---- > [!definition] Definition. ([[normal curvature]]) > Let the $\text{II}_{p}$ be the [[second fundamental form]] at $p \in S$, and $w \in T_{p}S$ be a unit [[tangent vector to a regular surface|tangent vector]], then $\text{II}_{p}(w,w)$ is called the **normal curvature along the direction of $w$ at $p$**. > [!equivalence] Let $T \in T_{p}S$. Suppose $\alpha:(-\varepsilon,\varepsilon) \to S$ is a [[parameterization by arc length|unit-speed]] [[parameterized curve|curve]] in $S$ such that $\alpha(0)=T$. Assume that at $\alpha(0)$ the [[curvature of parameterized curve|curvature]] of $\alpha$ is $\kappa=\kappa(0)$ and the [[unit normal vector to a parameterized curve|unit normal vector]] to $\alpha$ is $N=N(0)=\alpha''(0)$, then $\text{II}_{p}(T,T)=\kappa N \cdot n_{p}$, where $n_{p}$ denotes the [[Gauss map]] (surface normal) at $p$. See [[normal and geodesic curvature of a curve in a surface]]. > [!intuition] > How does [[normal curvature]] use the [[second fundamental form]] tell us about the geometry of $S$? ![[CleanShot 2024-04-07 at 19.01.41.jpg|300]] Imagine that the curve $\alpha$ is traced out by a driver driving along $S$. How might the passengers feel acceleration—feel an applied force? > 1. One way is by the driver slamming on the accelerator... but that just changes the speed at which $\alpha$ is traced. This does not affect the geometry of $\alpha$, and tells us nothing about the geometry of $S$. That's why we enforce $\alpha$ to be [[parameterization by arc length|unit-speed]]. > 2. Another way is by the driver manically steering the while left and right. This *does* affect the geometry of $\alpha$, but still doesn't tell us about the geometry of $S$. >3. A third way is if the driver keeps their feet and hands steady, allowing the road to account for the force/acceleration experienced by passengers. > The [[normal curvature]] uses the [[second fundamental form]] to 'sense' this third component. To see this, note that $\kappa N$ accounts for the 'total force' that the passengers feel— whether it is from turning the steering wheel (2) or from the surface geometry (3). By dotting with (i.e., projecting onto) the surface normal $n_{p}$, we exclude the influence of (2)... a driver cannot drive 'normal' (i.e., into or out of) to the road, after all). We are left with purely the influence of (3), the geometry of the surface, on the acceleration felt. > [!basicexample] > These examples build off those in [[second fundamental form]] and/or [[Gauss map]]. > We know that for the unit [[sphere]] $\mathbb{S}^{2}$ with outward normal: $dN_{p}=I_{2}$ for all $p$. Hence the normal curvature along any direction $w \in T_{p} \mathbb{S}^{2}$ is $\text{II}_{p}(w,w)=-dN_{p}(w) \cdot w=-w \cdot w=-1.$ On the [[cylinder]] we have seen that at $p=(x,y,z)$, in the chart $X(u,v)=(\cos u, \sin u, v)$ with [[orthonormal basis]] $\{ X_{u}, X_{v} \}$ the differential $dN_{p}$ has [[matrix]] $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}.$ So the [[normal curvature]] along the direction $X_{u}$ is given by $\text{II}_{p}(X_{u}, X_{u})=-dN_{p}(X_{u}) \cdot X_{u}= -\begin{bmatrix} 1 & 0 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} 1 \\ 0 \end{bmatrix}=-1$ and the [[normal curvature]] along the direction $X_{v}$ is given by $\text{II}_{p}(X_{v}, X_{v})=-dN_{p}(X_{v}) \cdot X_{v}=-\begin{bmatrix} 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} 0 \\ 1 \end{bmatrix}=0.$ For a general unit normal vector $w=\cos \theta X_{u}+\sin \theta X_{v}$ we have $\text{II}_{p}(w,w)=-dN_{p}(\cos \theta X_{u}+ \sin \theta X_{v}) \cdot (\cos \theta X_{u} + \sin \theta X_{v})$ $-\begin{bmatrix} \cos \theta & \sin \theta \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} \cos \theta \\ \sin \theta \end{bmatrix}=-\cos ^{2} \theta.$ Note that this implies that for all $w$ we have $-1 \leq \text{II}_{p}(w,w) \leq 0$, which is relevant for computing [[principal curvature]] on [[cylinder]]. > Note that $X_{u}$ and $X_{v}$ naturally have unit norm here; if this were not the case then we would have to normalize them before carrying out the computation. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` {"Examples":["[[normal curvature]]"],"Nonexamples":["None yet"],"Constructions":[],"Specializations":[],"Generalizations":["[[second fundamental form]]","[[Gauss curvature]]","[[mean curvature]]","[[principal curvature]]"],"Properties":[],"Sufficiencies":["[[second fundamental form#^82a614]]"],"Equivalences":[]} {"Examples":[],"Nonexamples":[],"Constructions":[],"Specializations":["[[principal curvature]]","[[Gauss curvature]]","[[mean curvature]]"],"Generalizations":[],"Properties":["[[second fundamental form#^82a614|Used in the construction of normal curvature]]","[[Gauss curvature#^e5d9a1|Used in the computation of Gauss curvature]]","[[mean curvature#^7c154c|Used in the computation of mean curvature]]","[[principal curvature#^e0b64a|Used in the computation of principal curvature]]"],"Sufficiencies":[],"Equivalences":[]} {"Examples":[],"Nonexamples":[],"Constructions":[],"Specializations":[],"Generalizations":[],"Properties":[],"Sufficiencies":[],"Equivalences":[]} {"Examples":[],"Nonexamples":[],"Constructions":[],"Specializations":[],"Generalizations":[],"Properties":[],"Sufficiencies":[],"Equivalences":[]} {"Examples":[],"Nonexamples":[],"Constructions":[],"Specializations":[],"Generalizations":[],"Properties":[],"Sufficiencies":[],"Equivalences":[]} {"Examples":[],"Nonexamples":[],"Constructions":[],"Specializations":[],"Generalizations":[],"Properties":[],"Sufficiencies":[],"Equivalences":[]} {"Examples":[],"Nonexamples":[],"Constructions":[],"Specializations":[],"Generalizations":[],"Properties":[],"Sufficiencies":[],"Equivalences":[]} {"Examples":[],"Nonexamples":[],"Constructions":[],"Specializations":[],"Generalizations":[],"Properties":[],"Sufficiencies":[],"Equivalences":[]} {"Examples":[],"Nonexamples":[],"Constructions":[],"Specializations":[],"Generalizations":[],"Properties":[],"Sufficiencies":[],"Equivalences":[]} {"Examples":[],"Nonexamples":[],"Constructions":[],"Specializations":[],"Generalizations":[],"Properties":[],"Sufficiencies":[],"Equivalences":[]}