Slides — Curves

After our long journey to understand exterior calculus (and its discrete counterpart), we will start putting these tools to work to manipulate real curves and surfaces. This lecture studies smooth and discrete curves, which illustrate many of the important features of geometry embedded in \(\mathbb{R}^n\).

Curve Slides

8 thoughts on “Slides — Curves”

  1. [Warning: advanced!] There were some questions in class today about the general definition of the wedge product of two vector-valued differential forms. Wikipedia has a nice explanation here; the key statement is this one:

    “In general, the wedge product of two \(E\)-valued forms is not another \(E\)-valued form, but rather an \((E\otimes E)\)-valued form. However, if \(E\) is an algebra bundle (i.e., a bundle of algebras rather than just vector spaces) one can compose with multiplication in \(E\) to obtain an \(E\)-valued form.”

    In the context of curves and surfaces in \(\mathbb{R}^3\), we will be working primarily with \(\mathbb{R}^3\)-valued differential forms. Nominally, the wedge product of two such forms is \((\mathbb{R}^3 \otimes \mathbb{R}^3)\)-valued, but since we have the cross product on \(\mathbb{R}^3\) (i.e., we have an “algebra bundle” rather than just a vector space) we obtain just \(\mathbb{R}^3\)-valued forms.

    Likewise, there will be a variety of scenarios where we take the wedge product of a real-valued differential form and a vector-valued differential form. Here we can just compose with the usual scalar-vector product to obtain a vector-valued form. A trivial example is when \(\phi\) is a real-valued scalar function, and \(\alpha\) is an \(\mathbb{R}^n\)-valued \(k\)-form. Then \(\phi \alpha\) is exactly what you’d expect: “scale up” the k-form by the scalar function. The reason this works out formally is that we’re just taking the wedge product \(\phi \wedge \alpha\) and then applying the usual scalar-vector product.

    1. So is it suggesting that “vector-valued” is technically a misnomer, and it really should be “(graded) algebra-valued” even though few would use this jargon?

      1. Yeah, algebra-valued would be ok. Though often vector spaces have all sorts of structure (inner products and so forth) and we still refer to them as “vector spaces.”

  2. Why is the curvature formula as given in slide 36 correct? It seems to always equal $$-\langle N, \frac d{ds}T\rangle=-1/|\frac d{ds}T|\langle \frac d{ds}T,\frac d{ds}T\rangle=-1/|\frac d{ds}T|(|\frac d{ds}T|^2)=-|\frac d{ds}T|$$

    1. @intrepidowl: There’s a subtle error here: in the first step, you implicitly assume that

      \[ N = \frac{dT/ds}{|dT/ds|}. \]

      In the slides, however, we define N as

      \[ N = \mathcal{J} T. \]

      Are these two expressions always the same? No, and the reason is because of curvature! :-). In other words, we know that

      \[ dT/ds = \kappa N, \]

      where the sign of \(\kappa\) can be negative or positive, i.e., the derivative of the tangent can be pointing to the “left” or “right” of the curve. But \(\mathcal{J}T\) is always pointing “left.”

  3. In the example of computation of the $T(s)$ and $N(s)$, I had the following questions:

    1. On Slide 35, the formula for computing $T(s)$, it is given as $T(s) = \frac{\partial{\gamma}}{\partial{s}}$ and is not normalized by the length of the curve. Should we normalize the tangent vector or not?

    2. On computing curvature k(s), the formula is given as $k(s) = -dot(N,\frac{\partial{T(s)}}{\partial{s}})$, but on slide 37, the negative sign is ignored. Shouldn’t k(s) be equal to a?

    1. 1. Good point: perhaps this slide should say more explicitly that the space curve is arc-length parameterized. As we discussed for plane curves, an arc-length parameterization is nice because you do not have to keep dividing by \(|\gamma^\prime|\). So this is usually the default assumption (just like we will typically assume that a curve is continuous and regular).

      2. As often happens in differential geometry, one simply needs to define a convention and stick with it. Unfortunately, I’m afraid I have not always done a good job of ensuring there is a universal convention that holds throughout all the slides and course notes! But this is honestly good practice anyway, since every author uses different conventions. So for instance, it would be acceptable in your homework to say, “I am using the convention that curvature is defined as \(\kappa := \langle N, dT/ds \rangle\)” or “I am using the convention that curvature is defined as \(\kappa := -\langle N, dT/ds \rangle\)”. As long as you are self-consistent (even within the scope of a single problem!) we will be happy.

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