Finding positive orientation for a surface

In summary, the conversation discusses the process of finding the normal to a parameterized surface by crossing the vectors ra and rb. The question arises of how to determine the positive orientation of the normal, which is arbitrary and requires additional information. An example is given of a surface with a specific parameterization and the correct method of finding its orientation towards the origin.
  • #1
Miike012
1,009
0
If I parameterized a surface by the vector function r(a,b) I would then proceed to find the normal to the surface by crossing ra and rb. But how would I determine which normal raXrb or rbXra has the positive orientation?

Im assuming you would first need to know if the concave side of the surface is facing away or towards the origin? But from there I don't know.
 
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  • #2
I don't see how it could be related to second order derivatives. Consider e.g. just moving around in the XY plane. Let's say the cross product is defined with a right-hand corkscrew rule, so (1,0,0)x(0,1,0) = (0,0,1). If increasing a moves r in the +ve x-direction and increasing b moves r in the +ve y-direction then raxrb is in the +ve z direction. But if you swap the sign of ra it will be in the negative z-direction.
 
  • #3
The distinction between "positive" and "negative" orientation for a surface is purely arbitrary. A surface can have either of two "orientations", the normal vector pointing in either of two directions out of the surface. But to designate one of those as the "postive orientation" and the other as "negative orientation" requires some other information.

Could you give us the exact wording of an example. Usually such "surface orientation" probles will say "oriented by postive z component" or in the case of a closed surface oriented by "outward" or "inward" normals.
 
  • #4
HallsofIvy said:
The distinction between "positive" and "negative" orientation for a surface is purely arbitrary. A surface can have either of two "orientations", the normal vector pointing in either of two directions out of the surface. But to designate one of those as the "postive orientation" and the other as "negative orientation" requires some other information.

Could you give us the exact wording of an example. Usually such "surface orientation" probles will say "oriented by postive z component" or in the case of a closed surface oriented by "outward" or "inward" normals.


I parameterized the surface
x = x
y = x^2+ z^2
z=z

Wrong picture look at the picture in post # 5
Now to find the orientation towards the origin do I do rxXrz or rzXrx?
 

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  • #5
Correct problem
 

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FAQ: Finding positive orientation for a surface

What is the process for finding positive orientation for a surface?

The process for finding positive orientation for a surface involves determining the direction of the surface's normal vector. This can be done by visually inspecting the surface or by using mathematical calculations.

Why is it important to find the positive orientation for a surface?

Finding the positive orientation for a surface is important because it helps to determine the direction in which the surface is facing. This information is crucial for various applications, such as in computer graphics, engineering, and physics.

How can I determine the normal vector of a surface?

The normal vector of a surface can be determined by taking the cross product of two tangent vectors on the surface. These tangent vectors can be found by taking partial derivatives of the equations that define the surface.

Can the positive orientation of a surface change?

Yes, the positive orientation of a surface can change depending on the context in which it is being observed. For example, if the surface is being used in a different coordinate system or if the surface is being viewed from a different angle, the positive orientation may appear to change.

Are there any real-world applications for finding positive orientation for a surface?

Yes, there are many real-world applications for finding positive orientation for a surface. For example, in computer graphics, determining the positive orientation of a surface is important for rendering realistic 3D images. In engineering, knowing the positive orientation of a surface can help with designing and constructing structures. In physics, the positive orientation of a surface is crucial for understanding the direction of forces and vectors acting on the surface.

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