Applied Mathematics concerning CAD Faces

This comprehensive page is a practical collection about CAD faces with layman explanations of mathematics and additional code fragments. It deals with vectors, angles, normals, 2D projections, rotations and more. The best news is that even BricsCAD Lite supports both Lisp and 3D commands like RuleSurf. Even with limited resources you can build great solutions. So: “Let’s face it! 😉

I made FlatMesh in an attempt to create blanks from ruled surfaces. While the basics are relative easy – translating 3D coordinates to 2D – it turned out that the exceptions are the real math challenge. I couldn’t find a this page like this, in order to clarify and understand the subject. So this page was born. I hope it is of help. I think it is pretty complete. However, if you think something is wrong, missing or can be done better, just drop a comment.

Definitions

Face: A face is, by mathematical definition, a flat polygon and almost always a triangle in a CAD environment. A (CAD) face has three edges and vertices. Complex 3D surfaces can be simulated by using 2D triangular faces. Faces with four vertices could be suitable for orthogonal based surfaces (like a cube) but are almost never suitable for complex 3D surfaces.
3DFace: This is both the entity name and command name in CAD systems like BricsCAD and AutoCAD. A 3DFace entity can be visible as a tetragon or a triangle. The use of tetragons has the advantage of clarity. On the other hand, in 3D, the four corners of each tetragon are (almost) never in one plane. To fix that, for a better visual, each tetragon consists of two adjacent triangles i.e. faces. The common edge of these two faces forms an invisible diagonal of a visible tetragon – the visible sides of the two triangles give the impression of a tetragon but, again, upon examination, the corners (almost) never lie in a plane.
Mesh: A mesh is a collection of triangular and or tetragonal 3Dface entities. A mesh forms a surface object.

Vector: A vector is a direction with a size or magnitude.
- A vector V is represented as
  - with a direction $x,y,z$
  - and a length or magnitude $\|V\|$
- The edges of faces can be considered as vectors like $\vec {AC}$ and $\vec {AB}$ .
- Knowing the coordinates of the vertices, we can do calculations on these vectors, on these faces. That is where our journey in mathematics start!

Mathematical solutions

Let’s start a bit steep to refresh memory, once on the way it gets easy 😉

Calculate the magnitude of a vector

The magnitude is the size, or more precise, the exact length of a vector. For vector $\vec v$ , magnitude is noted as $\|v\|$ .

For vector $\vec{A}$ with value $x,y,z$ , the magnitude or length $\|A\|$ is $\sqrt{x^2+y^2+z^2}$ according to the Pythagorean theorem.

For vector $\vec{AB}$ with coordinates $A=Ax,Ay,Az$ and $B=Bx,By,Bz$ , the magnitude $\|AB\|$ is:

$\sqrt{(Bx-Ax)^2+(By-Ay)^2+(Bz-Az)^2}$ .

Calculate the dot product of two vectors

The dot product is an operation on two vectors $\vec a$ and $\vec b$ with angle $\theta$ in between, resulting in a scalar (number, not a vector).

The dot product by itself has little meaning but is important as a value for other formulas. For example, the law of cosines has the dot product as its basis.

Dot product is written as: $\vec a \cdot \vec b$

There are two common ways to calculate the dot product:

$\vec a \cdot \vec b = \|a\| \|b\| \cos (\theta)$
$\vec a \cdot \vec b = xa*xb+ya*yb+za*zb$

This also means that:

$\theta = \arccos \Big( \frac {xa*xb+ya*yb+za*zb} {\|a\| \|b\|} \Big)$
See “Calculate the angle between two vectors” for more and caveats.

More…

Calculate the cross product of two vectors

The cross product is an algebraic operation on two vectors $\vec a$ and $\vec b$ resulting in a normal vector $\vec c$ perpendicular to both $\vec a$ and $\vec b$ .

Cross product is written as: $\vec a \times \vec b = \vec c$

Calculation:

If vectors $\vec a$ and $\vec b$ consist of coordinates $(xa,ya,za)$ and $(xb,yb,zb)$ , then the cross product is:
$\vec a \times \vec b = \vec c =$
$(ya*zb-za*yb,za*xb-xa*zb,xa*yb-ya*xb)$

Valuable for solving equations: The magnitude of the resulting vector, $\|\vec c\|$ , equals the area of the parallelogram of $\vec a$ and $\vec b$ .

Normal: A line, ray or vector perpendicular to something. In this case we mean a vector perpendicular to a plane. The side of the plane on which the normal starts is determined by the right hand rule. Therefore the order of $a$ and $b$ is important, $a \times b \ne b \times a$ and $a \times b = - ( b \times a )$ . More…

When not using the right hand rule, the sign (+,-) of the coordinates of $\vec c$ should be inverterted.

More…

Calculate the unit vector of a vector

A unit vector of a vector has the same direction but has a length or magnitude of 1. It can be retrieved by dividing each x, y and z of a vector by the magnitude of that vector, resulting in a new vector.

Calculate the angle between two vectors

In 2D, it is easy to comprehend. 3D is harder. For example grep a pencil in each hand and position and point them randomly before you. What on earth is the angle? Now bring the origins together. At that point, you can imagine a plane through the three points and it becomes clear there is an angle. Then the question becomes: What is the angle? And: Do you want the angle from pencil A to pencil B or the other way around? Finally, you can look at the plane from two sides, which changes the answers too. So there is enough complexity.

Using arccosine

The angle $\alpha$ between $\vec {AB}$ and $\vec {AC}$ …

$\alpha = \arccos \Big( \frac {\vec {AB} \cdot \vec {AC}} {\|AB\| \|AC\|} \Big)$

Example

AB = 1,2,3
AC = 4,5,6
|AB| =SQRT(1^2+2^2+3^2) = 3.74
|AC| =SQRT(4^2+5^2+6^2) = 8.77
AB.AC =1*4+2*5+3*6 = 32
acos((AB.AC)/(|AB|*|AC|)) = acos(32/(3.74*8.77)) = 0.226 rad = 12.9 degrees

Here, the angle range is 0 to $\pi$ . Although technically correct, that may not always be what you want – in cases with small angles there is a lot of cumulative error. What else is there?

Using arctangent

The same angle again but by using atan2.

Atan2 is arctan, arctangent, inverse tangent, of value x and y. But be careful, syntaxis, x and y order, can be different. For example, spreadsheet: atan2(x,y), C language: atan2(y,x) and CAD Lisp language: atan(y,x). Also keep in mind that angles range from 0 to $\pi$ radians.

How do we get x and y?

y is the magnitude of the cross product of vector AB and AC
$y = \| \vec{AB} \times \vec{AC} \|$
x is the dot product of vector AB and AC.
$x = \vec{AB} \cdot \vec{AC}$

Again, the angle $\alpha$ between $\vec {AB}$ and $\vec {AC}$ …

$\alpha = atan2 (y,x) = atan2 ( \| \vec{AB} \times \vec{AC} \| , \vec{AB} \cdot \vec{AC} )$

In addition written out in sheet notation for vector AB and AC and angle alpha in radians:

AB = (x1,y1,z1)
AC = (x2,y2,z2)
x =x1*x2+y1*y2+z1*z2
y =sqrt((y1*z2-z1*y2)^2+(z1*x2-x1*z2)^2+(x1*y2-y1*x2)^2)
alpha =atan2(x,y)

Is a vector angle obtuse, perpendicular or acute?

Continuing with the previous… Instead of, or in addition to, calculating an angle, the dot product is enough to say whether the angle is obtuse, perpendicular or acute. It is depending on the sign of $m = \vec {AB} \cdot \vec {AC}$ . More specific:

If m is positive, the angle is acute.
If m is zero, the angle is right, perpendicular.
If m is negative, the angle is obtuse.

A rectangle contains two adjacent faces. We can say something about the angle between the faces by first calculating the normal vectors of each face.

Calculate the normal of a face

The normal vector is the cross product of vector AB times vector AC. According to the right hand rule, it is written as $\vec {AB} \times \vec {AC}$

Example

AB = (1,2,3)
AC = (4,5,6)
ABAC = (2*6-3*5,1*6-3*4,1*5-2*4) = (-3,6,-3)

The opposite normal vector is ACAB, being ABAC with opposite signs, i.e. (3,-6,3).

The magnitude of normal ABAC is area $\|AB\| \|AC\| \cos (\angle CAB)$ . So you can tell something about angle CAB if you have the magnitudes of all three vectors.

Calculate 2D projection of a face – Pythagorean

With three coordinates A, B and C known of a 3D face, we can calculate a projection in 2D. Alternative, law of cosines can be used here too.

Calculating C based on A is (0,0) and B is (p,0)

Summary

Known:
Distances p, q, r
Constraints:
Point A = (0,0)
Point B = (p,0)
Point C is above axis AB
First Cy is positive
z coordinates always 0
Requested:
Coordinates of C.
Answer:
C = ((p^2+r^2-q^2)/(2*p),sqrt(r^2-t^2))
Subsequent Cy values can be negative too, i.e. -sqrt(r^2-t^2)

Rationale

A: (x,y,z)
C: (x',y',z')

Variant 1  Variant 2 ( is OR)
 CAB <= pi()/2  CAB > pi()/2 =>
x <= x'  x > x'

r^2 = t^2+s^2 =>
s^2 = r^2-t^2

q^2 = (p-t)^2+s^2  (p+t)^2+s^2
= (p-t)^2+r^2-t^2  (p+t)^2+r^2-t^2
= p^2-2pt+t^2+r^2-t^2  p^2+2pt+t^2+r^2-t^2
= p^2-2pt+r^2  p^2+2pt+r^2 =>
2pt = p^2+r^2-q^2  -p^2-r^2+q^2 =>
t = (p^2+r^2-q^2)/(2*p)  (p^2+r^2-q^2)/(-2*p)
s^2=r^2-t^2
s=sqrt(r^2-t^2)  s=-sqrt(r^2-t^2)

With two answers for t it is tempting to just code both values. Annoying and itching is the fact I cannot say anything beforehand about CAB.

Hmm, is that so? Climbing from the bottom – the resulting formula’s – up, there is fragment p^2+r^2-q^2. Wait, Pythagoras! That should be 0 when CAB is 90 !

More specific:

One problem solved, one problem created! Let’s see, If p^2+r^2-q^2 < 0then t = (p^2+r^2-q^2)/(-2*p) and else t = (p^2+r^2-q^2)/(2*p). So t is always a positive value and whether it is on the left or right side of x=0 (point A) depends on the value of p^2+r^2-q^2.

More specific, as an x-coordinate, t = (p^2+r^2-q^2)/(2*p). And that is always true! Great, and even my old brains survived this.

A final note about s. It is a square root so the argument can be both positive and negative. It won’t bother us for the assumed first C, but it will bother us for subsequent C’s and all D’s.

Calculate 2D projection of an adjacent face

Getting the coordinate D is similar to finding C.

One segment of the RuleSurf result is build out of two triangles or faces. Using FlatMesh means in most cases: Start with A and B and calculate C and D, then treat those resulting C and D as a new set of A and B and start allover, until the complete mesh is processed. In most cases, the first picture is in play. However, sometimes the second picture can be in play. In addition, even C can be under axis AB but focus is on D for proper explanation.

A,B,C and D are 3D points. There is a plane ABC. D is somewhere relative to this plane. We need to know where D lies. More specific, in a planar view to ABC, is D under or above axis BC? To get a better impression, look at it from a different angle:

This question seems not easy to answer, but, as long as we talk about angles, we can say that:

If the angle between the face normals is acute, D lies above BC.
If the angle between the face normals is obtuse, D lies under BC.

After calculating the normals as explained before, the dot product is all we need to answer the question. Just remember to use the right hand rule.

That was actually the hard part. D is more or less calculated the same way as C.

However, we need to use the axis CB (not AB) as the base, as the mirror line for when D gets under CB. Finally, C and B were rotated around A, D should also be rotated properly. That is where starts to play a role. Rotating…

Calculate rotation of a coordinate in 2D

Seriously written for mathematicians, so don’t read https://en.wikipedia.org/wiki/Rotation_(mathematics). Pff… Okay, it boils down to this:

Given is coordinate pair x,y and angle α. Get x',y' by using:

x'=x*cos(α)-y*sin(α)
y'=y*cos(α)+x*sin(α)

In order not to get unexpected results, angle α is positive when counter-clockwise – in this example it is a positive value.

For calculating a rotation in 3D you can consider two rotations, one in the xy-plane and one in the yz-plane.

Translations of coordinates

This one is easy, translations like command Move is just adding static values to x, y and z values.

Programming solutions

The purpose of this code is educational. It works and aims to show the structure. Unless you experience slow processing, it should be sufficient.

In order to avoid interference with functions from third parties, functions below are prefixed with “lv:” as in Lib Vector.

acos

Calculate arccosine in AutoCAD. BricsCAD has a native function acos.

Returns the arccosine of a number in radians. This code can be used for (acos ...):

(acos
  num1
)

Arguments

Num1 is an integer or real.

Return Values

The arccosine of num1, in radians.

Examples

(acos (/ pi 4))
0.667457216028384

asin

Calculate arcsine in AutoCAD. BricsCAD has a native function asin.

Returns the arcsine of a number in radians. This code can be used for (asin ...):

(asin
  num1
)

Arguments

Num1 is an integer or real.

Return Values

The arcsine of num1, in radians.

Examples

(asin (/ pi 4))
0.903339110766513

lv:dist3d

3D distance between point 1 and 2.

In Lisp there is a function (distance a b) but it is tempting to use your own function because (distance a b) treats all points 2D when it encounters one 2D point. An alternative function (lv:dist3d point1 point2) works always 3D and calculates “the square root of the sum of the squares of the delta x, y and z values”.

With the distance between point1 and point2, point1 is x,y,z and point2 is x',y',z', the formula in spreadsheet notation is:
dist3d=sqrt(((x'-x)^2)+((y'-y)^2)+((z'-z)^2))

(lv:dist3d
  point1 point2
)

Arguments

Point1 and point2 are lists containing x,y,z coordinates.

Return Values

The 3D distance between point1 and point2 as a real.

Examples

:(setq pt2 (list 4 0 0))
(4 0 0)
: (setq pt1 (list 0 3 0))
(0 3 0)
: (lv:dist3d pt1 pt2)
5.0

lv:vector

Vector between point 1 and point 2

For the vector coordinates, point 2 is subtracted from point 1.

(lv:dist3d
  point1 point2
)

Arguments

Point1 and point2 are lists containing x,y,z coordinates.

Return Values

Resulting vector as a list of reals and or integers.

Examples

: (setq point1 (list 5.0 0.0 0.0))
(5.0 0.0 0.0)
: (setq point2 (list 0.0 3.0 0.0))
(0.0 3.0 0.0)
: (lv:vector point1 point2)
(-5.0 3.0 0.0)

lv:cross-product

Cross product of vector1 and vector2

Take care of the order of vector1 and vector2, it affects the sign of the coordinates.

(lv:cross-product
  vector1 vector2
)

Arguments

Vector1 and vector2 are lists containing x,y,z coordinates.

Return Values

The cross product as a list of reals and or integers.

Examples

: (setq v1 (list 0.0 3 0))
(0.0 3 0)
: (setq v2 (list 4 0 0))
(4 0 0)
: (lv:cross-product v1 v2)
(0 0.0 -12.0)

lv:dot-product

Dot product of vector1 and vector2

(lv:dot-product
  vector1 vector2
)

Arguments

Vector1 and vector2 are lists containing x,y,z coordinates.

Return Values

The dot product as a real or integer.

Examples

: (setq v1 (list 1 3 0))
(1 3 0)
: (setq v2 (list 4 0 0))
(4 0 0)
: (lv:dot-product v1 v2)
4

lv:magnitude

Magnitude of vector vc

(lv:magnitude
  vc
)

Arguments

Vector is a list containing x,y,z coordinates.

Return Values

The magnitude, vector length, as a real.

Examples

: (setq vc (list 0.0 4.0 3.0))
(0.0 4.0 3.0)
: (lv:magnitude vc)
5.0

lv:unit-vector

Unit vector of a vector vc.

(lv:unit-vector
  vc
)

Arguments

Vector is a list containing x,y,z coordinates.

Return Values

The unit vector of vector vc as a list of reals.

Examples

: (setq vc (list 4 0 0))
(4 0 0)
: (lv:unit-vector vc)
(1.0 0.0 0.0)

lv:acute

Check if angle between vector a and b is acute, according to right hand rule.

lv:arop

Determine if angle between vector a and b is acute, right or obtuse (aro).

Function lv:arop returns string “a”, “r” and “o” for resp. an acute, right or obtuse angle between vector a and b.

(lv:arop
  vector1 vector2
)

Arguments

Vector1 and vector2 are lists containing x,y,z coordinates.

Return Values

Function lv:arop returns string “a”, “r” and “o” for resp. an acute, right or obtuse angle between vector a and b.

Examples

: (setq vector1 (list 1.0 0.0 0.0))
(1.0 0.0 0.0)
: (setq vector2 (list 0.0 1.0 0.0))
(0.0 1.0 0.0)
: (lv:arop vector1 vector2)
"r"

lv:vector-ang

Angle between vector1 and vector2.

Calculation is based on right hand rule and function acos.

(lv:vector-ang
  vector1 vector2
)

Arguments

Vector1 and vector2 are lists containing x,y,z coordinates.

Return Values

Angle in radians as a real.

Examples

: (setq vector1 (list 1.0 0.0 0.0))
(1.0 0.0 0.0)
: (setq vector2 (list 0.0 1.0 0.0))
(0.0 1.0 0.0)
: (/ (lv:vector-ang vector1 vector2) pi 0.5)
1.0

lv:rot-pt

Rotation of point pt over angle ang

Angle ang in radians, sign is positive when counter clock wise. Point pt: only x and y are evaluated.

(lv:rot-pt
  pt ang
)

Arguments

Point pt is a list containing x,y or x,y,z coordinates and Angle ang is a real or integer.

Return Values

A list containing resulting x,y coordinates

Examples

: (setq pt (list 6.5 0.0 0.0))
(6.5 0.0 0.0)
: (setq ang pi)
3.14159265358979
: (lv:rot-pt pt ang)
(-6.5 0.0)

Lisp code as one file

The following can be put in a file “lib-vector.lsp” and can be loaded by dragging or proper loading in BricsCAD (and AutoCAD).

;; Vector mathematic functions, (c) 2023 NedCAD
;; Functions: lv:function-name, lv: Library Vector.
 
(setq lib-vector T) ; for testing, (if (not lib-vector) (load ...))
 
;; Adding functionality for AutoCAD: function acos and asin
(if (not acos)(defun acos (r / ) (atan (sqrt (- 1 (expt r 2))) r)))
(if (not asin)(defun asin (r / ) (atan r (sqrt (- 1 (expt r 2))))))
 
(defun lv:dist3d (a b / )
  (sqrt (apply '+ (mapcar '(lambda (k l) (expt (- l k) 2)) a b)))
)
 
(defun lv:vector (a b / )
  (mapcar '(lambda (k l) (- l k)) a b)
)
 
(defun lv:cross-product (a b / lsa lsb lsc lsd xa xb ya yb za zb)
  (setq
    lsa (list (setq ya (cadr a)) (setq za (caddr a)) (setq xa (car a)))
    lsb (list (setq zb (caddr b)) (setq xb (car b)) (setq yb (cadr b)))
    lsc (list yb zb xb) lsd (list za xa ya)
  )
  (mapcar '(lambda (o p q r) (- (* o p) (* q r))) lsa lsb lsc lsd)
)
 
(defun lv:dot-product (a b / )
  (apply '+ (mapcar '(lambda (k l) (* k l)) a b))
)
 
(defun lv:magnitude (a / )
  (sqrt (apply '+ (mapcar '(lambda (b) (expt b 2.0)) a)))
)
 
(defun lv:unit-vector (a / ma)
  (setq ma (lv:magnitude a))
  (mapcar '(lambda (b) (/ b ma)) a)
)
 
(defun lv:acute (a b / )
   (if (> (lv:dot-product a b) 0.0) T)
)
 
(defun lv:arop (a b / dp)
  (setq dp (lv:dot-product a b))
  (cond ((> dp 0.0) "a") ((< dp 0.0) "o") ("r"))
)
 
(defun lv:vector-ang (a b / )
   (acos (/ (lv:dot-product a b) (* (lv:magnitude a) (lv:magnitude b))))
)
 
(defun lv:rot-pt (a g / sg cg xa ya)
   (setq sg (sin g) cg (cos g) xa (car a) ya (cadr a))
   (list (- (* xa cg) (* ya sg)) (+ (* ya cg) (* xa sg)))
)
 
(princ)

Definitions

Mathematical solutions

Calculate the magnitude of a vector

Calculate the dot product of two vectors

Calculate the cross product of two vectors

Calculate the unit vector of a vector

Calculate the angle between two vectors

Using arccosine

Using arctangent

Is a vector angle obtuse, perpendicular or acute?

Calculate the normal of a face

Calculate 2D projection of a face – Pythagorean

Summary

Rationale

Calculate 2D projection of an adjacent face

Calculate rotation of a coordinate in 2D

Translations of coordinates

Programming solutions

acos

asin

lv:dist3d

lv:vector

lv:cross-product

lv:dot-product

lv:magnitude

lv:unit-vector

lv:acute

lv:arop

lv:vector-ang

lv:rot-pt

Lisp code as one file

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