# Abstract Nonsense

## Space Curves

Point of Post: In this post we extend our notions of curvature and similar matters from plane curves to space curves.

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Motivation

Although we have been, up until this point, discussing properties of plane curves, the majority of our focus shall be on curves (and surfaces) living in space (i.e. in $\mathbb{R}^3$).  One would hope that the theory of such curves, in particular the theory of their curvature, is as nice as it was for plane curves. Unfortunately, this is not the case. As we shall see, we can define curvature for a curve in $\mathbb{R}^3$ very analogously to how we defined it for curves in $\mathbb{R}^2$, but what we shall see is that unlike the $\mathbb{R}^2$ case, curvature isn’t the only important invariant (in the sense that a space curve is not, except for an isometry, determined by its curvature). Indeed, we will see that the embedded circle $\left\{(x,y,0):x^2+y^2\right\}$ and the helix $\left\{(\frac{1}{2}\cos(t),\frac{1}{2}\sin(t),\frac{1}{2}t):t\in\mathbb{R}\right\}$ have the same curvature functions (constant functions, equal to $1$) yet are clearly not differing by an isometry. Clearly the problem is that curvature doesn’t account for ‘upward or downward’ motion and consequently isn’t sensitive enough to differentiate between curves in $\mathbb{R}^3$. We are thus necessitated to create another statistic for curves which, intuitively at least, measures the up and down movement. This statistic is known as the torsion of the curve and measures what the previous sentence (the intuition of up and down movement) wishes–the extent to which the curve does not stay within a given plane. We shall then see that these two statistics completely determine, as always up to an isometry, space curves–the torsion controlling the movement in and out of planes, and the curvature controlling the movement within each plane.

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Curvature and Torsion

For a unit speed curve $\gamma:I\to\mathbb{R}^3$ we define the curvature $\kappa_\gamma$ to be equal to $\kappa_\gamma(t)=\|\gamma''(t)\|$. Note now, that similar to the case for plane curves, one has that $\gamma'(t)$ and $\gamma''(t)$ are perpendicular, and so letting $\displaystyle \bold{n}_\gamma(t)=\frac{1}{\kappa_\gamma(t)}\gamma''(t)$ one has that $\gamma'(t),\bold{n}_\gamma(t)$ are perpendicular unit vectors. Thus, letting $\bold{b}_\gamma(t)$ be defined as $\bold{b}_\gamma(t)=\gamma'(t)\times \bold{n}_\gamma(t)$ we have that $(\bold{t},\bold{n},\bold{b})$ (where we have omitted the $\gamma$ subscript and called $\gamma'(t)$, $\bold{t}$ which is historically consistent) forms an ordered orthonormal basis which satisfies the right hand rule that taking the cross product of any two consecutive basis vectors gives the third. This ordered basis is known as the Frenet frame for $\gamma$. Now, recalling that the cross product is bilinear and recalling how derivatives interact with bilinear functions we may easily deduce that

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$\displaystyle \bold{b}'(t)=\bold{t}'(t)\times \bold{n}(t)+\bold{t}'(t)\times\bold{n}'(t)=\bold{t}\times\bold{n}'(t)$

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where we have used the obvious facts that $\bold{t}\parallel \bold{n}'$ and $\bold{t}'\parallel \bold{n}$. Thus, $\bold{b}'(t)$ is perpendicular to both $\bold{b}(t)$ and $\bold{t}(t)$, and so we may conclude that $\bold{b}(t)=c(t)\bold{n}(t)$ for some $c(t)\in\mathbb{R}$. We define the torsion of the curve $\gamma$, denoted $\tau(t)$, to be the constant $-c(t)$. Note that for this definition to have made sense, we needed that $\kappa(t)$ is non-zero.

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Just as the case for curvature, we’d like to extend the idea of torsion to arbitrary regular curves with non-zero curvature. To do this we do what is natural and define the torsion for a regular curve $\gamma:I\to\mathbb{R}^3$ with non-zero curvature to be the torsion of any of its unit speed reparamaterization. One can easily check that the torsion is invariant under which unit speed reparametrization one chooses (this should be clear from the fact that $\mu=\pm\nu+d$ for any two unit speed reparamaterizations $\mu,\nu$). In fact, just as the case for curvature there is a general (albeit, very messy, formula) for the torsion of a curve. Namely:

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Theorem: Let $\gamma:I\to\mathbb{R}^3$ be a regular curve with nowhere vanishing curvature. Then, the torsion $\tau(t)$ is given by the following formula:

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$\displaystyle \frac{\left(\gamma'(t)\times\gamma''(t)\right)\cdot\gamma'''(t)}{\|\gamma'(t)\times\gamma''(t)\|^2}$

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The proof of this can be found in any half-decent book on basic differential geometry (cf. [2]). As stated, the torsion of a curve measures precisely how much a curve stays within a given plane. This is exemplified by the following obvious ‘corollary’ of this idea:

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Theorem: Let $\gamma:I\to\mathbb{R}^3$ be a regular curve with nonwhere vanishing curvature. Then, $\gamma$ is contained within a plane if and only if the torsion, $\tau(t)$, is zero for all $t\in I$.

Proof: Since torsion and the property of ‘being in a plane’ are invariant under reparamaterization we may obviously assume that $\gamma$ is unit speed. So, now suppose that $\gamma(I)$ is contained in some plane of the form $\Pi=\left\{v\in\mathbb{R}^3:v\cdot N=c\right\}$ where $N$ is a constant unit vector, and $c\in\mathbb{R}$. We see then that $\gamma(t)\cdot N=c$ for all $t\in I$ and so, differentiating this, gives $\gamma'(t)\cdot N=0$ for all $t\in N$. Similarly, differentiating again gives $\kappa\bold{n}\cdot N=\gamma''(t)\cdot N=0$, and so $\bold{n}\cdot N=0$ (since $\kappa$ is non-zero). Note though that this says, using the Frenet frame notation, that $\bold{t},\bold{n}$ are parallel to $N$ and so $cN=\bold{b}$ for some $c\in\mathbb{R}$. Note though that since $\bold{b}$ is a unit vector, we must have that $c=\pm 1$ and so $\bold{b}(t_0)=\pm N$ for every fixed $t_0\in I$, but since $\bold{b}(t)$ is continuous we must have that either $\bold{b}(t)=N$ for all $t\in I$ or $\bold{b}(t)=-N$ for all $t\in I$. Either way, we find that $\bold{b}'(t)=\bold{0}$ and so evidently $\tau(t)=0$ for all $t\in I$.

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Conversely, suppose that $\tau(t)=0$, then we have that $\bold{b}'(t)=\bold{0}$ for all $t\in I$ and so $\bold{b}(t)$ is a constant vector. Note then that $(\gamma\cdot \bold{b}(t))'=\gamma'(t)\cdot\bold{b}(t)=0$ (since $\gamma(t)\perp\bold{b}(t)$ and so clearly $\gamma\cdot\bold{b}(t)$ is constant, say equal to $c$. Clearly then, $\gamma$ sits inside the plane $\left\{v\in\mathbb{R}^2:v\cdot\bold{b}(t)=c\right\}$. $\blacksquare$

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As a corollary, we may combine the above theorem with our previous knowledge concerning constant curvature and circles, we get the following:

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Corollary: Let $\gamma:I\to\mathbb{R}^3$ be a regular curve with nonvanishing curvature and everywhere zero torsion. Then, $\gamma(I)$ is contained in a circle.

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Perhaps the most important theorem about space curves, is the one which we discussed about in the motivation, concerning their characterization. Namely, we have seen that the way a curve ‘moves’ (i.e. the way its curvature acts) dictates the entirety of the curve’s being, if we are dealing with plane curves. Now, since curves in $\mathbb{R}^3$ are just plane curves with an extra dimension one would hope that curvature along with the way the curve moves in this third dimension should characterize the curve in much the same way it did for plane curves. Using our new found terminology, one would hope that for a space curve, the only invariant, what characterizes it (up to isometry) is its curvature and torsion. In fact, this is true, and one of most beautiful, complex, yet intuitive aspects of the geometry of spaces curves. Putting this in rigorous words, perhaps to the death of the intuition, we have the following:

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Theorem(Fundamental Theorem of Space Curves): Let $\gamma:I\to\mathbb{R}^3$ be a unit speed curve with nowhere vanishing curvature. If $\mu:I\to\mathbb{R}^3$ is another unit speed curve with nowhere vanishing curvature such that $\kappa_\gamma=\kappa_\mu$ and $\tau_\gamma=\tau_\mu$, then there exists $M\in\text{Iso}(\mathbb{R}^3)$ with $\gamma=M\circ\mu$.

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While the theorem is beautiful, the proof is not. To my inexpert eye, it seems like a lot of symbol pushing, and so we omit the proof here. A perfectly fine proof can be found in [2].

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References:

[1] Carmo, Manfredo Perdigão Do. Differential Geometry of Curves and Surfaces. Upper Saddle River, NJ: Prentice-Hall, 1976. Print.

[2] Pressley, Andrew. Elementary Differential Geometry. London: Springer, 2001. Print.

[3]  Montiel, Sebastián, A. Ros, and Donald G. Babbitt. Curves and Surfaces. Providence, RI: American Mathematical Society, 2009. Print.