A Transversion-Based Machine for Curvature

Fundamental Forms

One of the key properties of a manifold is its curvature.  For simple shapes such a sphere or a plane, curvature is constant, but for the most part it is a function dependent on position in the manifold.  Surfaces in 3D Euclidean space are 2-manifolds that can be overlaid with a 2D coordinate system.

The specific form of a surface is determined by instantaneous curvature at every point.  There are several different parallel concepts that fall under the term curvature, but usually what is referred to is the extrinsic curvature of a manifold as opposed to its intrinsic curvature.  Extrinsic curvature refers to the curvature of a manifold as it exists in an embedding space while intrinsic curvature refers to the inherent curvature of a manifold independent of any embedding space.

Surfaces have both intrinsic and extrinsic curvature unlike curves, which only have extrinsic curvature.  Thus, the particular shape of a surface in an embedding space can vary without changing its intrinsic curvature, so when constructing surfaces in 3D space, we need a measure of the properties of a surface that relates the extrinsic and intrinsic properties to each other.  In differential geometry, there are two important constructs that largely characterize the extrinsic and intrinsic properties of a surface: the first fundamental form and the second fundamental form.

The first fundamental form measures intrinsic properties through the inner product of the local tangent space.  At each point along a surface, there is a plane tangent to that point, which can be defined by the normal at that point.  This plane has two independent directions that can be thought of as 2D vectors in the tangent plane.  How these vectors relate to each other is precisely what the first fundamental form is measuring.  The inner (or dot) product simply measures how orthogonal the vectors are to each other and what their relative scale is.  The first fundamental form is thus a 2×2 matrix correlating the vectors to each other.

If a surface is given by \(X(u, v)\) where \(u\) and \(v\) are the parameterizing coordinates of the surface, then the first fundamental form can be calculated from the first derivatives of these coordinates \(X{_u}, X{_v}\).

\(I = \left[ \begin{array}{ rc c } E & F \\ F & G \end{array} \right] \)

where \(E = X{_u}\cdot{}X{_u}, F= X{_u}\cdot{}X{_v}, G= X{_v}\cdot{}X{_v}\).

The first fundamental form is often referred to as the metric tensor of the surface as it can be used to calculate arc lengths and areas at points along the surface.  The second fundamental form by contrast is also referred to as the shape tensor.  It describes how the extrinsic properties of a surface relate to its intrinsic properties.  It’s a quadratic form in the tangent space and can be calculated from the second derivatives of the surface coordinates and the normal to the surface.  Again, it’s a 2×2 matrix of the form

\(II = \left[ \begin{array}{ rc c } L & M \\ M & N \end{array} \right] \)

where \(M = X{_{uu}}\cdot{}\vec{n}, N = X{_{uv}}\cdot{}\vec{n}, M = X{_{vv}}\cdot{}\vec{n}\) and \(n = X{_u}\wedge{}X{_v}/|X{_u}\wedge{}X{_v}|\).

Taken with the first fundamental form, the second fundamental form defines the curvature of the surface and the shape operator from which the surface can be constructed.  Thus given these two matrices for every point on a surface, the surface itself can be reconstructed.

The Transversion Machine

Given what we know about differential geometry and how various properties of the surface at each point relate to its specific structure, the question now posed is are two-fold: What is the minimal information required to reconstruct a particular surface? and What kinds of structure can we setup to generate this information as intuitively as possible that will enable use to modulate and transform a given surface in new and interesting ways?

For the first question, it’s clear that somehow we need to construct a generator that enables the calculation of the first and second fundamental forms without much computational complexity either explicitly or implicitly.  In addition, the generator should be coordinate free, that is it should not refer directly to coordinates in space but instead according to local, differential structures.  As has been mentioned in previous posts, the best tools for coordinate free geometric calculations are geometric algebra and the related geometric calculus.

The Tangent Plane

For the first fundamental form, the representation is geometric algebra is straightforward.  It’s constructed form the first derivatives over the surface in the principal directions, forming a tangent plane to the surface as well as the normal vector.  In geometric algebra we can represent this construct with a bivector made up of the outer product of two vectors.  In 3D Euclidean space, the dual of the bivector gives the normal as well.

The Osculating Spheres

The second fundamental form is a bit trickier to represent.  First, its geometric meaning needs to be properly expressed.  What the second fundamental form essential describes is how far a nearby point is from the tangent plane along the vertical distance.

If a point (x+dx, x+dy) is a distance D form the tangent plane, then D is given by \(D^2 = Ldx^2 + Mdxdy + Ndy^2\) where L, M, and N are the second fundamental form coefficients.

So the question now is how to relate the way the surface curves away from the tangent plane to some kind of relation in geometric algebra to the tangent plane we’re using to represent the first fundamental form.  Essentially, what we’re after here is a way to ascribe the curvature of the surface to a geometric entity.  When working with plane curves, the curvature of the surface is inversely related to the radius of a circle tangent to the surface at that point.  This circle is called the osculating circle.  Similar to how a tangent vector describes a differential movement along a curve, an osculating circle describes a differential curvature.  For surfaces, the same property is given by an osculating sphere where the radius of the sphere is inversely related to the curvature at a given point.

From our observations about the relationship between curvature and the osculating sphere, we should be able to use a sphere to represent the second fundamental form, but there’s still an open question about how to use it to calculate where exactly the surface will curve to efficiently and an equally tricky issue related to how to handle positive, negative and zero curvature cases.

For both problems, we can find a solution in the 5D conformal model of geometric algebra.  In this model, objects in geometric algebra can be reflected (inverted) in a sphere by treating the sphere as a versor.  If we place a sphere of a certain radius tangent to our tangent plane and reflect it into the sphere, the tangent vectors will be precisely where the curvature described by the radius of the sphere places them.  In other words, given an osculating sphere, the tangent plane can simply be inverted in it to warp it onto the surface described by the first and second fundamental forms.

While spherical inversion is a discrete operation, spherical transversion is a continuous operation that generates loxodromic motions.  The transversion operation is also known as the special conformal transformation.  When properly defined, the transversion operator can smoothly interpolate a sphere through an infinite radius (i.e. a plane) to a sphere of equal radius tangent to it but reflected over the plane generated when its radius was taken to infinity.  

In the picture above, the blue circle represents the transversing sphere while the black circle is the transversed sphere.  Its trajectory is plotted by the red circles until it reaches infinite radius (the grey line) and finally by the green circles as it returns back to the original radius but reflected over the grey line.  The transformation in this case is defined by a unit dual sphere at the origin along with a translation in the horizontal direction.  For a smooth transformation, the translation’s magnitude is such that it takes 20 steps for the black circle to become the final inner green circle.  The general transversion operation is given by \(V = STS^{-1}\) where S is a dual sphere and T a translation.  In the above case, \(S = o  - \frac{1}{2}\infty\) and \(T = e^{-\frac{e_1\wedge{}\infty}{2*20}}\).

In the general case, the transversing sphere and the transversed sphere can be of any radius with the constraint that the center of the transversing sphere is tangent to the transversed sphere.  Additionally, the direction of translation must be along the line from the center of the transversed sphere to the center of the transversing sphere.  When the radii of the two sphere changes, so does the rate at which the sphere is transversed.  The scaling factor relating the radii of the two sphere is \(\frac{(R_1)^2}{R_2}\) where \(R_1\) is the radius of the transversing sphere and \(R_2\) is the radius of the transversed sphere.  To compare, the image below has \(R_1 = 2\) and \(R_2 = \frac{1}{2}\).

In the image above, we can think of the grey line as the tangent plane and the red and green circles as representing the possible osculating spheres of different radii.  For curves, when the curvature is positive, the osculating circle is to one side of the curve.  When it’s negative, it’s on the other.  The same principle applies to surfaces and the osculating sphere.  Thus, we can say that the red and green circles represent osculating spheres related to positive and negative curvatures respectively and since the transversion operation smoothly interpolates from one to the other through the plane of zero curvature, we can smoothly vary the osculating sphere over the surface by defining a differential transversion operation that continuously varies the spheres from one point on the surface to the next.

Operating the Machine

Given the osculating spheres and the tangent plane as described above, the transversion machine is nearly complete, but there remain a few details to work out.  First, each independent direction over the surface has its own rate of curvature and thus its own osculating sphere.  Second, the operation of the machine is as follows:

  1. Reflect the principal vectors defining the tangent plane are inverted into their corresponding osculating spheres.
  2. A transformation for each  inversion is constructed such that the osculating spheres are rotated and translated to be tangent to the location where the tangent vectors were inverted.
  3. The radius of curvature of each osculating sphere is updated
  4. Repeat from step 1.

It’s important to note that at step 2, there are two movements: one in the u direction and one in the v direction where u and v are the principle directions.  Thus, at each point the machine can be thought of as being duplicated and moved in different directions to form the surface.  Once the surface is closed, the machine stops.

Defining the Parameters of the Machine

Given the transversion machine, there is a question of how to drive it.  The parameters defining the continuous changes in the radii of curvature of the two osculating spheres needs to be defined somehow.  This is where the constraints needed to generate a closed surface are applied and will be the topic of another post.  Essentially, it’s a problem of going from a topology where the curvature parameters are given at the vertices and smoothly interpolating those values over the faces such that a closed surface results.

This entry was posted in Space and Geometry and tagged , , , . Bookmark the permalink.

Leave a Reply