Second Fundamental Form (Gaussian Geometry Application) | Mathematics | Page 1

lost flax May 22, 2024, 8:00 PM

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I'm trying to understand this notation Dv(Fu) * Fu = v Fuu.
I understand that v is the normal vector and Fu is the derivative of F with respect to u. I just don't understand the notation v(Fu) if v is a three dimensional vector and not a function of Fu (?) given by:

quasi zephyrBOT May 22, 2024, 8:00 PM

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lost flax I'm trying to understand this notation Dv(Fu) * Fu = v Fuu. I understand that v ...

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bitter waspBOT May 22, 2024, 8:02 PM

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khulpu

lost flax May 22, 2024, 8:13 PM

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F(u,v) is a parameterization of a surface S.

full talon May 22, 2024, 8:42 PM

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lost flax I'm trying to understand this notation Dv(Fu) * Fu = v Fuu. I understand that v ...

L is the shape opertator of $F_u$ dotted with $F_u$

bitter waspBOT May 22, 2024, 8:42 PM

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Omegabet_

full talon May 22, 2024, 8:43 PM

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by definition, the shape operator is the negative covariant derivative of the normal field in the direction of the vector

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the $-D\nu(F_u)$ is thus just the definition of the shape operator

bitter waspBOT May 22, 2024, 8:43 PM

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Omegabet_

full talon May 22, 2024, 8:45 PM

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as for why the equality with the other expression, for any curve $\alpha\subseteq M$, $\alpha''\cdot U=S(\alpha')\cdot\alpha'$ ($U$ the normal field, $S$ the shape operator). To which it follows

bitter waspBOT May 22, 2024, 8:45 PM

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Omegabet_

lost flax May 23, 2024, 9:54 AM

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I still didn't understand.

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Because after doing the cross product we get a three-dimensional vector and from what I understood the derivative of the normal will then have two vector components in u' and v'.

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Also in another literature it is written as Nu = aXu + bXv meaning the partial derivative of N to u can be written as a sum of a linear combination of Xu and Xv, suggesting the derivative of N is in the tangent plane (?).

full talon May 23, 2024, 1:17 PM

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lost flax Because after doing the cross product we get a three-dimensional vector and from...

By definition, $X_u\times X_v$ for a patch $X$ is normal to $T_p(M)$

bitter waspBOT May 23, 2024, 1:17 PM

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Omegabet_

full talon May 23, 2024, 1:18 PM

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hence at each point of M we get a unit normal field $U:=\frac{X_u\times X_v}{\norm{X_u\times X_v}}$

bitter waspBOT May 23, 2024, 1:18 PM

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Omegabet_

full talon May 23, 2024, 1:19 PM

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to which we define the shape operator/LMN wrt this U

full talon May 23, 2024, 1:26 PM

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lost flax Also in another literature it is written as Nu = aXu + bXv meaning the partial d...

also which N lol, there's the Frenet N and the Patch computation N

lost flax May 23, 2024, 1:37 PM

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full talon also which N lol, there's the Frenet N and the Patch computation N

yes, writers are using different letters lol. N as the normal vector

full talon May 23, 2024, 1:37 PM

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Normal frenet normal, or normal normal field?

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cause afaik you dont write T,N,B wrt a patch

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since they're defined wrt a curve

lost flax May 23, 2024, 1:38 PM

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Now I read (via GPT) that derivatives of N belong in the tangent because they give the direction of the normal vector

full talon May 23, 2024, 1:38 PM

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dont consult GPT for math

lost flax May 23, 2024, 1:39 PM

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I normally try not to, just went there for a quick search because I was extremely confused

full talon May 23, 2024, 1:39 PM

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ok

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well again, N from frenet, or N from EFGLMN?

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cause N from LMN is a scalar

lost flax May 23, 2024, 1:40 PM

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Frenet I believe

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We don't use LMN here, just capital EFG or lowercase

full talon May 23, 2024, 1:40 PM

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then you dont talk about patches unless you specify if its the u-curves or v-curves

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you just defined LMN in the picture...

lost flax May 23, 2024, 1:41 PM

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I took that pic from the web

full talon May 23, 2024, 1:41 PM

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Yeah

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so?

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why would you use a picture that's not relevant to your question

lost flax May 23, 2024, 1:41 PM

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I wanted to share my book but it's in Portuguese and thought most wouldn't understand

full talon May 23, 2024, 1:41 PM

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anyway, you dont define the Frenet frame wrt a patch, you define it wrt a curve

lost flax May 23, 2024, 1:42 PM

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It is relevant because it's the same concept just a different language but again the notation is slightly different from what we use

full talon May 23, 2024, 1:42 PM

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Unless you're horribily referring to $\alpha(\cdot):=X(\cdot,v_0)$

bitter waspBOT May 23, 2024, 1:42 PM

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Omegabet_

full talon May 23, 2024, 1:42 PM

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the u-curves of X

lost flax May 23, 2024, 1:42 PM

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You're making me even more confused I must confess.

full talon May 23, 2024, 1:43 PM

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yeah cause you're not being clear

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I have to try and piece together what you're even asking...

lost flax May 23, 2024, 1:43 PM

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Let me share a ss from the book

lost flax May 23, 2024, 1:43 PM

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full talon I have to try and piece together what you're even asking...

I can understand

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This is N. The hat means cross product.

full talon May 23, 2024, 1:44 PM

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Ok, so N is a unit normal field

lost flax May 23, 2024, 1:49 PM

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yes

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X is a parametrisation of the surface R3 -> R2

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With the coordinates u and v

full talon May 23, 2024, 1:50 PM

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yes

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X is a patch

lost flax May 23, 2024, 1:51 PM

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Yeah

full talon May 23, 2024, 1:51 PM

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but N_u isnt defined, since (u,v) lies in U

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and N is a vector field on S (specifically on X(U), but regardless a vector field on S)

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so N doesnt have domain U

lost flax May 23, 2024, 4:30 PM

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Nu is the derivative of N wrt u?

full talon May 23, 2024, 5:52 PM

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lost flax

I assume it means $N_u:=(N\circ X)_u$

bitter waspBOT May 23, 2024, 5:52 PM

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Omegabet_

full talon May 23, 2024, 5:52 PM

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since $N\circ X:U\to T_p(\mathbb{R}^3)\cong\mathbb{R}^3$

bitter waspBOT May 23, 2024, 5:52 PM

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Omegabet_

lost flax May 23, 2024, 6:42 PM

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full talon I assume it means $N_u:=(N\circ X)_u$

which means N times the derivative of X wrt to u (according to composite theorem)?

full talon May 23, 2024, 6:47 PM

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NoX is a map from R^2 to R^3, ie f(u,v):=(f_1(u,v),...,f_3(u,v))

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but yes, f_u would just be (f1_u,...,f3_u)

lost flax May 24, 2024, 4:48 PM

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full talon but yes, f_u would just be (f1_u,...,f3_u)

hey, I just had that moment of clarity and feel stupid of how obvious this was since Xu and Xv are orthogonal to N thus the derivative of the dot product is also zero. Thank you for the patience though!

austere steepleBOT May 24, 2024, 4:48 PM

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@torn pewter has given 1 rep to @full talon

full talon May 24, 2024, 4:48 PM

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lost flax hey, I just had that moment of clarity and feel stupid of how obvious this was s...

Yep, that's a thing lol

lost flax May 24, 2024, 4:50 PM

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+close

#Second Fundamental Form (Gaussian Geometry Application)