How measure the angle between two surfaces of 3D solid ?

[AIberto]

Hi all

 

How measure the angle between two surfaces of 3D solid ?

and

How measure the distance(The shortest) between two surfaces of 3D solid ?

Edited December 1, 2015 by AIberto

[lrm]

The angle between two surfaces can be found by using the following vector equation:

 

angle = arcsin( (Na cross Nb) / (|Na| |Nb|) )

 

where Na and Nb are the normal vectors of the two surfaces.

 

To get a normal vector of a flat surface assuming p1, p2, and p3 are three, non-collinear points on the surface

 

N = ( vector from p1 to p2 ) cross (vector from p1 to p3)

 

 

Here's a function I wrote for Autolisp to computer the cross product of two vectors. I'm sure someone will offer a more elegant solution.

 

;;; Compute the cross product of 2 vectors a and b
;;
(defun cross (a b / crs)
 (setq	crs (list
      (- (* (nth 1 a) (nth 2 b))
	 (* (nth 1 b) (nth 2 a))
      )
      (- (* (nth 0 b) (nth 2 a))
	 (* (nth 0 a) (nth 2 b))
      )
      (- (* (nth 0 a) (nth 1 b))
	 (* (nth 0 b) (nth 1 a))
      )
    )				;end list
 )					;end setq c
)					;end cross
;;

~Lee

[Lee Mac]

Alternatively, you could follow the dot-product route to calculate the angle, i.e.:

(defun dihedral ( n1 n2 )
   (acos (apply '+ (mapcar '* n1 n2)))
)

;; ArcCosine  -  Lee Mac
;; Args: -1 <= x <= 1

(defun acos ( x )
   (if (<= -1.0 x 1.0)
       (atan (sqrt (- 1.0 (* x x))) x)
   )
)

Evaluate the above with the unit normal vectors for each plane to return the dihedral angle between the planes - calculate the normal vectors using the cross-product as Lee has described above.

[AIberto]

lrm & Lee , Thank a lot.

Can you give a Example for test ? Thanks!

[lrm]

I am not sure how much detail you want for an example. You can create a box with a 45° chamfered edge to verify the calculation. How familiar are you with AutoCAD's VisualLISP?

~Lee

[lrm]

The following will calculate the dihedral angle between two planes from 3 points on each of the planes.

(defun c:diangle (/ p1 p2 p3 n1 n2 ang)
; calulate dihedral angle between two planes
;  from 3 points defining each plane
; get 3 points plane 1 
 (setq	p1 (getpoint "\nSelect point #1 for plane 1")
p2 (getpoint "\nSelect point #2 for plane 1")
p3 (getpoint "\nSelect point #3 for plane 1")
 )
 (setq n1 (normal3p p1 p2 p3)) ; normal vector plane 1
				; get 3 points plane 2 
 (setq	p1 (getpoint "\nSelect point #1 for plane 2")
p2 (getpoint "\nSelect point #2 for plane 2")
p3 (getpoint "\nSelect point #3 for plane 2")
 )
 (setq n2 (normal3p p1 p2 p3));normal vector plane 2
				; calculate angle in degrees
 (setq ang (/ (* 180.0 (dihedral n1 n2)) 3.1415926535))
 (princ "\nThe dihedral angle is: ")
 (princ ang)
 (princ "°")
 (princ)
)
;;; unit normal vector from 3 points on plane
(defun normal3p (p1 p2 p3 / n d)
				; calculate normal vector
 (setq n (cross (mapcar '- p2 p1) (mapcar '- p3 p1)))
				; calculate unit vector normal
 (setq d (distance '(0 0 0) n))
 (setq n (mapcar '/ n (list d d d)))
)  

;;; cross product of 2 vectors a and b
(defun cross (a b / crs)
 (setq	crs (list
      (- (* (nth 1 a) (nth 2 b))
	 (* (nth 1 b) (nth 2 a))
      )
      (- (* (nth 0 b) (nth 2 a))
	 (* (nth 0 a) (nth 2 b))
      )
      (- (* (nth 0 a) (nth 1 b))
	 (* (nth 0 b) (nth 1 a))
      )
    )				;end list
 )					;end setq crs
)					;end cross

; Lee Mac's dihedral and acos functions
(defun dihedral ( n1 n2 )
   (acos (apply '+ (mapcar '* n1 n2)))
)
;; ArcCosine  -  Lee Mac
;; Args: -1 <= x <= 1
(defun acos ( x )
   (if (<= -1.0 x 1.0)
       (atan (sqrt (- 1.0 (* x x))) x)
   )
)

;;

[AIberto]

The following will calculate the dihedral angle between two planes from 3 points on each of the planes.

(defun c:diangle (/ p1 p2 p3 n1 n2 ang)
; calulate dihedral angle between two planes
;  from 3 points defining each plane
; get 3 points plane 1 
 (setq	p1 (getpoint "\nSelect point #1 for plane 1")
p2 (getpoint "\nSelect point #2 for plane 1")
p3 (getpoint "\nSelect point #3 for plane 1")
 )
 (setq n1 (normal3p p1 p2 p3)) ; normal vector plane 1
				; get 3 points plane 2 
 (setq	p1 (getpoint "\nSelect point #1 for plane 2")
p2 (getpoint "\nSelect point #2 for plane 2")
p3 (getpoint "\nSelect point #3 for plane 2")
 )
 (setq n2 (normal3p p1 p2 p3));normal vector plane 2
				; calculate angle in degrees
 (setq ang (/ (* 180.0 (dihedral n1 n2)) 3.1415926535))
 (princ "\nThe dihedral angle is: ")
 (princ ang)
 (princ "°")
 (princ)
)
;;; unit normal vector from 3 points on plane
(defun normal3p (p1 p2 p3 / n d)
				; calculate normal vector
 (setq n (cross (mapcar '- p2 p1) (mapcar '- p3 p1)))
				; calculate unit vector normal
 (setq d (distance '(0 0 0) n))
 (setq n (mapcar '/ n (list d d d)))
)  

;;; cross product of 2 vectors a and b
(defun cross (a b / crs)
 (setq	crs (list
      (- (* (nth 1 a) (nth 2 b))
	 (* (nth 1 b) (nth 2 a))
      )
      (- (* (nth 0 b) (nth 2 a))
	 (* (nth 0 a) (nth 2 b))
      )
      (- (* (nth 0 a) (nth 1 b))
	 (* (nth 0 b) (nth 1 a))
      )
    )				;end list
 )					;end setq crs
)					;end cross

; Lee Mac's dihedral and acos functions
(defun dihedral ( n1 n2 )
   (acos (apply '+ (mapcar '* n1 n2)))
)
;; ArcCosine  -  Lee Mac
;; Args: -1 <= x <= 1
(defun acos ( x )
   (if (<= -1.0 x 1.0)
       (atan (sqrt (- 1.0 (* x x))) x)
   )
)

;;

 

Many thanks ,lrm .

This code must pick 6 point ...Can only pick 2 face ?

How pick 2 face ? maybe can use : Solidedit ---> face--->copy.............

At end can remove that face...

Maybe need an option. choose Angle is "Acute" or "obtuse" .

[lrm]

Alberto, You are welcome. Glad to be of help.

~Lee