\\alpha''(s) = T'$$
Curvature Function ($\kappa > 0$) for a unit speed curve is
$$\kappa(s)=|T'(s)| $$
If $\kappa \neq 0$ then we can say
$$T'(s) = \kappa(s)N(s)$$
Def: Osculating Plane - The plane spanned by T and N the tangent and normal vectors
$$B(s) = T(s) \times N(s)$$
Pf: T and B' are orthogonal. Start by taking the derivative of $0 = T \cdot B$
$$0 = T' \cdot B + T \cdot B' = \kappa N \cdot B + T \cdot B' = T \cdot B' = 0$$
Because B' is parallel to N:
$$B'(s) = -\tau(s)N(s)$$
Because $N = B \times T$
$$N'=-\kappa T + \tau B$$
### Fernet Formulas For Unit Speed
$$T' = \kappa N \\
N'=-\kappa T + \tau B \\
B' = -\tau N$$
****Theorem:**** $\kappa = 0 \ \Longleftrightarrow \alpha$ is a straight line
****Theorem:**** $\tau = 0 \ \Longleftrightarrow \alpha$ lies in a plane
****Theorem:**** $\tau = 0, \ \kappa\ = \frac{1}{R}\ \Longleftrightarrow \alpha$ is a circle
### The fundamental Theorem on Curves
****Theorem:**** Given any smooth functions $\kappa(s) > 0$ and $\tau(s)$, there exists a regular, smooth unit-speed curve, $\alpha$ having $\kappa$ and $\tau$ as its curvature and torsion. This curve is unique in $\mathbb{R}^3$!
## Arbitrary Speed Curves
### Fernet Formulas for Arbitrary Speed Curves
$$T' = \kappa vN \\
N'=-\kappa vT + \tau vB \\
B' = -\tau vN$$
Formulas to compute curvature $\kappa$ and torsion $\tau$
$$\kappa = \frac{|\alpha' \times \alpha''|}{v^3} \\
\\tau = \frac{(\alpha' \times \alpha'')\cdot \alpha'''}{|\alpha' \times \alpha''|^2}$$
# Surfaces
Definitions:
* ****Regular**** mapping $x: D \rightarrow \R^3 $ if $x_u \times x_v \neq 0\ \forall\ u, v\in D$
* ****Coordinate Patch:**** is a one-to-one mapping x of some open set D in $\mathbb{R}^2$ into $\mathbb{R}^3$
* ****Surface:**** A ⊂ M of $\mathbb{R}^3$ such that each point of M has a neighborhood in M that is contained in the image of some patch
* ****Atlas:**** A collection of patches whose images cover all of M
* Curves with constant u on M are called Parallels
* Curves with constant v on M are called Meridians
### Implicit Function Theorem
If $f(x,y,z)$ is smooth and let $S = {(x,y,z)\in \mathbb{R}^3: f(x,y,z) = 0}$ if $p \in S$ is a point where $f_z \neq 0$, then there is a neighborhood D of p in which the set $S\ \cap\ D$ is a graph of a smooth function $z = g(x,y)$
### Important Corollary
A sufficient condition for S to be a surface is that $f_x, f_y, f_z$ do not vanish at any point of S
$\implies$if we get $\nabla f = 0$ then something went wrong
# Curves and Tangent Vectors
The coordinate curves of a parameterized surface $x(u,v)$ are $x_u, x_v$ and together they span the tangent plane $T_p(M)$. We define the ****Unit Normal Vector**** as follows
$$U= \frac{x_u \times x_v}{|x_u \times x_v|}$$
Note: $x_u, x_v$ are not necessarily unit vectors and not necessarily orthogonal
### Directional Derivative
We say that $\nabla_v f$ is the derivative of f in the v direction
# Shape Operator
$$S_p(v) = -\nabla_vU = -v_1U_u-v_2U_v$$
If M is a plane then U is constant $\implies\ S_p(v) = 0$ for all v
If M is a sphere of radius $R$ then $S_p(v) = \frac{-v}{R}$
### Fundamental Theorem of Shape Operator
****Theorem:**** $S_p$ is a self-adjoint linear operator from $T_p(M)$ to itself
****Corollary:****
$$S_p(x_u)\cdot x_v = U \cdot x\_{uv} \\ S_p(x_u)\cdot x_u = U \cdot x\_{uu} \\ S_p(x_v)\cdot x_v = U \cdot x\_{vv}$$\_
****Lemma:**** For any curve $\alpha(t)$ in $M$, $\alpha'' \cdot U = S(\alpha')\cdot \alpha'$
If a curve has unit speed then the following is true by fernet formulas
$$\alpha'' \cdot U = \kappa N \cdot U$$
Def: For a unit speed curve in M, the ****Normal Curvature**** k(T) is:
$$\kappa N \cdot U = S_p(T) \cdot T$$
The Real EigenValues of $S_p$, $k_1 \geq k_2$ are the ****Principal Curvatures**** of M at P
They have the Eigenvectors $u_1, u_2$ called the ****Principal Directions****
The number $K = k_1k_2 = det(S_p)$ is the ****Gaussian Curvature****
The number $H = \frac{1}{2}(k_1+k_2)=\frac{1}{2} Tr(S_p)$ is the ****Mean Curvature****
M is ****Flat**** if K = 0
M is ****Minimal**** if H = 0
If $k_1 = k_2$ then the point is ****Umbilic****
If $T\_\\alpha$ is $u_1$ or $u_2$ at ever point then $\alpha$ is a\_ *****line of curvature***** *or a* *****Principal Curve*****
If $k_n(T\_\\alpha) = 0,\ \alpha$ is\_ *****Asymptotic*****
Quantities that change sign if $U \rightarrow -U$: $S,\ k_i,\ H$
Note that $K$ stays the same
****Theorem:**** If every point of a connected surface M is ****umbilic**** then M is part of a plane or sphere
### Curvature Formulas
$$S(x_u) = ax_u + bx_v \\
S(x_v) = cx_u + dx_v$$
$$\Longrightarrow \left(\begin{array}{cc} a & b \\ c & d \end{array}\right)$$
$$E = x_u \cdot x_u \\
F = x_u \cdot x_v \\
G = x_v \cdot x_v \\m = S(x_u) \cdot x_v = U \cdot x\_{uv} \\
n = S(x_v) \cdot x_v = U \cdot x\_{vv} $$
$$S = \begin{bmatrix} a & b \\
d & e
\\end{bmatrix} = \begin{bmatrix} E & F \\
F & G
\\end{bmatrix}^{-1}\begin{bmatrix} l & m \\
m & n
\\end{bmatrix}$$
$$K = \frac{ln-m^2}{EG-F^2}$$
$$H = \frac{Gl+En-2Fm}{2(EG-F^2}$$
$$k_1, k_2 = H \pm \sqrt{H^2-K}$$
## Special Cases
### Monge Patch
${x}(u,v) = \[u,v,f(u,v)\]$
$$E=1+f_x^2,\ F = f_xf_y,\ G = 1 + f_y^2$$
Let $U = (-f_x,-f_y,1)/W$ where $W = \sqrt{1+f_x^2+f_y^2}$
$$l = \frac{f\_{xx}}{W},\ m = \frac{f\_{xy}}{W},\ n = \frac{f\_{yy}}{W}$$
Gaussian and Mean Curvature
$$K = \frac{f\_{xx}f\_{yy} - f\_{xy}^2}{W},\ H = \frac{\[(1+f_x^2)f\_{yy}-2f\_{x}f\_{y}f\_{xy} + (1 + f_y^2)f{xx}\]}{2W^3}$$
### Surface of Revolution
$x(u,v) = \[g(u), h(u)cosv,h(u)sinv\]$
$$E = (g')^2 + (h')^2,\ F = 0,\ G = h^2$$
$$l=\frac{g''h'-h''g'}{\sqrt{(g')^2+(h')^2}},\ m=0,\ n=\frac{hg'}{\sqrt{(g')^2+(h')^2}}$$
$$K=\frac{g'(g''h'-h''g')}{h\[(g')^2+(h')^2\]^2}$$
If $(g')^2 + (h')^2 = 1$ then:
$$K = -\frac{h''}{h},\ H = \frac{1}{2}\[(g''h'-h''g')+\frac{g'}{h}$$
If g(u) = u then
$$K = -\frac{h''}{h\[1+(h')^2\]^2},\ H=\frac{1}{2}\[\\frac{-h''}{\[1+(h')^2\]^{3/2}}+\frac{1}{h\[1+(h')^2\]^{1/2}}\]$$
# Importance of the Metric
$$\left(\begin{array}{cc} E & F \\ F & G \end{array}\right),\ metrics\ {\ E,F,G\ }$$
Letting
$$v=v_1x_u+v_2x_v \\
w=w_1x_u+w_2x_v \\
v \cdot w = v_1w_1E + (v_1w_2+v_2w_1)F + v_2w_2G \\
\|v|^2 = Ev_1^2+2Fv_1v_2 + Gv_2^2 \\
\\angle x_u, x_v \implies \theta=cos^{-1}(F/\sqrt{EG})$$
With $\alpha(t) = x(u(t),v(t))$
$$S=\int_a^b\sqrt{\alpha'\cdot\alpha'}dt = \int_a^b\sqrt{E(u')^2+2Fu'v'+G(v')^2}dt$$
$$|x_u\times x_v|^2 = |x_u|^2|x_v|^2sin^2\theta = |x_u|^2|x_v|^2(1-cos^2\theta) = EG-F^2$$
$$A = \int\int\_{R}\sqrt{EG-F^2}dudv$$
# Isometries
# Geodesics
We can split the acceleration of a curve, $C: \alpha(t) = x\[u(t),v(t)\]$ where the first term is the normal component of acceleration, $\alpha$ and the second component is the orthogonal projection of $\alpha$ onto the tangent plane to M at
$$\alpha'' = (\alpha''\cdot U)U+\alpha''{tan}$$
For unit speed, $|\alpha''| = \kappa(t)$
# Formulas
### Fernet Formulas
$$ T' = \kappa N \\
N'=-\kappa T + \tau B \\
B' = -\tau N $$
$$U= \frac{x_u \times x_v}{|x_u \times x_v|}$$
$$T' = \kappa vN \\
N'=-\kappa vT + \tau vB \\
B' = -\tau vN$$