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steepest slope direction on the plane

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Old   July 6, 2005, 23:24
Default steepest slope direction on the plane
  #1
Patrick
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The plane is given by three points(P1, P2, P3). It's 3D. How to get the the steepest slope direction vector on this plane? It looks like a high school geometry problem. But I forget those things. thanks!
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Old   July 7, 2005, 04:33
Default Re: steepest slope direction on the plane
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Karthik
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Hi Patrick ,

Create two vectors, Vector 1 = P2-P1 Vector 2 = P3-P1 The cross product of those two vectors will give you vector which represents the normal direction ( steepest gradient).

The above answer is for a single plane with its point representations. When an entire flow field is taken into consideration, the implementation differs depending upon the problem taken into consideration.

cheers karthik
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Old   July 7, 2005, 11:18
Default Re: steepest slope direction on the plane
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Patrick
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That's normal direction. What I mean is the steepest slope alone the plane. Thank you anyway.
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Old   July 7, 2005, 14:49
Default Re: steepest slope direction on the plane
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Richard
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If h is the height of the plane above a reference plane, the steepest slope is in the direction of grad(h).
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Old   July 7, 2005, 16:40
Default Re: steepest slope direction on the plane
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Mani
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Unless you specify some reference point, line, or plane, the expression 'steepest slope' has no meaning. Let's assume that your reference plane is the x-y plane at z=0, so you are interested in the steepest descent to zero altitude (z). The steepest descent from any point in space to this plane is simply described by a unit vector in z-direction (either positive or negative, depending on the position of that point). Now, all you need to do is to project that vector onto your plane. In general: Given a reference plane, find the normal vector on that reference plane, and project it onto your plane.
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Old   July 8, 2005, 01:36
Default Re: steepest slope direction on the plane
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Karthik
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Hi Patrick,

Find the gradient of the flow paramater ( eg velocity ) to determine its steepest slope. Consider the stencil arrangement of 9 cells in 2D and 27 in 3D around the partcular node of a cell in which the steepest slope has to be calculated.

cheers Karthik
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Old   July 8, 2005, 13:36
Default Re: steepest slope direction on the plane
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Patrick
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"Given a reference plane, find the normal vector on that reference plane, and project it onto your plane." But how to project? In which direction? That is the problem.

Now I derived the expression my self. Suppose the unit normal vector n(A,B,C). Here A,B,C is the vector component. (I use y not z as the elevation direction, sorry) Then the steepest slope in the plane is: (A*B/sqrt(A^2+C^2), -sqrt(A^2+C^2), C*B/sqrt(A^2+C^). (check the signs, not so sure)

For example: A plane parallel to the z axis, and has a angle of 45 degree with XZ plane. The unit normal vector(1/sqrt(2), 1/sqrt(2), 0), then the steepest slope direction (-1/sqrt(2), 1/sqrt(2),0).

Thank you all guys!
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Old   July 8, 2005, 15:05
Default Re: steepest slope direction on the plane
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Mani
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"But how to project? In which direction? That is the problem."

The projection of a vector onto a plane is uniquely defined. There is no ambiguity about "directions". And that's exactly what you're doing with your formula.
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