Siggraph Course Mesh Parameterization: Theory and Practice Differential Geometry Primer

Parameterization surface parameter domain mapping and Generally speaking, a parameterization is a bijective mapping between a surface and a parameter domain. If the surface and the parameter domain have similar topology, then such a bijective mapping is guaranteed to exist. For the moment, we will only consider the case of the surface being in R3 and the domain being a planar patch in R2. If the surface is a triangular mesh, the problem of computing such a mapping is referred to as mesh parameterization.

Example – Cylindrical Coordinates A first simple example of such a parameterization are cylindrical coordinates. Imagine an open cylinder (without bottom and top) being cut open along its side and unrolled in the plane. This gives a rectangle and there exists a natural bijective mapping between the points in this rectangle and the points on the surface of the cylinder. This mapping is a parameterization of the cylinder surface and very handy, because it allows us to capture the cylinder by using only two coordinates u and v in the parameter domain instead of the three coordinates x, y, and z in space.

Example – Orthographic Projection Another example is the orthographic projection, which can be used to “flatten” a half-sphere onto a disk. The mapping simply projects each surface point orthogonally into the plane and vice versa.

Example – Stereographic Projection A third well-known example is the stereographic projection. The mapping from the surface to the parameter domain can be constructed as follows: For any point on the northern hemisphere, consider the ray that connects it with the south pole. The corresponding parameter point is the intersection of that ray with the equatorial plane. In this example, a small computation also reveals the inverse mapping that maps from the circle to the northern hemisphere.

Example – Mappings of the Earth usually, surface properties get distorted orthographic ∼ 500 B.C. stereographic ∼ 150 B.C. Mercator 1569 Lambert 1772 The stereographic projection is one of the oldest ways to generate a map of the Earth and goes at least back to the Greek Hipparchus (~190–120 B.C.). In contrast to the even older orthographic projection, which was already known to the Egyptians, it has the advantage of being conformal, i.e. it preserves angles. Indeed, any infinitesimally small angle measured in the map is exactly the same as if measured on the corresponding point on the surface of the Earth. This property of being conformal, unfortunately comes at the price that the mapping does not preserve areas, an effect which is even better to see in the conformal cylindrical Mercator projection that goes back to the Flemish cartographer Gerardus Mercator (1512–1594). Obviously, Antarctica and Greenland seem far too big in proportion to regions near the Equator. The first who invented an equiareal projection was Johann Heinrich Lambert (1728–1777), but although this mapping gets the areas right, it quite drastically distorts angles and shapes in general. conformal (angle-preserving) equiareal (area-preserving)

Distortion is (almost) Inevitable Theorema Egregium (C. F. Gauß) “A general surface cannot be parameterized without distortion.” no distortion = conformal + equiareal = isometric requires surface to be developable planes cones cylinders A natural question to ask then is: can there be a mapping of the Earth that is both conformal and equiareal? Such a mapping is then called isometric. Unfortunately, the answer is negative. It was Carl Friedrich Gauß (1777–1855) who could prove that for general surfaces, any parameterization distorts either angles or areas – and usually both. The only surfaces that can be parameterized without distortions are those with zero Gaussian curvature, which are therefore called developable surface. Examples of such surfaces are planes, cones, and cylinders.

What is Distortion? parameter point surface point small disk around image of under shape of D But what actually is distortion? Well, let us take a look at some parameter point in the parameter domain. The parameterization maps this point to a corresponding surface point. And if we now consider a small disk around the parameter point, then it is likewise mapped to some region around the surface point. Usually, this region will not be a disk again – instead, it is a distorted disk, like here in the example of the orthographic parameterization. And it is the shape of this region that tells us if and how the parameterization locally distorts the metric space. Note that the stereographic parameterization distorts the disk much less – almost indistinguishable. f (D)

Linearization Jacobian of tangent plane at Taylor expansion of first order approximation of In order to better study the shape of the distorted disk, let us linearize the parameterization. The partial derivatives of the parameterization are vectors in space and span the tangent plane of the surface at the surface point. By expanding the parameterization in its Taylor series and considering only its first two terms, we can approximate the parameterization with a linear function that maps all points in the vicinity of the parameter point into the tangent plane around the surface point.

Infinitesimal Dis(k)tortion small disk around image of under shape of ellipse semiaxes and behavior in the limit This approximating function then maps the unit disk around the parameter point into an ellipse with semiaxes σ1 and σ2 around the surface point. As the radius of the disk goes to zero, two things happen: First, the ellipse scales linearly with the radius, that is, its shape stays the same and its semiaxes shrink at the same rate as the radius of the disk. Second, the shape of the ellipse that lives in the tangent plane gets closer and closer to the shape of the distorted disk that lives on the surface. By combining both observations, we can say that the parameterization maps an infinitesimally small disk around the parameter point into an infinitesimally small ellipse with semiaxes σ1 and σ2 around the surface point. Thus, σ1 and σ2 capture the local distortion of the mapping.

Linear Map Surgery Singular Value Decomposition (SVD) of with rotations and and scale factors (singular values) These local distortion parameters σ1 and σ2 turn out to be the singular values of the Jacobian. Indeed, we can decompose the Jacobian matrix into a product of very special matrices, that is, two rotations that enclose a diagonal stretch matrix. Thus, the Jacobian deforms the disk around the parameter point in the following way: First, it is rotated in the parameter plane and changes only its orientation, but not its shape. Second, it is stretched along the u and v direction with the factors σ1 and σ2 to form an ellipse. Moreover, the 3×2 matrix Σ also embeds this ellipse into three dimensional space by introducing a third and perpendicular dimension. Third, the ellipse is now rotated into the tangent plane at the surface point, so that the new third dimension becomes the surface normal.

Notion of Distortion isometric or length-preserving conformal or angle-preserving equiareal or area-preserving everything defined pointwise on We are now ready to formally define the notion of distortion, and in particular distinguish three cases: If the stretch factors are both 1, then the mapping locally resembles a rotation which does not distort anything and keeps all local distances invariant. This is what we call isometric or length-preserving. If both values are identical, then we have a local rotation plus uniform scaling which does not distort angles, but uniformly stretches lengths and areas. In this case, the mapping is called conformal or angle-preserving. If the product of the stretch factors is 1, then the area of the distorted ellipse is equal to the area of the circle and likewise for any other shape. The parameterization is then called equiareal or area-preserving. Note that these are local properties and it can easily happen that a mapping is isometric at some parameter point but neither conformal nor equiareal at another.

Example – Cylindrical Coordinates ⇒ isometric As an example, let us go back to the cylindrical coordinates. For this parameterization, it is quite easy to compute the derivative and we can see immediately that this is an isometric mapping. This corresponds to the statement made earlier, that cylinders are developable surfaces for which an isometric mapping exists.

Example – Orthographic Projection with a ⇒ neither conformal nor equiareal In case of the orthographic projection, a simple calculation reveals that it is neither conformal nor equiareal.

Example – Stereographic Projection with ⇒ conformal In case of the stereographic projection, we need to work a bit harder, but in the end we find the previously made statement confirmed: it is indeed a conformal mapping. However, notice that the size of the singular values varies for different parameter values u and v and this corresponds exactly to the local changes of area.

Computing the Stretch Factors first fundamental form eigenvalues of singular values of and In order to compute the singular values of the Jacobian, let us use a small trick. If we multiply the Jacobian with its transpose, then we get a 2×2 matrix, the first fundamental form of the mapping. This first fundamental form is also called the metric tensor. It is a positive definite and symmetric matrix with two real eigenvalues. The singular values of the Jacobian (the stretch factors of the mapping) are just the square roots of these eigenvalues.

Measuring Distortion local distortion measure has minimum at isometric measure conformal measure overall distortion Now that we know how to compute the singular values, we can proceed and continue to measure the distortion of a parameterization. First, we need some local measure that take two stretch factors and maps them to some real value. Often, this local measure is constructed such that it is minimal either for a locally isometric mapping or for all mappings that are locally conformal. The overall distortion of the mapping is then usually defined by taking the average of the local distortions at all points in the parameter domain.

Examples – Conformal Measures Conformal energy MIPS energy [Pinkall & Polthier 1993] [Lévy et al. 2002] [Desbrun et al. 2002] Here are some examples of local conformal measures. The conformal energy just takes the square of the difference between the stretch factors. Note that this can give a small value even if the stretch factors are relatively far apart – provided both values are small. Thus, this mapping tends to favor maps that locally shrink the metric. For example, E(1,1/2) = 1/8 = 1/4 E(2,1). The MIPS energy measures the condition number of the Jacobian matrix with respect to the Frobenius norm, which turns out to be ratio of the stretch factors plus its inverse. Being symmetric in this way, it penalizes shrinking in the same way as stretching. For example, E(1,1/2) = 5/2 = E(2,1). [Hormann & Greiner 2000]

Examples – Isometric Measures Green-Lagrange deformation tensor Combined energy [Maillot et al. 1993] The Green-Lagrange deformation tensor measures how far the first fundamental form is from the identity (measured in the Frobenius norm). The minimal value E = 0 is obtained for isometric mappings. Like the conformal energy, it tends to favor shrinking. For example, E(1,1/2) = 1/4 = 1/4 E(2,1). The energy of Degener et al. is a combination of the MIPS energy on the previous slide and the analogue of the MIPS energy for measuring area distortion. By modifying the parameter θ, one can mediate between angle and area distortion. Like the MIPS energy it is symmetric with respect to shrinking and stretching. The minimal value E = 2θ+1 is obtained for isometric mappings. A Good heuristic for the exponent that tends to give a nice trade-off between area and angle distortion is θ ≈ 3. [Degener et al. 2003]

Examples – Other Measures Dirichlet energy Stretch energies ( , , and symmetric stretch) [Pinkall & Polthier 1993] [Eck et al. 1995] There are also some local measures that are not motivated by measuring the conformality of isometry of a mapping. The Dirichlet energy measures the average of the squares of the singular values, which (by the Binomial Theorem) is always greater than or equal to the product of both terms. By the way, the difference between the two is the conformal energy. This energy is minimal for harmonic functions and has been used, for example, to compute minimal surfaces. Finally, several so-called stretch energies have been proposed: The first is just the square root of the Dirichlet energy and can also be considered the local root mean square stretch of the mapping. The second measures the maximal stretch (which is equivalent to taking the 2-norm of the Jacobian). The third takes the maximum of the maximal and the inverse of the minimal stretch, in order to somehow symmetrize the previous energy. Note that this measure actually is globally minimal for isometric mappings. [Sander et al. 2001] [Sorkine et al. 2002]

Piecewise Linear Parameterizations piecewise linear atomic maps distortion constant per triangle overall distortion We will come back to these measures later when we will use them to compute parameterizations that minimize them. Note that all this machinery simplifies considerably in case of mesh parameterizations. In this setting, the parameterization is a piecewise linear mapping that maps parameter triangles to surface triangles, and the distortion is constant for each triangle. Thus, the integral average of the overall distortion simplifies to a sum.

Beyond Distortion surface normal surface area independent of the particular parameterization intrinsic surface properties

Curvature second fundamental form Gaussian curvature mean curvature