Four point plane to Point+Vector Plane?

thomthom

http://code.google.com/apis/sketchup/docs/ourdoc/geom.html#introduction

A plane can be represented as either an Array of a point and a vector, or as an Array of 4 numbers that give the coefficients of a plane equation.

How does one convert a plane defined by four points to a plane defined by [ point3d, vector3d ] ?

"coefficients of a plane equation"

remus

Just a small correction, when you say "a plane defined by 4 points" you really mean a plane defined by 4 numbers (which are the coefficients of the equation of the plane.)

thomthom

hmm... always read "numbers" as "Point3d"s...
I never used or encountered anything other than the [ point3d, vector3d ] variant, but I want to make a method to ensure that's the format I get planes in.

thomthom

http://local.wasp.uwa.edu.au/~pbourke/geometry/planeeq/

@unknownuser said:

The standard equation of a plane in 3 space is Ax + By + Cz + D = 0

@unknownuser said:

The normal to the plane is the vector (A,B,C).

So given a plane of [a,b,c,d], the normal would be Geom::Vector3d.new(a,b,c)?
But what is d ? And I don't see where I derive any point3d from this - to fit [ point3d, vector3d ] ...

thomthom

Think it makes sense now....

Did some testing:


p1 = [200,200,200]
[200, 200, 200]
p2 = [400,200,200]
[400, 200, 200]
p3 = [400,400,200]
[400, 400, 200]
p4 = [200,400,200]
[200, 400, 200]
Geom.fit_plane_to_points(p1,p2,p3,p4)
[-0.0, -0.0, 1.0, -200.0]

remus

d is just a constant, so unless theres an error in the docs (i.e. they mean 'you can define a plane with a vector and a constant', not unlikely) then youll have to find a point on the plane.

thomthom

This seem to work.


	# Return a plane in the format [ point3d, vector3d ]
	def self.normalize_plane(plane)
		return plane if plane.length == 2
		a, b, c, d = plane
		v = Geom;;Vector3d.new(a,b,c)
		p = ORIGIN.offset(v.reverse, d)
		return [p, v]
	end

When I test it:


p1 = [200,200,200]
[200, 200, 200]
p2 = [400,200,200]
[400, 200, 200]
p3 = [400,400,200]
[400, 400, 200]
p4 = [200,400,200]
[200, 400, 200]
plane = Geom.fit_plane_to_points(p1,p2,p3,p4)
[-0.0, -0.0, 1.0, -200.0]
TT_Lib;;Geom3D.normalize_plane(plane)
[Point3d(0, 0, 200), Vector3d(0, 0, 1)]

AdamB

Its just not efficient. A plane is commonly defined by a direction and a distance along that direction (the "D" constant). Keep in mind a plane is infinite so you just need "a point" - any point on the plane to define it.

So explicitly storing the Point3d is (groan) pointless. If you want a point on the plane from its plane equation you just do:

plane_normal.scale(D)

ie You just need to store 4 values not 6 as you are doing.

AdamB

Also you really don't want 4 points as input as they may not be coplanar. 3 points are guaranteed to be coplanar.

thomthom

Yea - I'm not dealing with points. As remus pointed out to me, it was four numbers. I just wanted a method that'd convert a plane defined as four numbers into [ point3d, vector3d ] - as the Geom module says planes can be in either format.