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OpenSCAD User Manual/Mathematical Functions

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Trigonometric functions

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The trig functions use the C Language mathematics functions, which are based in turn on Binary Floating Point mathematics, which use approximations of Real Numbers during calculation. OpenSCAD's math functions use the C++ 'double' type, inside Value.h/Value.cc,

A good resource for the specifics of the C library math functions, such as valid inputs/output ranges, can be found at the Open Group website math.h & acos

cos

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Mathematical cosine function of degrees. See Cosine

Parameters

<degrees>
Decimal. Angle in degrees.
Usage example: 
 for(i=[0:36])
    translate([i*10,0,0])
       cylinder(r=5,h=cos(i*10)*50+60);
 
OpenSCAD Cos Function‎

sin

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Mathematical sine function. See Sine

Parameters

<degrees>
Decimal. Angle in degrees.
Usage example 1: 
 for (i = [0:5]) {
  echo(360*i/6, sin(360*i/6)*80, cos(360*i/6)*80);
   translate([sin(360*i/6)*80, cos(360*i/6)*80, 0 ])
    cylinder(h = 200, r=10);
 }
Usage example 2: 
 for(i=[0:36])
    translate([i*10,0,0])
       cylinder(r=5,h=sin(i*10)*50+60);
 
OpenSCAD Sin Function

tan

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Mathematical tangent function. See Tangent

Parameters

<degrees>
Decimal. Angle in degrees.
Usage example: 
 for (i = [0:5]) {
  echo(360*i/6, tan(360*i/6)*80);
   translate([tan(360*i/6)*80, 0, 0 ])
    cylinder(h = 200, r=10);
 }

acos

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Mathematical arccosine, or inverse cosine, expressed in degrees. See: Inverse trigonometric functions

asin

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Mathematical arcsine, or inverse sine, expressed in degrees. See: Inverse trigonometric functions

atan

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Mathematical arctangent, or inverse tangent, function. Returns the principal value of the arc tangent of x, expressed in degrees. atan cannot distinguish between y/x and -y/-x and returns angles from -90 to +90. See: atan2 and also Inverse trigonometric functions

atan2

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Mathematical two-argument atan function atan2(y,x) that spans the full 360 degrees. This function returns the full angle (0-360) made between the x axis and the vector(x,y) expressed in degrees.

Usage examples: 

atan2(5.0,-5.0);     //result: 135 degrees. atan() would give -45
atan2(y,x);          //angle between (1,0) and (x,y) = angle around z-axis

Other Mathematical Functions

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abs

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Mathematical absolute value function. Returns the positive value of a signed decimal number.

Usage examples: 

abs(-5.0);  returns 5.0
abs(0);     returns 0.0
abs(8.0);   returns 8.0

ceil

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Mathematical ceiling function.

Returns the next highest integer value by rounding up value if necessary.

See: Ceil Function 

echo(ceil(4.4),ceil(-4.4));     // produces ECHO: 5, -4

concat

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[Note: Requires version 2015.03]

Return a new vector that is the result of appending the elements of the supplied vectors.

Where an argument is a vector the elements of the vector are individually appended to the result vector. Strings are distinct from vectors in this case.

Usage examples: 

echo(concat("a","b","c","d","e","f"));          // produces ECHO: ["a", "b", "c", "d", "e", "f"]
echo(concat(["a","b","c"],["d","e","f"]));      // produces ECHO: ["a", "b", "c", "d", "e", "f"]
echo(concat(1,2,3,4,5,6));                      // produces ECHO: [1, 2, 3, 4, 5, 6]

Vector of vectors 

echo(concat([ [1],[2] ], [ [3] ]));             // produces ECHO: [[1], [2], [3]]

Note: All vectors passed to the function lose one nesting level. When adding something like a single element [x, y, z] tuples (which are vectors, too), the tuple needs to be enclosed in a vector (i.e. an extra set of brackets) before the concatenation. in the exmple below, a fourth point is added to the polygon path, which used to resemble a triangle, making it a square now: 

polygon(concat([[0,0],[0,5],[5,5]], [[5,0]]));

Contrast with strings 

echo(concat([1,2,3],[4,5,6]));                   // produces ECHO: [1, 2, 3, 4, 5, 6]
echo(concat("abc","def"));                       // produces ECHO: ["abc", "def"]
echo(str("abc","def"));                          // produces ECHO: "abcdef"

cross

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Calculates the cross product of two vectors in 3D or 2D space. If both vectors are in the 3D, the result is a vector that is perpendicular to both of the input vectors. If both vectors are in 2D space, their cross product has the form [0,0,z] and the cross function returns just the z value of the cross product:

cross([x,y], [u,v]) = x*v - y*u

Note that this is the determinant of the 2x2 matrix [[x,y],[u,v]]. Using any other types, vectors with lengths different from 2 or 3, or vectors not of the same length produces 'undef'.

Usage examples: 

echo(cross([2, 3, 4], [5, 6, 7]));     // produces ECHO: [-3, 6, -3]
echo(cross([2, 1, -3], [0, 4, 5]));    // produces ECHO: [17, -10, 8]
echo(cross([2, 1], [0, 4]));           // produces ECHO: 8
echo(cross([1, -3], [4, 5]));          // produces ECHO: 17
echo(cross([2, 1, -3], [4, 5]));       // produces ECHO: undef
echo(cross([2, 3, 4], "5"));           // produces ECHO: undef

For any two vectors a and b in 2D or in 3D, the following holds:

cross(a,b) == -cross(b,a)

exp

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Mathematical exp function. Returns the base-e exponential function of x, which is the number e raised to the power x. See: Exponent 

echo(exp(1),exp(ln(3)*4));    // produces ECHO: 2.71828, 81

floor

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Mathematical floor function. floor(x) = is the largest integer not greater than x

See: Floor Function 

echo(floor(4.4),floor(-4.4));    // produces ECHO: 4, -5

ln

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Mathematical natural logarithm. See: Natural logarithm

len

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Mathematical length function. Returns the length of an array, a vector or a string parameter.

Usage examples: 

str1="abcdef"; len_str1=len(str1);
echo(str1,len_str1);

a=6; len_a=len(a);
echo(a,len_a);

array1=[1,2,3,4,5,6,7,8]; len_array1=len(array1);
echo(array1,len_array1);

array2=[[0,0],[0,1],[1,0],[1,1]]; len_array2=len(array2);
echo(array2,len_array2);

len_array2_2=len(array2[2]);
echo(array2[2],len_array2_2);

Results: 

WARNING: len() parameter could not be converted in file , line 4
ECHO: "abcdef", 6
ECHO: 6, undef
ECHO: [1, 2, 3, 4, 5, 6, 7, 8], 8
ECHO: [[0, 0], [0, 1], [1, 0], [1, 1]], 4
ECHO: [1, 0], 2

This function allows (e.g.) the parsing of an array, a vector or a string.

Usage examples: 

str2="4711";
for (i=[0:len(str2)-1])
	echo(str("digit ",i+1,"  :  ",str2[i]));

Results: 

ECHO: "digit 1  :  4"
ECHO: "digit 2  :  7"
ECHO: "digit 3  :  1"
ECHO: "digit 4  :  1"

Note that the len() function is not defined and raises a warning when a simple variable is passed as the parameter.

This is useful when handling parameters to a module, similar to how shapes can be defined as a single number, or as an [x,y,z] vector; i.e. cube(5) or cube([5,5,5])

For example 

module doIt(size) {
	if (len(size) == undef) {
		// size is a number, use it for x,y & z. (or could be undef)
		do([size,size,size]);
	} else { 
		// size is a vector, (could be a string but that would be stupid)
		do(size);
	}
 }
 
doIt(5);	// equivalent to [5,5,5]
doIt([5,5,5]);	// similar to cube(5) v's cube([5,5,5])

let

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[Note: Requires version 2015.03]

Sequential assignment of variables inside an expression. The following expression is evaluated in context of the let assignments and can use the variables. This is mainly useful to make complicated expressions more readable by assigning interim results to variables.

Parameters 

let (var1 = value1, var2 = f(var1), var3 = g(var1, var2)) expression

Usage example: 

echo(let(a = 135, s = sin(a), c = cos(a)) [ s, c ]); // ECHO: [0.707107, -0.707107]

Let can also be used to create variables in a Function. (See also: "Let Statement")

log

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Mathematical logarithm to the base 10. Example: log(1000) = 3. See: Logarithm

lookup

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Look up value in table, and linearly interpolate if there's no exact match. The first argument is the value to look up. The second is the lookup table -- a vector of key-value pairs.

Parameters

key
A lookup key
<key,value> array
keys and values

There is a bug in which out-of-range keys return the first value in the list. Newer versions of Openscad should use the top or bottom end of the table as appropriate instead.

Usage example: Create a 3D chart made from cylinders of different heights. 

 function get_cylinder_h(p) = lookup(p, [
 		[ -200, 5 ],
 		[ -50, 20 ],
 		[ -20, 18 ],
 		[ +80, 25 ],
 		[ +150, 2 ]
 	]);
 
 for (i = [-100:5:+100]) {
 	// echo(i, get_cylinder_h(i));
 	translate([ i, 0, -30 ]) cylinder(r1 = 6, r2 = 2, h = get_cylinder_h(i)*3);
 }
 
OpenSCAD Lookup Function

max

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Returns the maximum of the parameters. If a single vector is given as parameter, returns the maximum element of that vector.

Parameters 

max(n,n{,n}...)
max(vector)
<n>
Two or more decimals
<vector>
Single vector of decimals [Note: Requires version 2014.06].

Usage example: 

max(3.0,5.0)
max(8.0,3.0,4.0,5.0)
max([8,3,4,5])

Results: 

5
8
8

min

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Returns the minimum of the parameters. If a single vector is given as parameter, returns the minimum element of that vector.

Parameters 

min(n,n{,n}...)
min(vector)
<n>
Two or more decimals
<vector>
Single vector of decimals [Note: Requires version 2014.06].

Usage example: 

min(3.0,5.0)
min(8.0,3.0,4.0,5.0)
min([8,3,4,5])

Results: 

3
3
3

mod

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Included in this document only for clarity. The 'modulo' operation exists in OpenSCAD as an operator %, and not as function. See modulo operator (%)

norm

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Returns the euclidean norm of a vector. Note this returns the actual numeric length while len returns the number of elements in the vector or array.

Usage examples: 

a=[1,2,3,4];
b="abcd";
c=[];
d="";
e=[[1,2,3,4],[1,2,3],[1,2],[1]];
echo(norm(a)); //5.47723
echo(norm(b)); //undef
echo(norm(c)); //0
echo(norm(d)); //undef
echo(norm(e[0])); //5.47723
echo(norm(e[1])); //3.74166
echo(norm(e[2])); //2.23607
echo(norm(e[3])); //1

Results: 

ECHO: 5.47723
ECHO: undef
ECHO: 0
ECHO: undef
ECHO: 5.47723
ECHO: 3.74166
ECHO: 2.23607
ECHO: 1

pow

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Mathematical power function.

As of version 2021.01 you can use the exponentiation operator ^ instead.

Parameters

<base>
Decimal. Base.
<exponent>
Decimal. Exponent.

Usage examples: 

for (i = [0:5]) {
 translate([i*25,0,0]) {
   cylinder(h = pow(2,i)*5, r=10);
   echo (i, pow(2,i));
 }
}

echo(pow(10,2)); // means 10^2 or 10*10
// result: ECHO: 100

echo(pow(10,3)); // means 10^3 or 10*10*10
// result: ECHO: 1000

echo(pow(125,1/3)); // means 125^(0.333...), which calculates the cube root of 125
// result: ECHO: 5

rands

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Random number generator. Generates a constant vector of pseudo random numbers, much like an array. The numbers are doubles not integers. When generating only one number, you still call it with variable[0].

Parameters

min_value
Minimum value of random number range
max_value
Maximum value of random number range
value_count
Number of random numbers to return as a vector
seed_value (optional)
Seed value for random number generator for repeatable results. On versions before late 2015, seed_value gets rounded to the nearest integer

Usage examples: 

// get a single number
single_rand = rands(0,10,1)[0];
echo(single_rand);

// get a vector of 4 numbers
seed=42;
random_vect=rands(5,15,4,seed);
echo( "Random Vector: ",random_vect);
sphere(r=5);
for(i=[0:3]) {
 rotate(360*i/4) {
   translate([10+random_vect[i],0,0])
     sphere(r=random_vect[i]/2);
 }
}
// ECHO: "Random Vector: ", [8.7454, 12.9654, 14.5071, 6.83435]

round

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The "round" operator returns the greatest or least integer part, respectively, if the numeric input is positive or negative.

Usage examples: 

round(5.4);
round(5.5);
round(5.6);
round(-5.4);
round(-5.5);
round(-5.6);

Results: 

5
6
6
-5
-6
-6

sign

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Mathematical signum function. Returns a unit value that extracts the sign of a value see: Signum function

Parameters

<x>
Decimal. Value to find the sign of.

Usage examples: 

sign(-5.0);
sign(0);
sign(8.0);

Results: 

-1.0
0.0
1.0

sqrt

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Mathematical square root function.

Usage example
translate([sqrt(100),0,0])sphere(100);

Infinities and NaNs

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How does OpenSCAD deal with inputs like (1/0)? Basically, the behavior is inherited from the language OpenSCAD was written in, the C++ language, and its floating point number types and the associated C math library. This system allows representation of both positive and negative infinity by the special values "Inf" or "-Inf". It also allow representation of creatures like sqrt(-1) or 0/0 as "NaN", an abbreviation for "Not A Number". Explanations can be found on the web, for example the Open Group's site on math.h or Wikipedia's page on the IEEE 754 number format. However, OpenSCAD is its own language so it may not exactly match everything that happens in C. For example, OpenSCAD uses degrees instead of radians for trigonometric functions. Another example is that sin() does not throw a "domain error" when the input is 1/0, although it does return NaN.

Here are some examples of infinite input to OpenSCAD math functions and the resulting output, taken from OpenSCAD's regression test system in late 2015.

0/0: nan sin(1/0): nan asin(1/0): nan ln(1/0): inf round(1/0): inf
-0/0: nan cos(1/0): nan acos(1/0): nan ln(-1/0): nan round(-1/0): -inf
0/-0: nan tan(1/0): nan atan(1/0): 90 log(1/0): inf sign(1/0): 1
1/0: inf ceil(-1/0): -inf atan(-1/0): -90 log(-1/0): nan sign(-1/0): -1
1/-0: -inf ceil(1/0): inf atan2(1/0, -1/0): 135 max(-1/0, 1/0): inf sqrt(1/0): inf
-1/0: -inf floor(-1/0): -inf exp(1/0): inf min(-1/0, 1/0): -inf sqrt(-1/0): nan
-1/-0: inf floor(1/0): inf exp(-1/0): 0 pow(2, 1/0): inf pow(2, -1/0): 0
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