Gaussian Curvature in Propagation Problems in Physics and Engineering

Abstract

The computation of the Gaussian curvature of a surface is a requirement in many propagation problems in physics and engineering. A formula is developed for the calculation of the Gaussian curvature by knowledge of two close geodesics on the surface, or alternatively from the projection (i.e., image) of such geodesics. The formula will be very useful for problems in general relativity, civil engineering, and robotic navigation.

1. Introduction

In many propagation problems in physics and engineering, it becomes necessary to compute the Gaussian curvature of a two-dimensional surface. In physics, this becomes necessary in the applications of general relativity, where it is sometimes desired to calculate the Gaussian curvature at a point in space from the observed geodesic paths of planets or light rays [1, 2]. In engineering, engineers who are involved in the design of structures such as geodesic domes frequently require a practical formula for computing the Gaussian curvature, where relations exist between the Gaussian curvature at any point on the surface of the structure and the stability of such a structure [3]. In certain other engineering applications, such as computer vision and robotic navigation, engineers sometimes find themselves facing the complicated problem of having to compute the Gaussian curvature of a surface in order to calculate 3-dimensional depth data (or range) [4–6].

From the basic principles of differential geometry, the Gaussian curvature 𝐺 at any point of a two-dimensional surface 𝑆 is given by𝐺=𝑘₁𝑘₂, (1.1) where 𝑘₁ and 𝑘₂ are the maximum and the minimum normal curvatures [6]. Unfortunately, in many practical situations, 𝑘₁ and 𝑘₂ are simply unknown. In the following section, we will derive a formula for computing the Gaussian curvature at any point on a surface by knowledge of two close geodesics on the surface, or alternatively from the projection (i.e., image) of such geodesics (this is very important in applications such as general relativity and robotic navigation, where no direct knowledge of the geodesics exists, but only an image of the geodesics is available). A simple test of the formula is given in Section 3 (the test shows that 𝐺, as computed from the formula, must vanish in an Euclidean 2-space). In Section 4, it is proven that the Gaussian curvature is a projective invariant and hence can be calculated from any projected image of two geodesics.

2. Calculation of the Gaussian Curvature from Geodesic Deviation

It is well known that any general 2-dimensional surface is topologically equivalent at any given point to a surface of revolution [6]. Hence, two close geodesics on the surface, when considered only within a small surface patch, can be treated as embedded in a surface of revolution. Such curves, however, will not necessarily be geodesics in the surface of revolution. Consider now a surface of revolution, where the smooth curve 𝑆 is a general curve that is embedded in the surface (Figure 1). 𝑆^′ is a copy of 𝑆 that is obtained by rotating 𝑆 through a small angle 𝜃.

Figure 1 

A general curve 𝑆 that is embedded in a surface of revolution and a copy 𝑆′ that is separated from 𝑆 by a small rotation.

𝑝^𝑟 is a position vector, defined over a circular ring passing by 𝑆-𝑆^′. Let us select two parameters 𝑢 and 𝑣, such that 𝑢 varies as we travel along the curve 𝑆, but 𝑣 is constant, and 𝑣 varies as we pass from one curve to another, but 𝑢 is constant. Obviously,𝑢=𝑠,𝑣=𝜃, (2.1) where 𝜃 is the rotation angle of the axis from 𝑆 to 𝑆^′. Given such parameters on any surface in space, it can be shown that [7]∇²_𝑠𝜃𝛿^𝜌𝑑𝜃−∇²_𝜃𝑠𝛿^𝜌𝑑𝜃=_{𝜁,𝜇,𝜈}𝑅^𝜌_𝜁𝜇𝜈𝛿^𝜁𝛿^𝜈𝑝^𝜇, (2.2) where 𝛿^𝑟 is the unit tangent vector to the curve, 𝑅^𝑎_𝑏𝑐𝑑 is the mixed curvature tensor [7], and the symbol ∇ is the covariant derivative operator [6, 7]. If the curve 𝑆 was a geodesic in the surface, we must have had [6, 7]∇_𝑠𝛿^𝑟=0, (2.3) since the covariant derivative of the unit tangent vector to a geodesic vanishes along the curve [6, 7]. Since 𝑆 is a general curve, however, then ∇_𝑠𝛿^𝑟 will be the components of a vector of finite length, normal to the vector 𝛿^𝑟 [6]. On the other hand, due to circular symmetry in a surface of revolution, the vector ∇_𝑠𝛿^𝑟, clearly, is parallel transported [6, 7] along a circular ring in the surface. Hence, we must conclude that∇_𝜃∇_𝑠𝛿^𝑟=∇²_𝜃𝑠𝛿^𝑟=0, (2.4) at any point on 𝑆. Further, given the parameters 𝑠 and 𝜃, it can be shown that [7] ∇²_𝑠𝜃𝛿^𝑟𝑑𝜃=∇²_𝑠𝑝^𝑟, (2.5) for any 2-dimensional surface. From (2.4) and (2.5), (2.2) is rewritten as∇²_𝑠𝑝^𝜌=_{𝜁,𝜇,𝜈}𝑅^𝜌_𝜁𝜇𝜈𝛿^𝜁𝛿^𝜇𝑝^𝜈, (2.6) where 𝜁, 𝜇, 𝜈=1,2. Moreover, it has be shown that [7]𝑅^𝑎_𝑏𝑐𝑑=𝐺𝛿^𝑎_𝑐𝑔_𝑏𝑑−𝛿^𝑎_𝑑𝑔_𝑏𝑐, (2.7) for a smooth 2-dimensional manifold, where 𝐺 is the Gaussian curvature, 𝛿^𝑎_𝑏 is the Kronecker delta, and 𝑔_𝑎𝑏 are the components of the metric tensor at any point on the surface. Substituting from (2.7) into (2.6) and carrying out the summation, we obtain∇²_𝑠𝑝^𝜌+𝐺𝑝^𝜌=𝐺𝛿^𝜌_𝜇,𝜈𝑔_𝜇𝜈𝛿^𝜇𝑝^𝜈, (2.8) where we have used the identity_𝜇,𝜈𝑔_𝜇𝜈𝛿^𝜇𝛿^𝜈=1. (2.9) Equation (2.8) is analogous to the equation of geodesic deviation [6, 7]. Once again, if 𝑆 was a geodesic in the surface, we must have had an orthogonality condition_𝜇,𝜈𝑔_𝜇𝜈𝛿^𝜇𝑝^𝜈=0, (2.10) and (2.8) would have reduced to the well-known equation of deviation of two geodesics in a Riemannian 2-manifold. Equation (2.8), in its given form, will not allow the computation of the Gaussian curvature 𝐺, since the metric tensor components, as well as all the covariant derivatives on the surface, are unknown. However, (2.8) can be further reduced as follows: for an infinitesimal rotation 𝑑𝜃,𝑝^𝑟=𝜕𝑥^𝑟𝜕𝜃𝑑𝜃. (2.11) Thus,𝜕𝑝^𝑟=𝜕𝑠𝜕𝛿^𝑟𝜕𝜃𝑑𝜃. (2.12) Now,𝛿^𝑟_𝑆′=𝛿^𝑟_𝑆+𝜕𝛿^𝑟_𝑆𝜕𝜃𝑑𝜃=𝛿^𝑟_𝑆+𝜕𝑝^𝑟,𝜕𝑠 (2.13) where 𝛿^𝑟_𝑆, 𝛿^𝑟_𝑆′ are unit tangent vectors at 𝑆 and 𝑆^′, respectively, separated by a rotation 𝑑𝜃 (Figure 2).

Figure 2 

Unit tangent vectors to 𝑆 and 𝑆′, respectively.

Given that, for any vector 𝑝^𝑟 on the surface [7],∇_𝑠𝑝^𝜌=𝜕𝑝^𝜌+𝜕𝑠_𝜇,𝜈Γ^𝜌_𝜇𝜈𝑝^𝜇𝜕𝑥^𝜈,𝜕𝑠 (2.14) where Γ^𝑎_𝑏𝑐 is a Christoffel symbol of the second kind, we can always select coordinates such that Christoffel symbols vanish at the origin [6, 7] (e.g., we can select coordinates on the surface, at the location of the vector 𝑝^𝑟). Then, the vector ∑_𝜇,𝜈Γ^𝜌_𝜇𝜈𝑝^𝜇𝛿^𝜈 is generally very small in the vicinity of the origin, and can be neglected (i.e., a linear approximation of ∇_𝑠𝑝^𝜌 is assumed here. This approximation will be further justified in the following discussion and in Section 3). Therefore, let𝜂^𝑟=𝛿^𝑟_𝑆′−𝛿^𝑟_𝑆=𝜕𝑝^𝑟𝜕𝑠≈∇_𝑠𝑝^𝑟. (2.15) Further, let us define the deviation angle, 𝜓, as the angle between the two unit tangent vectors 𝛿^𝑟_𝑆, 𝛿^𝑟_𝑆′, at any point along the curve 𝑆. Generally, the angle between two curves is given by [7]cos𝜓=_𝜇,𝜈𝑔_𝜇𝜈𝑑𝑥^𝜇⋅𝑑𝑠𝑑𝑥^𝜈𝑑𝑠^′, (2.16) but since 𝑠=𝑠^′ is the length of the curve, and having 𝛿^𝑟=𝑑𝑥^𝑟/𝑑𝑠, we can writecos𝜓=_𝜇,𝜈𝑔_𝜇𝜈𝛿^𝜇_𝑆𝛿^𝜈_𝑆′. (2.17) Hence, from (2.15) and (2.17),_𝜇,𝜈𝑔_𝜇𝜈𝜂^𝜇𝜂^𝜈=_𝜇,𝜈𝑔_𝜇𝜈𝛿^𝜇_𝑆′𝛿^𝜈_𝑆′+𝛿^𝜇_𝑆𝛿^𝜈_𝑆−2𝛿^𝜇_𝑆′𝛿^𝜈_𝑆=21−cos𝜓≈_𝜇,𝜈𝑔_𝜇𝜈∇_𝑠𝑝^𝜇∇_𝑠𝑝^𝜈. (2.18) We also see thatcos𝜓≈_𝜇,𝜈𝑔_𝜇𝜈𝛿^𝜇𝛿^𝜈+∇_𝑠𝑝^𝜈≈1+_𝜇,𝜈𝑔_𝜇𝜈𝛿^𝜇∇_𝑠𝑝^𝜈. (2.19) Now, consider (2.8) and the summation_𝜇,𝜈𝑔_𝜇𝜈𝑝^𝜇∇²_𝑠𝑝^𝜈+𝐺_𝜇,𝜈𝑔_𝜇𝜈𝑝^𝜇𝑝^𝜈=𝐺_𝛼,𝛽𝑔_𝛼𝛽𝑝^𝛼𝛿^𝛽_𝜇,𝜈𝑔_𝜇𝜈𝛿^𝜇𝑝^𝜈=𝐺_𝜇,𝜈𝑔_𝜇𝜈𝛿^𝜇𝑝^𝜈2, (2.20) and let𝑃=_𝜇,𝜈𝑔_𝜇𝜈𝑝^𝜇𝑝^𝜈 (2.21) denote the Euclidean norm of the vector 𝑝^𝑟; thus,𝑑𝑃𝑑𝑠²=2_𝜇,𝜈𝑔_𝜇𝜈𝑝^𝜇∇_𝑠𝑝^𝜈,𝑑²𝑑𝑠²𝑃²=2_𝜇,𝜈𝑔_𝜇𝜈𝑝^𝜇∇²_𝑠𝑝^𝜈+∇_𝑠𝑝^𝜇∇_𝑠𝑝^𝜈. (2.22)(2.23) Substitution from (2.18), (2.21), and (2.23) into (2.20) gives12𝑑²𝑃²𝑑𝑠²−21−cos𝜓+𝐺𝑃²=𝐺_𝜇,𝜈𝑔_𝜇𝜈𝛿^𝜇𝑝^𝜈2. (2.24) To evaluate the last term in (2.24), we rewrite (2.8) as𝑝^𝜌=𝛿^𝜌_𝜇,𝜈𝑔_𝜇𝜈𝛿^𝜇𝑝^𝜈−1𝐺∇²_𝑠𝑝^𝜌. (2.25) Now, from (2.22) and (2.25), we have𝑑𝑃²𝑑𝑠=2_𝜇,𝜈𝑔_𝜇𝜈𝛿^𝜇_𝛼,𝛽𝑔_𝛼𝛽𝑝^𝛼𝛿^𝛽−1𝐺∇²_𝑠𝑝^𝜇∇_𝑠𝑝^𝜈=2_𝜇,𝜈𝑔_𝜇𝜈𝑝^𝜇𝛿^𝜈_𝜇,𝜈𝑔_𝜇𝜈𝛿^𝜇∇_𝑠𝑝^𝜈−2𝐺_𝜇,𝜈𝑔_𝜇𝜈∇²_𝑠𝑝^𝜇𝛿^𝜈_𝑠′−𝛿^𝜈_𝑠. (2.26) Each of the components in the last term of (2.26) vanishes identically. To prove this, we evaluate each of the components for each of the curves, 𝑆 and 𝑆^′, by substitution from (2.6). We have _𝜇,𝜈𝑔_𝜇𝜈𝛿^𝜇∇²_𝑠𝑝^𝜈=_𝛼𝛿^𝛼_{𝜌,𝜁,𝜇,𝜈}𝑔_𝛼𝜌𝑅^𝜌_𝜁𝜇𝜈𝛿^𝜁𝛿^𝜇𝑝^𝜈=_{𝛼,𝜁,𝜇,𝜈}𝑅_{𝛼𝜁𝜇𝜈}𝛿^𝛼𝛿^𝜁𝛿^𝜇𝑝^𝜈=_{𝛼,𝜁,𝜇,𝜈}𝐺𝑔_𝛼𝜇𝑔_𝜁𝜈−𝑔_𝛼𝜈𝑔_𝜁𝜇𝛿^𝛼𝛿^𝜁𝛿^𝜇𝑝^𝜈. (2.27) A straightforward summation shows that the right-hand side of (2.27) vanishes. We therefore conclude that ∇²_𝑠𝑝^𝑟 is in the direction normal to the curve. In plus ∇_𝑠𝑝^𝑟 is in the direction of the tangent to the curve.

Finally, substitution from (2.19) into the first term of (2.26) gives𝑑𝑃²𝑑𝑠=2_𝜇,𝜈𝑔_𝜇𝜈𝑝^𝜇𝛿^𝜈cos𝜓−1, (2.28) or_𝜇,𝜈𝑔_𝜇𝜈𝑝^𝜇𝛿^𝜈=−𝑑𝑃²/𝑑𝑠2,1−cos𝜓 (2.29) and hence (2.24) is further reduced to()𝑑𝐺=1/2²𝑃²/𝑑𝑠²−21−cos𝜓(𝑑𝑃²/𝑑𝑠)/2(1−cos𝜓)²−𝑃², (2.30) where 𝐺 is the Gaussian curvature of the surface at the location of the vector 𝑃. In the following section, we prove that the Gaussian curvature given by (2.30) must vanish in an Euclidean 2-space. In Section 4, it is further proven that 𝐺 is a projective invariant and hence can be calculated from any projected image of the curves 𝑆 and 𝑆^′.

3. Investigation of the Behavior of 𝐺 in an Euclidean Space

Here, we illustrate by a simple example that the Gaussian curvature 𝐺, given by (2.30), must vanish in an Euclidean 2-space.

Consider a right circular cone, shown in Figure 3.

Figure 3 

A right circular cone and the corresponding geometry.

𝑃 is the Euclidean norm of the position vector, and 𝜃 is the rotation angle (as discussed in the above text).

From the Figure 3, we see that𝑃²=2𝑟²(1−cos𝜃),𝑟=𝑠sin𝛼, (3.1) where 𝑠 is the length of the generator. Hence,𝑃²=2(1−cos𝜃)𝑠²sin²𝛼,𝑑𝑃²𝑑𝑠=4𝑠sin²𝛼(1−cos𝜃). (3.2) Thus𝑑²𝑃²𝑑𝑠²=4sin²𝛼(1−cos𝜃). (3.3) For an infinitesimal rotation, cos𝜃 is expressed by the first two terms of its power series, that is,𝜃cos𝜃≈1−²2!, (3.4) and thus, (3.3) is written as𝑑²𝑃²𝑑𝑠²=2sin²𝛼𝜃². (3.5) Furthermore, we can see that𝑟²𝜃²=2𝑠²1−cos𝜓, (3.6) where 𝜓 is the deviation angle, or=11−cos𝜓2sin²𝛼𝜃². (3.7) From (3.5) and (3.7), we have 𝑑²𝑃²𝑑𝑠²=41−cos𝜓. (3.8) By comparison of (2.30) and (3.8), we immediately see that 𝐺 must vanish in an Euclidean 2-space. This proves the correctness of (2.30).

4. Proof That the Gaussian Curvature 𝐺 Is a Projective Invariant

Now, we will reach our final goal by demonstrating that 𝐺, formulated by (2.30), can be measured directly in the image plane.

We rewrite (2.8) as∇𝐺=−^2𝑠𝑝^𝜌𝑝^𝜌−𝛿^𝜌∑_𝜇,𝜈𝑔_𝜇𝜈𝛿^𝜇𝑝^𝜈. (4.1) From (2.27), we saw that ∇²_𝑠𝑝^𝑟 is a vector in the direction normal to the curve. Now, by using (2.9) and taking the summation_𝜇,𝜈𝑔_𝜇𝜈𝛿^𝜇𝑝^𝜈−_𝜇,𝜈𝑔_𝜇𝜈𝛿^𝜇𝛿^𝜈_𝜇,𝜈𝑔_𝜇𝜈𝛿^𝜇𝑝^𝜈=0, (4.2) it is easy to see that the denominator in (4.1) is also a vector in the direction normal to the curve.

If we now let∇²_𝑠𝑝^𝜌=𝛼𝑣^𝜌,𝑝^𝜌−𝛿^𝜌_𝜇,𝜈𝑔_𝜇𝜈𝛿^𝜇𝑝^𝜈=𝛽𝑣^𝜌, (4.3) where 𝛼, 𝛽 are scalars, and 𝑣^𝜌 is a vector in the direction normal to the curve, then𝛼𝐺=−𝛽. (4.4) Now, consider the orthographic projection of the two vectors in (4.3), written as∇²_𝑠𝑝^𝜌=_𝜎𝐽^𝜌_𝜎∇²_𝑠𝑝^𝜎=𝛼_𝜎𝐽^𝜌_𝜎𝑣^𝜎,𝑝^𝜌−𝛿^𝜌_𝜇,𝜈𝑔_𝜇𝜈𝛿^𝜇𝑝^𝜈=_𝜎𝐽^𝜌_𝜎𝑝^𝜎−𝛿^𝜎_𝜇,𝜈𝑔_𝜇𝜈𝛿^𝜇𝑝^𝜈=𝛽_𝜎𝐽^𝜌_𝜎𝑣^𝜎, (4.5) where 𝐽^𝑎_𝑏 is a transformation Jacobian between a coordinate system on the surface and a coordinate system in the image plane.

If measured in the image plane, the Gaussian curvature 𝐺 is now given by𝐺=−∇²_𝑠𝑝^𝜌𝑝^𝜌−𝛿^𝜌∑_𝜇,𝜈𝑔_𝜇𝜈𝛿^𝜇𝑝^𝜈𝛼=−𝛽, (4.6) as we can easily see from (4.5).

Hence,𝐺=𝐺. (4.7) The Gaussian curvature is therefore a projective invariant. It should be noted that, while orthographic projection is assumed, the image plane may be placed in any arbitrary position with respect to the curve 𝑆, and for all such positions, the Gaussian curvature 𝐾 holds the same numerical value. Equation (2.30) should be used in the image plane to obtain a correct measurement of 𝐾.