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sample v for inclusion in parametric equation

asked 2020-04-17 14:06:41 -0500

superfly gravatar image

updated 2020-04-17 21:28:35 -0500

supra56 gravatar image

I am working through the following equation in the plane-line intersection point. But the question is smaller than this. I am confused about v, the direction vector:

t = - (dot(n,p) + d) / dot (n, v)

I know one contributing equation is:

p = po + t * v

I know that v represents the direction vector. What do I calculate v from :

  • 2 different points on the plane?
  • position vectors on the planes?
  • other?
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answered 2020-04-28 14:48:35 -0500

superfly gravatar image

updated 2020-04-28 14:59:21 -0500

One vector of the line should be a direction vector, and this is v. The second point Po can be anywhere on the line that together with the direction vector, will satisfy the line equation that will together with plane equation will create an intersection point (see above).

ex. p = (0,0,0) + t(4,3,5) in this case Po = origin (0,0,0) and direction vector (4,3,5) create a ray from origin outward.

Parametric eq from the ray: X = 4t +0 Y = 3t+0 Z = 5t+0

Plane Equation: 4x + 3y - 1z = 4 solve for t 4(4t) + 3(3t) -1(5t) = 4 16t + 9t -5t = 4 t = 4/20 = 1/5

Find x,y,z X = (4)(1/5) = 4/5 y = (3)(1/5) = 3/5 Z = (5)(1/5) = 5/5 = 1.

If you changed Po then the direction vector would change.

Alternative Plane Equation n.p + d = 0 sub for p = p0+tv n.(po+tv)+d = 0 n.po+n.tv+d = 0 t(n.v) = -d-n.po t = -d-n.po/n.v t = -(d+n.po)/n.v solve for t n = (4,3,-1) v =

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Asked: 2020-04-17 14:06:41 -0500

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Last updated: Apr 28