Calculating an elliptical surface and formulating this surface in 3d

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In summary, the conversation discusses a problem involving a sphere with four selected points on the Earth's surface and the process of merging them with shortest curves to create an elliptical surface. The question is how to calculate the area of this surface and formulate it in 3D coordinates. The suggested solution is to find the (x,y,z) coordinates and use a surface area integral to calculate the area.
  • #1
erencan144
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Hello.
Let's think that we have a sphere that shown in the picture above. The user will select 4 different point the Earth's surface. Then I must merge this points with shortest curves, then I got a surface. (Like picture 2) Because of the world's surface, our area is elliptical. How can I calculate the area of this elliptical surface and formulating this elliptical surface in 3d coordinates (x,y,z).

I would be very grateful, if you tell me how can I deal with this problem.

Sorry for poor English.

picture 1

http://img705.imageshack.us/img705/6263/worldre.png

picture 2

http://img836.imageshack.us/img836/3679/world2u.png
 
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  • #2
Well you need to know the (x,y,z) coordinates first then just find the area using a surface area integral.
 

1. How do you calculate the surface area of an elliptical shape?

Calculating the surface area of an elliptical shape involves using the formula A = πab, where a and b are the semi-major and semi-minor axes of the ellipse. These values can be obtained by measuring the distance from the center of the ellipse to the furthest points on its x and y axes.

2. What is the difference between an elliptical surface and a circular surface?

An elliptical surface is a three-dimensional shape with a curved surface that resembles an ellipse, while a circular surface is a flat two-dimensional shape with a curved perimeter. Elliptical surfaces have two different radii, while circular surfaces only have one.

3. How can an elliptical surface be formulated in 3D space?

To formulate an elliptical surface in 3D space, you can use the parametric equations x = a cos(t), y = b sin(t), z = 0, where a and b are the semi-major and semi-minor axes of the ellipse, and t is the parameter that varies from 0 to 2π. This will generate a 3D surface that resembles an ellipse when plotted.

4. Can an elliptical surface have negative values for its axes?

Yes, an elliptical surface can have negative values for its axes. This will result in an ellipse that is oriented in a different direction compared to a positive value for the same axis. For example, a negative value for the semi-major axis will result in an ellipse that is longer in the y direction and shorter in the x direction.

5. What real-world objects have elliptical surfaces?

There are many real-world objects that have elliptical surfaces, such as eggs, some fruits like watermelons and lemons, some types of lenses, and some celestial bodies like the orbits of planets and moons. Elliptical surfaces are also commonly used in architecture and design, such as in the shape of domes and arches.

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