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# Orienting boundary with surface

Determining the proper orientation of the boundary given the orientation of the surface. Created by Sal Khan.

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• Isn't it easier to use a right-hand rule? Curling four fingers across the direction of the curve with your thumb pointing to the positive normal vector direction? Or are there cases where this does not apply? • You can do that for every contour. The right-hand rule its just a more intuitive way to know how a cross product will look like. Positive it your thumb its on the positive direction of z or x or y, depending on which axis you're working and negative if it points to the negative direction.Some call it the Screwdriver rule, if the screwdriver is turning a screw and it would take it out of the surface or point or whatever the resultant vector is positive and vice-versa for the negative situation.

Did i made it more clear to you?
• Hello,
I am struggling with following question for so many days now. It would be great if you could help.
If I have a Normal at a point on a surface on the surface boundary and a tangent at the same point on the curve, how do I check whether the curve is in the positive orientation with surface? In short, how can I verify that the surface is really on the left or not from the curve tangent and surface normal information?
In case of planar surfaces, it is quite easier to solve because surface normal is constant all over the surface. On a non planar surface it is more difficult because the normal changes. e.g. in case of cylinder which is cut by two circular curves (one at z=0 and one at z = h) the curve orientations are required to be opposite of each other.

Thank you very much in advance.

Regards,
Anup • If the concern is making sure the curve is positively orientated I believe you could find the "curl" with a line integral. If the line integral turned out positive the surface path would be counterclockwise which is positive by convention and tell you the surface was on your left. If the value was negative then the path of the line integral is clockwise making the surface on your right.
• So, it's just the right hand rule? Using the thumb as N and wrapping fingers is where the circulation is going? N positive = counterclockwise. N negative = clockwise? • For the simple case of a "normal" surface it's just the right hand rule right.
Image for example a cylinder with open end on both ends.

So now you can't you the right hand rule. The normal vector of the sides of the cylinder is pointing radial outwards. On you have to orientate the TWO boundary curves (because in this case you have 2 boundary Curves) according to the rule Sal explained. And the two curved will have opposite orientation. Hope this makes sense to you.
• This should be in the Tips & Thanks section, but since people rarely view that section I will leave this in the Questions.

No need to watch this video
Just watch the video after this :

It's basically the identical video but it has an added insight at the end (using the "corkscrew" method to visualize direction). Personally, I prefer the right hand rule but any of these methods work • When he says that if you orient your head in the direction of the normal vector and you were to walk along the path, the surface would be on your left (-), does that imply that you are outside of the surface?
(1 vote) • a question about direction. in a 3d cordinate system, if we have a curve, what is the normal at a point going to be like. is it a plane perpendicular to the tangent.
well i know for a surface we can find a normal at a point p, and than its tangent plane by simple formula <p-x>.<n> = 0.
(1 vote) • this may be in the context of geometry, does integrating ∯1.n̂ds just gives the area enclosed by the curve.
(1 vote) 