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Parameterization of Surfaces
Adrian Down
September 28, 2006
1 Review
1.1 Motivation
We need to develop the machinery necessary to parameterize surfaces. We could imagine constructing a surface as a composition of a number of curves in three dimensions. However, this approach is difficult because of the com- plicated geometry of the surface in three dimensions. A better approach is to consider the surface as a map from a two-dimensional space to R^3. This para- metric description has the advantage that we can gain information about the surface by doing computations in the image in R^2 using the tools of regular multivariable calculus.
1.2 Parameterization
We will consider the domain of a a surface to be a region in R^2 denoted as U with axes u 1 and u 2. We can define a surface with a condition similar to the normality condition used for curves,
Definition (Simple surface). A simple surface (or coordinate patch) is a function x : U → R^3 such that,
|x 1 × x 2 | 6 = 0 ∀ (a, b) ∈ U
Note. Recall the notation for derivatives,
xi =
dx dui^
dx^1 du^1
dx^2 du^2
dx^3 du^3
Differentiation can be carried out componentwise.
This definition ensures that x 1 and x 2 are both nonzero and are not parallel. Drawing these vectors, it can be seen that at any point on a surface, they define a plane tangent to the surface. Since these vectors are not linearly dependent, they form a basis for this space. However, we will have to do more work to show that these vectors form a basis for this tangent space. To form a basis for R^3 , we will need a third linearly independent vector. It is the normal vector, defined as,
n =
x 1 × x 2 |x 1 × x 2 |
Note. n is well-defined, since |x 1 × x 2 | 6 = 0 by the assumption of regularity.
2 Reparameterization
2.1 Motivation
Suppose the image of a curve x : U → R^3 can be equivalently described by y : V → R^3 , which maps from a different region of R^2. We would like to see how to transform between these two coordinate systems. This is useful for at least two reasons. Firstly, any property that we claim is geometric must be invariant under transformation, so we formulas for transformation to check which properties of a surface are actually geometric. Secondly, it may sometimes be possible to simplify calculations on a surface by going to an appropriate coordinate system.
2.2 Jacobian
Definition (Coordinate transformation). Let U and V be open subsets ⊂ R^2. A coordinate transformation f : V → U is a smooth, bijective (one-to-one and onto) mapping with a smooth inverse f −^1 ≡ g : U → V.
Notation. f is a function of the variables in the V coordinates,
f = f (v^1 , v^2 ) = f
f 1
v^1 , v^2
, f 2
v^1 , v^2
The inverse g ≡ f −^1 is a function of the coordinates in U,
g = g(u^1 , u^2 ) = g
g^1
u^1 , u^2
, g^2
u^1 , u^2
We can also identify J(g) in the above equation for ∂f^
i ∂uj^. Thus, J(f )J(g) = δij
To check if the Jacobian matrix is invertible, we should verify that it has nonzero determinant. However, we have found that the Jacobian matrix for f times the Jacobian matrix of g is equal to the identity matrix, and so J−^1 (f ) = J(g).
2.3 Transforming the tangent plane and normal
Definition (Reparameterization). Let f be a coordinate transformation from V to U. If x is a simple surface, then y = x ◦ f is a reparameterization of x.
The next theorem says that the tangent plane and the normal are un- changed under coordinate transformation, meaning that the tangent plane is geometric. However, since the parameterization may change the orientation of the curve, the normal may change sign.
Theorem. If x : U → R^3 is a simple surface, f : V → U is a coordinate transformation, and y = x ◦ f , then
- the tangent plane to the simple surface x at P = x (f (a, b))is equal to the tangent plane to the simple surface y at P = y(a, b).
- the normal to the surface x at P is the same as the normal to the surface y at P up to a minus sign.
Proof. We will not present the full details of the proof. The main idea is that we must check how the cross product changes under reparameterization. A change of variables is accomplished with the Jacobian matrix, and so calculations omitted here would show,
∂y ∂v^1
×
∂y ∂v^2
= det(J(f ))
∂x ∂u^1
×
∂x ∂u^2
The full details of this derivation are presented in the book. Since x is a surface, the cross product of the derivatives of x is nonzero. We showed above that J(f ) is invertible, and hence has nonzero determinant. This determinant can take the values ±1. In the case that the determinant of the Jacobian matrix is equal to −1, the reparameterization changes the orientation of the surface.
3 Tangent vectors
3.1 Motivation
At first, this discussion of the tangent plane may seem to be a step back from what we have already done. We have already defined the tangent plane, and we have assumed thus far that the tangent vectors span this tangent plane. However, we have not yet proven this rigorously. We will move towards the study of geometric quantities, separating those that are intrinsic from those that are extrinsic. Intrinsic properties are those that can be calculated using only knowledge of the local properties of the surface. Extrinsic properties are those that can be measured with knowl- edge of the entire surface. Intrinsic properties are of interest because they represent those quantities that could be measured by a group of intelligent bugs crawling around on the surface with know knowledge of the larger three dimensional space in which the surface is embedded. We will define the first fundamental form of the curvature of a surface, which is related to the curvature that can be measured intrinsically, and the second fundamental form, which is related to the curvature that can be measured extrinsically.
3.2 Definition
With this distinction between intrinsic and extrinsic properties in mind, we would like to define tangent vectors in terms of quantities that can be mea- sured locally on a surface.
Definition (Tangent vector). A vector X is a tangent vector to a simple surface x : U → R^3 at a point P = x(a, b) if X is the velocity vector at P of some curve in x(U). Equivalently, X is a tangent vector if and only if for some > 0, ∃ a curve α : (−, ) → x(U) ⊂ R^3 such that
- α(0) = P
- d dtα (0) = X
Note. The curve α in the definition does not have to be parameterized by arc length.
Taking the derivative of β,
dβ dt
= r
dα dt ⇒
dβ dt
(0) = r
dα dt
(0) = rX
By the definition of a tangent vector β(t) satisfies the requirements such that rX is a tangent vector. Hence a scalar multiple of a tangent vector remains in the space of tangent vectors. To prove the vector sum of two tangent vectors is also a tangent vector, we take a similar strategy of considering the curves in R^2 that define the two tangent vectors. We construct a new tangent vector that is the vector sum of the original two tangent vectors by combining the curves in R^2. We can translate between R^2 and the surface by the function x. Suppose we have two tangent vectors X and Y. It is possible to find functions αi^ and βi^ such that,
α = x
α^1 (t), α^2 (t)
β = x
β^1 (t), β^2 (t)
such that
α(0) = β(0) = P dα dt
(0) = X
dβ dt
(0) = Y
We will define a new curve in R^2 by combining the components of these curves. We can map this sum to the surface using the function x. Next time we will provide the full details of this step and show that the curve which is the sum corresponds to a tangent vector.
