[Closed] Geometrical calculations : points, lines, planes : intersections, distances, angles

nice stuff

takes me back to highschool math days…

line and line:

Line-Line Intersection : intersection of two lines AB and CD:

fn lineLineIntersect pA pB pC pD = (
	local a=pB-pA
	local b=pD-pC
	local c=pC-pA
	local cross1 = cross a b
	local cross2 = cross c b
	pA + ( a*( (dot cross2 cross1)/((length cross1)^2) ) )
	)

or by using vectors:

fn lineLineIntersect pA a pB b = (
	local c=pB-pA
	local cross1 = cross a b
	local cross2 = cross c b
	pA + ( a*( (dot cross2 cross1)/((length cross1)^2) ) )
	)

see also : Eric W. Weisstein. From MathWorld–A Wolfram Web Resource.
http://mathworld.wolfram.com/Line-LineIntersection.html

Line-Line Distance : distance between two skew lines

fn lineLineDist pA a pB b = (
	local c=pB-pA
	local crossAB=cross a b
	abs (dot c crossAB) / (length crossAB)
	)

see also : Eric W. Weisstein. From MathWorld–A Wolfram Web Resource.
http://mathworld.wolfram.com/Line-LineDistance.html

Line-Line // Distance : distance between two parallel lines
use “Point-Line Distance” : pA-pB is the first line and pC is a point of the second line

Line-Line Angle : angle between two lines:

fn getVectorsAngle vAB vCD = (
	acos (dot (normalize vAB) (normalize vCD))
	)

or

fn lineLineAngle pA pB pC pD = (
	local vAB=pB-pA
	local vCD=pD-pC
	local angle=acos (dot (normalize vAB) (normalize vCD))
	if angle<90.0 then angle else (180.0-angle)
	)

see also : Eric W. Weisstein. From MathWorld–A Wolfram Web Resource.
http://mathworld.wolfram.com/Line-LineAngle.html

plane and point:

Point-Plane Projection : find the point on the plane ABC which is the projection of the point D

fn pointPlaneProj pA pB pC pD = (
	local nABC=normalize (cross (pB-pA) (pC-pA))
	pD+((dot (pA-pD) nABC)*nABC)
	)

Point-Plane Distance : find the distance between a plane and a point

fn pointPlaneDist pA pB pC pD = (
	local nABC=normalize (cross (pB-pA) (pC-pA))
	length ((dot (pA-pD) nABC)*nABC)
	)

see also : Eric W. Weisstein. From MathWorld–A Wolfram Web Resource.
http://mathworld.wolfram.com/Point-PlaneDistance.html

plane and line:

Line-Plane Intersection : intersection between a line and a plane

fn planeLineIntersect planePoint planeNormal linePoint lineVector = (
	local lineVector=normalize lineVector
	local d1=dot (planePoint-linePoint) planeNormal
	local d2=dot lineVector planeNormal
	if abs(d2)<.0000000754 then ( if abs(d1)>.0000000754 then 0 else -1 )
		else ( linePoint + ( (d1/d2)*lineVector ) )
	)

This script is based on an original script of Joshua Newman.
return 0 for parrallel (no intersections) results
and -1 for co-incident (infinate) results

see also : Eric W. Weisstein. From MathWorld–A Wolfram Web Resource.
http://mathworld.wolfram.com/Line-PlaneIntersection.html

Line-Plane Distance : distance between a plane and a line //
use “Point-Plane Distance” : where ‘point’ is a point of the line

Vector perpendicular to a line into a Plane : the perpendicular of AB into the plane ABC

vectorPerp=cross vectorAB (cross vectorAB vectorAC)
-- or
vectorPerp=cross (cross vectorAB vectorAC) vectorAB

see also : Eric W. Weisstein. From MathWorld–A Wolfram Web Resource.
http://mathworld.wolfram.com/Perpendicular.html

Thanks for the aknowelgement PP.

Still can’t get around using the tolerence then eh?

Great resource BTW, keep up the good work!

J¬

This is great, thanks.

This should be a sticky!

Thanks guys

jman: The tolerance seems to work for me, but I use rather the function in cases where the line is not parallel to the plan. Can you give an example where the tolerance is an issue?

Well, the continuation:

plane and plane:

Plane-Plane Intersection : find the line which is the intersection of 2 planes
p1 : a point of the plane 1
n1 : the normal of the plane 1

fn planePlaneIntersect p1 n1 p2 n2 = (
	-- n1, n2 are normalized
	local lineVector = cross n1 n2
	local proj1=(dot n1 p1)*n1
	local proj2=(dot n2 p2)*n2
	local perp1=cross n1 (normalize lineVector)
	local perp2=cross n2 (normalize lineVector)
	local cr = cross (proj2-proj1) perp2
	local intersectionPoint = proj1 + (perp1*( (dot cr lineVector) / ((length lineVector)^2)) )
	ray intersectionPoint lineVector
	)

I think that it is not the easiest solution. If you know a better means…
see also : Eric W. Weisstein. From MathWorld–A Wolfram Web Resource.
http://mathworld.wolfram.com/Plane-PlaneIntersection.html

Plane perpendicular to a Plane and a Line : find the plane which is perpendicular to the plane n and to the line AB
It’s the plane ABC where C=A+n
see also : Eric W. Weisstein. From MathWorld–A Wolfram Web Resource.
http://mathworld.wolfram.com/ParallelPlanes.html

Plane-Plane Angle : find the angle between two planes n1 and n2 (normals)

fn getVectorsAngle n1 n2 = ( acos (dot (normalize n1) (normalize n2)) )