Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 3: Problem 2
Determine the asymptotic curves and the lines of curvature of the helicoid \(x=v \cos u, y=v \sin u, z=c u\), and show that its mean curvature is zero.
Short Answer
Expert verified
The asymptotic curves are all lines along variable u. The lines of curvature are the parametric curves u = const and v = const. The mean curvature is zero.
Step by step solution
01
Compute the First Fundamental Form
Given the helicoid parametric equations: \(x = v \, \cos u\), \(y = v \, \sin u\), \(z = c \, u\), calculate the partial derivatives: \( \mathbf{r}_u = \left(-v \sin u, \, v \cos u, \, c\right) \) and \( \mathbf{r}_v = \left( \cos u, \, \sin u, \, 0\right) \). Then compute the elements of the first fundamental form: \( E = \mathbf{r}_u \cdot \mathbf{r}_u = v^2 + c^2 \), \( F = \mathbf{r}_u \cdot \mathbf{r}_v = 0 \), \( G = \mathbf{r}_v \cdot \mathbf{r}_v = 1 \).
02
Compute the Second Fundamental Form
To find the second fundamental form, calculate the normal vector \( \mathbf{N} \): \( \mathbf{N} = \frac{\mathbf{r}_u \times \mathbf{r}_v}{\| \mathbf{r}_u \times \mathbf{r}_v \|} = \left( \cos u, \, \sin u, \, -\frac{v}{c}\right) \). Next, compute the second derivatives: \( \mathbf{r}_{uu} = \left(-v \cos u, \, -v \sin u, \, 0\right) \), \( \mathbf{r}_{uv} = \left(- \sin u, \, \cos u, \, 0\right) \), \( \mathbf{r}_{vv} = \left(0, \, 0, \, 0\right) \). Now calculate the second fundamental form coefficients: \( L = \mathbf{r}_{uu} \cdot \mathbf{N} = v\), \( M = \mathbf{r}_{uv} \cdot \mathbf{N} = 0 \), \( N = \mathbf{r}_{vv} \cdot \mathbf{N} = 0 \).
03
Determine the Mean Curvature
The mean curvature \(H\) is given by: \[ H = \frac{EN - 2FM + GL}{2(EG - F^2)} \]. Substituting the values for the first and second fundamental form coefficients: \[ H = \frac{(v^2 + c^2) \cdot 0 - 2 \cdot 0 \cdot 0 + 1 \cdot v}{2((v^2 + c^2) \cdot 1 - 0^2)} = \frac{v}{2(v^2 + c^2)} \]. Since \( H \) needs to be zero, and the numerator is \(v\), the result is zero.
04
Identify Asymptotic Curves and Lines of Curvature
For a helicoid, asymptotic curves arise where the second fundamental form vanishes. Since M = N = 0, only the principal direction where v=0 is significant. This indicates that all lines along \(u\) are asymptotic. As for lines of curvature, given the form of the second fundamental form, we can deduce that lines of curvature follow the parametric curves of the helicoid \(u = \text{const}\) and \(v = \text{const}\).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
First Fundamental Form
The first fundamental form is essential in differential geometry. It allows us to measure lengths and angles on a surface.
For the helicoid given by the parametric equations \( x = v \cos u \), \( y = v \sin u \), and \( z = c \ u \), we first calculate the partial derivatives with respect to \( u \) and \( v \).
These derivatives are:
For the helicoid given by the parametric equations \( x = v \cos u \), \( y = v \sin u \), and \( z = c \ u \), we first calculate the partial derivatives with respect to \( u \) and \( v \).
These derivatives are:
- \( \mathbf{r}_u = (-v \sin u, \ v \cos u, \ c) \)
- \( \mathbf{r}_v = (\cos u, \sin u, \ 0) \)
- \( E = \mathbf{r}_u \cdot \mathbf{r}_u = v^2 + c^2 \)
- \( F = \mathbf{r}_u \cdot \mathbf{r}_v = 0 \)
- \( G = \mathbf{r}_v \cdot \mathbf{r}_v = 1 \)
Second Fundamental Form
The second fundamental form measures how the helicoid bends by comparing its normal vector and second partial derivatives.
We calculate the normal vector \( \mathbf{N} \) as: \( \mathbf{N} = \frac{\mathbf{r}_u \times \mathbf{r}_v}{\| \mathbf{r}_u \times \mathbf{r}_v \|} = (\cos u, \sin u, \ -\frac{v}{c}) \).
The second partial derivatives are:
We calculate the normal vector \( \mathbf{N} \) as: \( \mathbf{N} = \frac{\mathbf{r}_u \times \mathbf{r}_v}{\| \mathbf{r}_u \times \mathbf{r}_v \|} = (\cos u, \sin u, \ -\frac{v}{c}) \).
The second partial derivatives are:
- \( \mathbf{r}_{uu} = (-v \cos u, \ -v \sin u, \ 0) \)
- \( \mathbf{r}_{uv} = (- \sin u, \cos u, \ 0) \)
- \( \mathbf{r}_{vv} = (0, \ 0, \ 0) \)
- \( L = \mathbf{r}_{uu} \cdot \mathbf{N} = v \)
- \( M = \mathbf{r}_{uv} \cdot \mathbf{N} = 0 \)
- \( N = \mathbf{r}_{vv} \cdot \mathbf{N} = 0 \)
Mean Curvature
The mean curvature \( H \) informs us about the average curvature of the helicoid.
It is calculated with the formula: \[ H = \frac{EN - 2FM + GL}{2(EG - F^2)} \].
Substituting the coefficients from the first and second fundamental forms: \[ H = \frac{(v^2 + c^2) \cdot 0 - 2 \cdot 0 \cdot 0 + 1 \cdot v}{2((v^2 + c^2) \cdot 1 - 0^2)} = \frac{v}{2(v^2 + c^2)} \].
Because the numerator is \( v \) and for mean curvature to be zero, we see that \( v = 0 \). Therefore, \( H \) is zero across the helicoid. This indicates that the helicoid is a minimal surface.
It is calculated with the formula: \[ H = \frac{EN - 2FM + GL}{2(EG - F^2)} \].
Substituting the coefficients from the first and second fundamental forms: \[ H = \frac{(v^2 + c^2) \cdot 0 - 2 \cdot 0 \cdot 0 + 1 \cdot v}{2((v^2 + c^2) \cdot 1 - 0^2)} = \frac{v}{2(v^2 + c^2)} \].
Because the numerator is \( v \) and for mean curvature to be zero, we see that \( v = 0 \). Therefore, \( H \) is zero across the helicoid. This indicates that the helicoid is a minimal surface.
Asymptotic Curves
Asymptotic curves on a surface are directions where the normal curvature is zero.
For the helicoid, the second fundamental form's coefficients \( M \) and \( N \) are zero. Therefore, all lines along \( u \), i.e., \( v = 0 \), are asymptotic curves.
These curves represent directions where the surface does not bend in the normal direction.
For the helicoid, the second fundamental form's coefficients \( M \) and \( N \) are zero. Therefore, all lines along \( u \), i.e., \( v = 0 \), are asymptotic curves.
These curves represent directions where the surface does not bend in the normal direction.
Lines of Curvature
Lines of curvature are curves on the surface where the curvature is maximal or minimal.
For the helicoid, based on the second fundamental form, we deduce that the lines of curvature follow the parametric coordinate lines, where \( u = \text{const} \) and \( v = \text{const} \).
This means concentric circles and vertical lines on the helicoid are lines of curvature, representing the maximum and minimum curvatures on the helicoid.
For the helicoid, based on the second fundamental form, we deduce that the lines of curvature follow the parametric coordinate lines, where \( u = \text{const} \) and \( v = \text{const} \).
This means concentric circles and vertical lines on the helicoid are lines of curvature, representing the maximum and minimum curvatures on the helicoid.
