Analytical Cartography

Using elementary vector calculus and  linear algebra, the authors obtain formulas for the Mercator and Lambert projections.  The inherent distortion of these projections is then investigated graphically using Mathematica.

by Clayton T. Hester and Thomas L. Miller

A map-projection is a representation of positions on the Earth onto a flat surface. Since there is inevitably some distortion, a variety of projection schemes have been developed for particular applications. Historically though, the most important application of cartography has been the construction of navigational charts, and for this purpose, perhaps the most crucial information a map conveys regards direction. Conformal maps are those that preserve angles, ie, relative directions at every point, and for this reason, these are most appropriatefor navigation and are the most familiar in appearance. Because conformality is a local property, it may be translated into a condition on the derivative of a transformation, and this is the basis of our constructions. This derivation is a nice application of elementary linear algebra, differential equations and vector calculus, but it really comes alive when it is applied graphically. We will compare different mapping schemes and study their inherent distortion by plotting various regions of the United States. We will also rotate regions on the globe to study the distortion inherent in various projections.  In particular, we will ``Mess with Texas'': how far north must Texas be rotated so that its projection is the same size as that of Alaska?  

Two common conformal projections are based on projections of points on a spherical globe onto cylinders or cones that are then unfurled to obtain a flat map.  These,  the Mercator and Lambert projections respectively, are the focus of this note. The figure below shows the 48 contiguous states on the globe (blue) and their Mercator projection. (Along with other graphs of the United States, this is based on state boundary coordinates obtained from the  US Geological Survey, [5])

When we unroll the cylinder, we obtain the following  familiar-looking map below.

We can also compare projections.  The following is an equi-rectangular projection of the ``lower 48''; that is, a simple plot of latitude against longitude with the same constant scale for each coordinate.

Notice how the latitudinal and longitudinal lines form a regular grid, but, when compared to the familiar Mercator projection above, this map appears to be compressed in the North-South direction.  Also notice how the grid in the Mercator projection in Figure 2 varies with latitude;  this North-South stretch corresponds to the conformal adjustment of the Mercator projection.  At large latitudes this results in considerable distortion. We will examine this in more detail in Sections 4.1.1 and 4.2.1, but for an example now, consider the Mercator projection of Alaska:

The other projection we discuss is constructed by mapping regions on the Earth onto a cone that intersects the globe at two specified latitudes. This, the Lambert projection, results in much less North-South distortion than that of Mercator, and because of this greater accuracy over large regions, Lambert projections were the principal types once used for air navigational charts. Of course, the use of large-scale navigational charts has now been largely supplanted by GPS systems; nevertheless,  ships still carry maps, and their construction was of great historical signifigance.

In Figure 6 below, we display Mercator and Lambert projections of Alaska.  A comparison with the blue representation in Figure 4 shows the Lambert projection to be much more faithful to the shape of Alaska on the globe than that of Mercator.

Similar constructions for these as well as other mapping schemes can be found in [4].  For general information about  cartography, see [3].

Vector Calculus Preliminaries

We will describe our map projections in terms of transformations between surfaces in (ℝ^3).  The surfaces of interest are open patches of spheres, cylinders and cones and are examples of so-called regular surfaces in (ℝ^3).  For a systematic account of the elementary theory of such surfaces see [1] or [2], but we sketch the basic calculus that is needed.  If 𝒮 is a regular surface, then for every p ∈ 𝒮 there is neighborhood V in (ℝ^3) and a one-to-one, continuously differentiable function F defined on a region U of the plane (ℝ^2) so that 𝒮 ⋂ V = F(U).  What's more, there is a continuously differentiable function G : V -> (ℝ^3)so that G(F(u)) = u for all  u ∈ Uand F(G(p)) = pfor every p ∈ 𝒮 ⋂ 
          V.  

As an example, consider the following patch on a spherical globe of radius ρ. The section of  the globe containing North America has parameterization

\(TraditionalForm\`\(\(\((x, y, 
          z)\) = \(S(θ, φ) = \((ρ\ \(cos(φ)\)\ \(cos(θ)\), ρ\ \(cos(φ)\)\ \(sin(θ)\), ρ\ \(sin(φ)\)\ )\)\)\)\(,\)\)\)

where the longitude θ is the angle east of the Prime Meridian (Greenwich), -π < θ < 0, and the latitude φ is the angle north of the Equator, 0 < φ < π/2.  The inverse mapping is

(θ, φ) = (S^(-1))(x, 
            y, z) = (-π/2 - arctan(x/y),  
            arctan(z/(Sqrt[(x^2) + (y^2)]))),

where arctan denotes the principal branch of the arctangent, -π/2 < arctan s < π/2, and where z > 0, y < 0 and (x^2) + (y^2) + (z^2) = (ρ^2).  Clearly (S^(-1))extends to a region in V ⊂ (ℝ^3). 

If F is a continuously differentiable function from a region U in (ℝ^n) into (ℝ^m), then the derivative of F at p is a linear transformation from (ℝ^n)into (ℝ^m) which we denote by D F(p). We identify this linear transformation with its matrix representation with respect to the standard bases for (ℝ^n)and (ℝ^m). This is the Jacobian matrix for F: if ((Subscript[y, 1]), (Subscript[y, 2]),  ... , (Subscript[y, m])) = 
                F((Subscript[x, 1]), (Subscript[x, 2]),  ... , (Subscript[x, n])), and u = ((Subscript[u, 1]), (Subscript[u, 2]), …, 
            (Subscript[u, n])) ∈ (ℝ^n), then

\(TraditionalForm\`\(\(\ \)\(D\ \(F(
          p)\) \((u)\) = \(\(\(\((∂\ y\_i\/∂\ x\_j)\)\_\(1 ≤ i ≤ m, 
1 ≤ 
                j ≤ 
                n\)\) \((u\_1, u\_2, …, u\_n)\)\^T\)\(.\)\)\)\)\)

By the chain rule, the Jacobian matrix of a composite function F∘G is simply the product of the Jacobian matrices of F and  G; that is,  D(F∘G) (p) = 
        D F(G(p))  D G(p).
For the spherical transformation S above, we have

D S(θ, φ)=ρ(({{\(\(-\ \(cos(φ)\)\)\(\ \)\(sin(θ)\)\(\ \)\), - sin(φ) cos(θ)}, {  cos(φ) cos(θ), - sin(φ) sin(θ)}, {0,  cos(φ)}})), D (S^(-1))(S(θ, φ))=(ρ^(-1))(({{-sec(φ) sin(θ), \(\(sec(φ)\) \(cos(θ)\)\), 0}, {-sin(φ) cos(θ), -sin(φ) sin(θ), cos(φ)}})), and

\(D\ \(\(S\^\(-1\)\)(
                  S(θ, φ))\)\ D\ \(S(θ, φ)\)\)=(({{1, 0}, {0, 1}}))=D ((S^(-1))∘
                      S) (θ, φ).

If 𝒮 is a regular surface with parameterization (x, y, z) = F(s, t) in a neighborhood of a point p = F((Subscript[s, 0]), (Subscript[t, 0])) ∈ 𝒮,  define the tangent space for 𝒮 at p to be (Subscript[T, 𝒮])(p)=range(D F((Subscript[s, 0]), (Subscript[t, 0]))). Then (Subscript[T, 𝒮])(p) has basis the columns (Subscript[F, s])((Subscript[s, 0]), 
          (Subscript[t, 0])) = ∂(F/∂)s ((Subscript[s, 0]), (Subscript[t, 0])) and (Subscript[F, t])((Subscript[s, 0]), 
          (Subscript[t, 0])) = ∂(F/∂)t ((Subscript[s, 0]), (Subscript[t, 0])) of D F((Subscript[s, 0]), (Subscript[t, 0])), and the plane tangent to 𝒮  at p is the translation of (Subscript[T, 𝒮])(p). 

Finally, suppose that W is an open subset of (ℝ^3) containing p ∈ 𝒮, and that Φ is a diffeomorphism on W; that is, suppose that Φ : 
          W -> (ℝ^3) is a continuously differentiable function whose inverse is also continuously differentiable on the open set Φ(W) in (ℝ^3).  Then  \(TraditionalForm\`𝒯\( = \&def\)Φ(𝒮 ⋂ W)\) is also a regular surface, and the restriction of D Φ(p) to (Subscript[T, 𝒮])(p) is a linear isomorphism between (Subscript[T, 𝒮])(p) and (Subscript[T, 𝒯])(Φ(p)). We will construct diffeomorphisms from the sphere onto cylinders or cones by ``factoring'' through the given parameterizations. Even though defining a diffeomorphism this way specifies the mapping only on the surface patch, the resulting map can be extended to a diffeomorphism between open sets in (ℝ^3).  In our examples there is a natural extension that can be written down explicitly, but because we're only interested  in the action on the tangent spaces, this step is in fact unnecessary.

Conformal Transformations

A diffeomorphism G : 𝒮 -> 𝒯 is conformal at a point p ∈ 𝒮 if G preserves angles between curves on 𝒮 that intersect at p.  If (Subscript[γ, 1])and (Subscript[γ, 2]) are curves on 𝒮 that intersect at p, then the angle between (Subscript[γ, 1]) and (Subscript[γ, 2]) is the angle between their respective tangents at p.  Since the tangents are arbitrary vectors in the tangent space (Subscript[T, 𝒮])(p),  the Cauchy-Schwartz formula and the chain rule allows us to restate the definition of conformality: G : 𝒮 -> 𝒯 is conformal p ∈ 𝒮 if and only if

\(⟨u, 
                  v⟩\/\(\(|\)\(|\)\(u\)\(|\)\(|\)\(\ \)\(|\)\(|\)\(v\)\(|\)\(|\)\)\)=\(⟨D\ \(G(p)\) u, \ 
                    D\ \(G(p)\) 
                      v⟩\/\(\(|\)\(|\)\(D\ \(G(p)\) 
                      u\)\(|\)\(|\)\(\ \)\(|\)\(|\)\(D\ \(G(p)\) 
                      v\)\(|\)\(|\)\)\) for all u,v ≠ 
          0 in (Subscript[T, 𝒮])(p).

It turns out to be easy to characterize conformal linear transformations on two-dimensional spaces.   The elementary proof of the following is omitted.

Theorem 1. Suppose that X and Y are two-dimensional subspaces of (ℝ^n), and that L : X -> Y is an invertible linear transformation. If  (Subscript[u, 1])and (Subscript[u, 2]) form an orthonormal basis for X, then L is conformal if and only if

L (Subscript[u, 1])⊥ L (Subscript[u, 2])   and   ||L (Subscript[u, 1])||=|| L (Subscript[u, 2])||.

Recall that a linear transformation is an isometry, that is, length-preserving, if it maps an orthonormal basis to an orthonormal basis. From Theorem 1, we see that the only difference between conformal and isometric linear transformations is a scaling coefficient. Applying Theorem 1 to the setting of differentiable functions, we obtain the following characterization of conformal transformations between surfaces.

Corollary. Let  G : 𝒮 -> 𝒯 be a diffeomorphism and let U(p) be a 3×2matrix whose columns form an orthonormal basis for the tangent space (Subscript[T, 𝒮])(p).  Then G is conformal at p if and only if the Jacobian matrix for G satisfies

(((D G(p) U(p)))^T) (D G(p) U(
                  p))=c(p) (({{1, 0}, {0, 1}}))

for some positive scalar c(p).

Proof. Let  (Subscript[u, k])=((Subscript[u, 1  k]),  (Subscript[u, 2  k]), (Subscript[u, 3  k])),  k = 1, 2, be an orthonormal basis for (Subscript[T, 𝒮])(p), and let U(p) be the matrix with columns (Subscript[u, 1])and (Subscript[u, 2]).  If V(p) = D G(p) U(p), then V(p) has columns (Subscript[v, k])=D G(p)(Subscript[u, k]), k = 1, 2, and

((V(p))^T) V(p)=(({{⟨(Subscript[v, 1]),(Subscript[v, 1])⟩, ⟨(Subscript[v, 1]),(Subscript[v, 2])⟩}, {⟨(Subscript[v, 2]),(Subscript[v, 1])⟩, ⟨(Subscript[v, 2]),(Subscript[v, 2])⟩}})).

Thus ((V(p))^T) V(p) is a multiple of the identity if and only if (Subscript[v, 1])⊥(Subscript[v, 2]) and ||(Subscript[v, 1])||=||(Subscript[v, 2])||. The result now follows from Theorem 1.

The Mercator and Lambert Projections

Recall the parameterization of the sphere of radius ρ, (Subscript[𝒮, ρ]),

\(TraditionalForm\`\(\(\((x, y, 
          z)\) = \(S(θ, φ) = \((ρ\ \(cos(φ)\)\ \(cos(θ)\), ρ\ \(cos(φ)\)\ \(sin(θ)\), ρ\ \(sin(φ)\)\ )\)\)\)\(,\)\)\)

where θ is the longitude of a point and φ its latitude. Since the columns of the corresponding Jacobian matrix (4)  are orthogonal,  we obtain a natural orthonormal basis for the tangent space at S(θ, φ).  Let 
(Subscript[u, 1])(θ, φ)=\(\(\(\(S\_θ\)(θ, φ)\)\(/\)\) || \(S\_θ\)(θ, φ) || \), (Subscript[u, 2])(θ, φ)=\(\(\(\(S\_φ\)(θ, φ)\)\(/\)\) || \(S\_φ\)(θ, φ) || \) and let U(θ, φ) be the matrix with columns (Subscript[u, 1])(θ, φ) and (Subscript[u, 2])(θ, φ).

(Subscript[u, 1])(θ, φ)=(-sin(θ),  cos(θ), 0), (Subscript[u, 2])(θ, φ)=\((\(-\(sin(φ)\)\)\ \(cos(θ)\), \ \(-\(sin(φ)\)\)\ \(sin(θ)\), cos(φ))\)

 and 
U(θ, φ)=D S(θ, φ)(({{(ρ^(-1)) sec(φ), 0}, {0, (ρ^(-1))}}))

It is convenient to describe the  cylinders and cones onto which the globe is projected in terms of cylindrical coordinates. In both the Mercator and Lambert projections, the longitude of a point is preserved and
conformality is obtained by adjusting the z or r coordinates of the projections.  In the case of the Mercator projection onto a cylinder of radius ρ, the projection of a point x=S(θ, φ) will have form

(Subscript[P, M])(x)=(ρ cos(θ), ρ sin(θ),  
            z(φ)).

For the Lambert projection onto a cone z = b + m r,  the transformation will have form

(Subscript[P, L])(x)=\(\((\(r(φ)\)\ \(cos(θ)\), \ \(r(φ)\)\ \(sin(θ)\), \ b + m\ \(r(φ)\))\)\(.\)\)

In either case, the functions r(φ)and z(φ) will be determined by differential equations based on the conformality condition (7).  Finally, we obtain our flat maps by isometrically ``unrolling'' the target cylinder and cone.

The Mercator Projection

Theorem 2.  The Mercator projection of a point on the globe with longitude θ and latitude φ is

M(θ, φ) = 
      c (θ,  
          ln(sec(φ) + 
              tan(φ))),

where c is a positive constant that determines the scale of the map.

Proof. Let (Subscript[𝒯, ρ]) be the circular cylinder with axis of symmetry the z-axis, and define R to be the usual parameterization of (Subscript[𝒯, ρ]),

R(θ, 
        z) = (ρ cos(θ), ρ sin(θ), 
          z).

Because our region of interest has longitudes -π < θ < 0, we use the branch of (S^(-1))given in (2).
Our projection onto the cylinder has form

(Subscript[P, M])(x)=R∘F∘(S^(-1))(x),

where (S^(-1))(x)=(θ, φ) and where  F(θ, φ) = (θ, 
          z(φ)) is the function to be determined by condition (7).
Let U(θ, φ) be the matrix (8) corresponding to the natural orthonormal basis for (Subscript[T, (Subscript[𝒮, ρ])])(
        S(θ, φ)), and let V(θ, φ) = 
        D (Subscript[P, M])(
            S(θ, φ)) U(θ, φ).  By the chain rule and (4) we compute:

V(θ, φ)=\(\((D\ \(R(
                  F(θ, φ))\)\ D\ \(F(θ, φ)\)\ D\ \(\(S\^\(-1\)\)(
                  S(θ, φ))\)\ )\) \(U(θ, φ)\)\)
       =(D R(θ, z(φ)) (({{1, 0}, {0, z' (φ)}})) D (S^(-1))(S(θ, φ))) D S(θ, φ) (({{(ρ^(-1)) sec (φ), 0}, {0, (ρ^(-1))}}))
     = D R(θ, z(φ)) (({{1, 0}, {0, z' (φ)}}))( D (S^(-1))(S(θ, φ)) D S(θ, φ) (({{(ρ^(-1)) sec (φ), 0}, {0, (ρ^(-1))}})))
 =(({{-ρ sin (θ), 0}, {ρ cos (θ), 0}, {0, 1}})) (({{1, 0}, {0, z' (φ)}})) (({{(ρ^(-1)) sec (φ), 0}, {0, (ρ^(-1))}}))  
= (({{-sec (φ) sin (θ), 0}, {\(\(\ \)\(sec \((φ)\)\ cos \((θ)\)\)\), 0}, {0, (ρ^(-1)) z' (φ)}})) .

Thus

\(\(V(θ, φ)\)\^T\ \(V(θ, φ)\)\)=(({{(sec^2)(φ), 0}, {0, (ρ^(-2)) (((z' (φ)))^2)}})), and so by the corollary to Theorem1, the transformation (Subscript[P, M]) is conformal at S(θ, φ)if and only if

\(TraditionalForm\`z' \((φ)\) = \(±ρ\)\ \(\(sec(φ)\)\(.\)\)\)

A moments thought shows that we should choose z' > 0, and since points on the equator should map to themselves, z(0) = 0.  The function z is therefore

z(φ)=ρ (∫_0^φ)sec(t) dt =ρ ln(sec(φ) + tan(φ)),

and so by (9)

\(TraditionalForm\`\(P\_M\)(
        S(θ, φ)) = \(\((ρ\ \(cos(θ)\), ρ\ \(sin(θ)\), \ ρ\ \(ln(
              sec(φ) + 
                tan(φ))\))\)\(.\)\)\)

The cylinder (Subscript[T, ρ]) is isometrically uncurled by the transformation Γ(
          R(θ, z)) = (ρ θ, z); thus

\(TraditionalForm\`M(θ, φ) = \(Γ(\(P\_M\)(
            S(θ, φ))) = \(\((ρ\ θ, \ ρ\ \(ln(sec(φ) + tan(φ))\))\)\(.\)\)\)\)

Since multiplication by a positive scalar constant preserves conformality, the theorem is established.

Examples

Figures (1), (2) and (4) are applications of Theorem 2 to the USGS coordinate data for state boundaries in the United States.  This data, along with the Mathematica code used to construct our maps,  is available at  http://www2.msstate.edu/~lmiller/cartography.
Below, we illustrate the North-South distortion of the Mercator projection by rotating Texas northward and comparing the resulting projection with those of Alaska and Texas at its true latitude.   The problem of describing the rotation of regions on the globe along arbitrary meridians is itself an instructive exercise in elementary linear algebra that is enhanced by the use of Mathematica.   Figure (7) shows Alaska and Texas along with copys of Texas rotated respectively by \(TraditionalForm\`\(15\^∘\), \ \(30\^∘\)\) and (45^∘) along the meridian θ = (99.54^∘)west.  Figure (8) displays the corresponding Mercator projections.

The distortion of the Mercator projection is more accurately protrayed by using regular rectangular patches rather than the complicated state boundaries. Let (Subscript[P, 0]) denote the set of points with latitude \(TraditionalForm\`\(-\(5\^∘\)\) ≤ φ ≤ \(5\^∘\)\)and longitude \(TraditionalForm\`\(-\(95\^∘\)\) ≤ θ ≤ \(85\^∘\)\)east; let (Subscript[P, α]) be the region obtained by rotating (Subscript[P, 0]) through an angle α along the (90^∘)meridian.  Figure (9) shows the pathches (Subscript[P, α]),  α = (0^∘), \(TraditionalForm\`\(20\^∘\), \ \(40\^∘\), \(60\^∘\)\) and (80^∘), and Figure (10) displays their respective Mercator projections.

The Lambert Projection

A Lambert projection with standard parallels  φ = a and φ = b, 0 ≤ a < b ≤ π/2, is the projection of the sphere (Subscript[𝒮, ρ]) onto a cone (Subscript[𝒯, a, b]) that intersects the sphere in the two circles  φ = a and φ = b. The cylindrical equation of (Subscript[𝒯, a, b]) is

z = ρ sin(
                      a) + ((sin(b) - sin(a))/(cos(b) - 
                          cos(a))) (r - ρ cos(a)),

and therefore a parameterization in cylindrical coordinates is

\(TraditionalForm\`\((x, y, 
        z)\) = \(R(θ, 
          r) = \(\((r\ \(cos(θ)\), \ 
            r\ \(sin(θ)\), ρ\ \(sin(
                  a)\) + \(\(sin(b) - sin(a)\)\/\(cos(b) - 
                      cos(a)\)\) \((r - ρ\ \(cos(
                        a)\))\))\)
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  = \(\((r\ \(cos(θ)\), \ 
              r\ \(sin(θ)\), ρ\ \(sin(
                    a)\) - \(cot(\(a + b\)\/2)\) \((r - ρ\ \(cos(
                          a)\))\))\)\(.\)\)\)\)\)

We will unroll the cone via the transformation

\(TraditionalForm\`\(Γ\_\(a, b\)\)(
          R(θ, 
            r)) = \((r\ \(csc(\(a + b\)\/2)\)\ \(sin(\(sin(\(a + b\)\/2)\) θ)\), \(-r\)\ \(csc(\(a + b\)\/2)\)\ \(cos(\(sin(\(a + b\)\/2)\) θ)\))\)\).

Lemma. The transformation (Subscript[Γ, a, b]) is an isometry from the cone (Subscript[𝒯, a, b]) into the plane.

Proof. If \(TraditionalForm\`G(θ, 
          r) = \((r\ \(csc(\(a + b\)\/2)\)\ \(sin(\(sin(\(a + b\)\/2)\) θ)\), \(-r\)\ \(csc(\(a + b\)\/2)\)\ \(cos(\(sin(\(a + b\)\/2)\) θ)\))\)\), then (Subscript[Γ, a, b]) = 
        G∘(R^(-1)).
Now we compute: R has Jacobian matrix

D R(θ, r)=(({{-r sin(θ), cos(θ)}, {r cos(θ), sin(θ)}, {0, -cot(((a + b)/2))}}))

and the tangent plane at  R(θ, r) has orthonormal basis (Subscript[u, 1])=(-sin(θ),  cos(θ), 0) and  (Subscript[u, 2])=sin(((a + b)/2)) (cos(θ),  
              sin(θ), -cot(((a + b)/2))). If U is the matrix with columns (Subscript[u, 1])and (Subscript[u, 2]), then

U=D R(θ, r) (({{(r^(-1)), 0}, {0, sin(((a + b)/2))}}))

G has Jacobian

D G(θ, r)=(({{r cos(θ sin(((a + b)/2))), \(\(csc(\(a + b\)\/2)\)\ \(sin(θ\ \(sin(\(a + b\)\/2)\))\)\)}, {r sin(θ sin(((a + b)/2))), \(\(\ \ \)\(\(-\(csc(\(a + b\)\/2)\)\)\ cos\\(left(θ\ \(sin(\(a + b\)\/2)\))\)\)\)}}))

If V = 
        D (Subscript[Γ, a, b])(R(θ, r)) U$, then

V=D G(θ, r)D (R^(-1))( R(θ, r))(D R(θ, r)  (({{(r^(-1)), 0}, {0, sin(((a + b)/2))}})))

=(({{cos(θ sin(((a + b)/2))), \(sin(θ\ \(sin(\(a + b\)\/2)\))\)}, { sin(θ sin(((a + b)/2))), -cos(θ sin(((a + b)/2))) }})),

and so (V^T) V=(({{1, 0}, {0, 1}})). By the corollary to Theorem 1, (Subscript[Γ, a, b]) is an isometry.

Theorem 3.  The Lambert projection from the sphere of (Subscript[𝒮, ρ]) onto the cone (Subscript[𝒯, a, b]) is given by 
 (Subscript[P, L])( S(θ, φ)) =  
        R(θ, r(φ)),  where R is the cylindrical parameterization of the cone (12) and

r(φ)=ρ((cos( a)  +  cos(b))/2)((( \(\(sec\ \((φ\ )\)\  + \ 
                        tan\ \((φ\ )\)\)\/\(sec(\(a\  + \ b\)\/2)\  + \ sec(\(a\  + \ b\)\/2)\)\)))^(-sin(((a  +  b)/2)))).

Therefore, the Lambert map projection with standard parallels φ = aand φ = b, 0 ≤  a < b ≤ π/2, is

\(TraditionalForm\`\(\(\ \)\(\(L\_\(a, b\)\)(
          S(θ, φ)) = \(\(Γ\_\(a, b\)\)(\ 
            R(θ, 
              r(φ)))
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  = \(\((\(r(φ)\) \(csc(\(a + b\)\/2)\)\ \(sin(\(sin(\(a + b\)\/2)\) θ)\), \ \(-\(r(φ)\)\) \(csc(\(a + b\)\/2)\)\ \(cos(\(sin(\(a + b\)\/2)\) θ)\))\)\(.\)\)\)\)\)\)

Proof.  The proof proceeds as in the case of the Mercator projection, and so we only give an outline. With U as in the proof of Theorem 2, and V = 
        D  (Subscript[P, L])(S(θ, φ)) U, we are led by the corollary to find the function r(φ) so that (V^T) V is a multiple of the identity. By the chain rule,

V=(D R(θ, r(φ)) (({{1, 0}, {0, r' (φ)}})) D (S^(-1))(S(θ, φ))) U,
and
(V^T) V=(({{(ρ^(-2)) 
                    r' (φ) (sec^2)(φ), 0}, {0, \(\(\(\(ρ\^\(-2\)\)(
                          r' \((φ)\))\)\^2\) \(\(csc\^2\)(\(a + b\)\/2)\)\)}})),

and our conformality condition is

\(TraditionalForm\`r' \((φ)\) = \(±\(sin(\(a + b\)\/2)\)\)\ \(r(φ)\)\ \(\(sec(φ)\)\(.\)\)\)

The normal to the cone at a point R(θ, ρ (( cos(a) + cos(b))/2)), midway between the standard parallels, 
is normal to the sphere at S(θ,((a + b)/2)). Since projections along normals are isometric, specifying 
\(TraditionalForm\`r(\(a + b\)\/2) = ρ\ \(cos(a)\  + \ cos(b)\)\/2\), gives us the best possible projection at this latitude. 
In order to preserve orientation, we need r to be a decreasing function of φ; thus the function we seek is the solution of the initial value problem

\(TraditionalForm\`\ \ \ \ r' \((φ)\) = \(-\(sin(\(a + b\)\/2)\)\)\ \ \(r(φ)\)\ \(sec(φ)\), \ 
    and\)
\(TraditionalForm\`r(\(a + b\)\/2) = ρ\ \(cos(a)\  + \ cos(b)\)\/2\)

The corollary to Theorem 1 guarantees that (Subscript[P, L]) is a conformal transformation from the sphere onto the cone, and the lemma above implies that  (Subscript[L, a, b]) = (Subscript[Γ, a, b])∘
          (Subscript[P, L]) is conformal.

Examples

Figure (6) above shows the Lambert projection of Alaska with standard parallels (45^∘)and (90^∘)north along side the corresponding Mercator projection. As a second example, Figure (11) contains the Lambert projection of the ``lower 48'' states corresponding to standard paralles (15^∘)and (60^∘);

Lambert projections are paricularly applicable for regions with relatively large ranges of latitude or for those at high latitudes.  Our final example is a Lambert projection of the eastern seaboard of the United States (standard parellels again φ =  (15^∘) and (60^∘)).  Notice the change in relative size of Maine.

References

1.  M. P.do Carmo, Differential Geometry of Curves and Surfaces, Prentice Hall,Englewood Cliffs,New Jersey,1976.

2.  A. Gray, Modern Differential Geometry of Curves and Surfaces, CRC Press, Boca Raton,Florida,1993.

3.  E. Raisz, Principles of Cartography, Mcgraw--Hill,New York,1963.

4.  P. Richardus and R. Adler, Map Projections, North Holland,New York,1972.

5. United States Geological Survey, http://mappings.usgs.gov

Acknowlegements.  This paper is based on work by the first author under the supervision of the second and was supported by an Undergraduate Reseach Award by the Northeast Mississippi Daily Journal, Fall, 1999.

About the Authors

Clayton T. Hester

Thomas L. Miller

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