holds for all t, the tangent planes are fixed along each generatrix; hence all tangent planes of the surface form a one-parameter family of planes. Conversely, any one-parameter family of planes envelopes a developable surface that may be written as a ruled surface (19.15), satisfying condition (19.16); see also Remark 10.

19.11 Nonparametric Surfaces

Let z = f(x, y) be a function of two variables as shown in Figure 19.14. The surface

is then called a nonparametric surface. From this, we immediately find

19.12 Composite Surfaces

A surface x = x(u, t) with global parameters u and t may be composed of patches or segments of different surfaces. Let x_– = x_–(t) denote the right boundary curve and x₊ = x₊(t) the left boundary curve of two such patches, connected along x = x(t), t ∈ [a, b], as illustrated in Figure 19.15. Both patches are tangent plane continuous if n_– = ±n₊ at all x(t). This may also be written

(19.18)

where α, β, γ are functions of t and the product αβ is nonvanishing.

The two patches are curvature continuous if they are tangent plane continuous and both Dupin’s indicatrices agree along the common boundary, in the sense of Remark 11 and as illustrated in Figure 19.16.

If the common boundary is C¹, both indicatrices have a pair of points in common that are opposed to each other in the direction of the boundary tangent. Although a conic section is defined by five points (see Section 12.5), or by its midpoint and three nondiametrical points, it can be shown that both Dupin’s indicatrices coincide if there exists a family of curvature continuous surface curves crossing the common boundary. In particular, this family may be one of asymptotic lines (Pegna and Wolter [459]) or any family of isolines (Boehm [74]) or even any family of unordered directions (Pegna and Wolter [459]). Moreover, if the boundary is C⁰ only at a point, there are two directions of curvature continuity there, and so no further conditions have to be met.