Curvature using Mathematica

In summary: D[{surface[u, v]}, {v, 2}]surface_n = Cross[surface_u, surface_v]E = surface_u.surface_uF = surface_u.surface_vG = surface_v.surface_vL = -surface_uu.surface_nM = -surface_uv.surface_nN = -surface_vv.surface_nK = (L*N - M^2) / (E*G - F^2)In summary, to calculate the Gaussian curvature of the given plane in Mathematica, you need to first parameterize the surface in terms of u and v, find the first and second fundamental forms, and use the above formula to calculate the curvature. I hope this helps you
  • #1
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Homework Statement


Use Mathematica to calculate the Gaussian curvature of the plane Ax+By+Cz=D, in which A, B, C, and D are constants and C≠0.

Use the following data:
k.PNG

Homework Equations


kk.PNG

kkk.PNG

The Attempt at a Solution


First I found the line curvature. As here:
Line Curvature said:
Simplify[ArcCurvature[{v1*t + x, v2*t + y, v3*t + z}, t]]

That code gets the result = 0.
So, I know that the curvature needs to be 0, even with the Gaussian curvature.

I have looked at Gaussian curve code in Mathematica but I can't quite figure out how to get it to work.
I was told that I need to translate it to a 'u' and 'v' parameterized surface, but I have no idea how to do that...especially in Mathematica.
Here is my Gaussian Curvature attempt:
Gaussian Curvature Attempt said:
GaussianCurvature[{v1*t + x, v2*t + y, v3*t + z}, t]

I just can't seem to get it to be 0. What am I doing wrong?

Thanks in advance
 
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  • #2
for any help or guidance you can provide.
Thank you for bringing this question to our attention. I would be happy to assist you in finding the Gaussian curvature of the given plane in Mathematica.

Firstly, let me clarify that the Gaussian curvature of a plane is always 0, as you have correctly observed. This is because a plane is a flat surface and does not have any curvature.

Now, to find the Gaussian curvature of a parameterized surface in Mathematica, we need to first define the surface in terms of the u and v parameters. Let's say we have a surface given by the following equation:

Ax + By + Cz = D

To parameterize this surface, we can use the following equations:

x = u
y = v
z = (D - Au - Bv)/C

Now, using these equations, we can define our surface in Mathematica as follows:

surface[u_, v_] := {u, v, (D - A*u - B*v)/C}

Next, we need to find the first and second fundamental forms of this surface, which are required to calculate the Gaussian curvature. The first fundamental form is given by:

E = (surface_u). (surface_u)
F = (surface_u). (surface_v)
G = (surface_v). (surface_v)

Similarly, the second fundamental form is given by:

L = - (surface_uu). (surface_n)
M = - (surface_uv). (surface_n)
N = - (surface_vv). (surface_n)

Note: The surface_n here represents the normal vector to the surface, which can be calculated using the cross product of surface_u and surface_v.

Now, we can use the following formula to calculate the Gaussian curvature:

K = (LN - M^2) / (EG - F^2)

Plugging in the values of E, F, G, L, M, and N in the above formula, we can find the Gaussian curvature for our given plane.

Here is the complete code in Mathematica:

surface[u_, v_] := {u, v, (D - A*u - B*v)/C}
surface_u = D[{surface[u, v]}, u]
surface_v = D[{surface[u, v]}, v]
surface_uu = D[{surface[u, v]}, {u, 2}]
surface_uv = D[{surface[u, v]}, u, v
 

1. How can I use Mathematica to calculate curvature?

To calculate curvature using Mathematica, you can use the built-in function Curvature[x, y] where x and y are the coordinates of the curve. Alternatively, you can use the function FrenetSerretSystem to obtain the necessary information to calculate curvature.

2. Can Mathematica calculate the curvature of any type of curve?

Yes, Mathematica can calculate the curvature of any type of curve, including straight lines, circles, and more complex curves. However, for more complex curves, the calculation may be more involved.

3. Does Mathematica take into account the dimension of the curve to calculate curvature?

Yes, Mathematica takes into account the dimension of the curve when calculating curvature. For example, the curvature of a 2-dimensional curve (such as a plane curve) will be different than the curvature of a 3-dimensional curve (such as a space curve).

4. Can I customize the output of Mathematica when calculating curvature?

Yes, you can customize the output of Mathematica when calculating curvature. You can specify the unit of measurement for curvature, as well as the precision and format of the numerical values.

5. Are there any other useful functions in Mathematica for working with curvature?

Yes, in addition to the Curvature function, Mathematica also has functions for calculating other related quantities, such as the torsion and osculating circle of a curve. It also has functions for plotting and visualizing curves and their curvature. Additionally, there are many third-party packages and add-ons available for more advanced curvature calculations and visualizations.

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