OpenSCAD User Manual/Mathematical Functions

Trigonometric functions
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
Mathematical cosine function of degrees. See Cosine

Parameters


 * &lt;degrees&gt; : Decimal. Angle in degrees.

sin
Mathematical sine function. See Sine

Parameters


 * &lt;degrees&gt; : Decimal. Angle in degrees.

tan
Mathematical tangent function. See Tangent

Parameters


 * &lt;degrees&gt; : Decimal. Angle in degrees.

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

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

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

atan2
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

abs
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
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
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
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
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
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
Mathematical natural logarithm. See: Natural logarithm

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

lookup
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
 * &lt;key,value&gt; 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.

max
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)


 *  : Two or more decimals
 * : Single vector of decimals.

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
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)


 *  : Two or more decimals
 * : Single vector of decimals.

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

norm
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
Mathematical power function.

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

Parameters


 * &lt;base&gt; : Decimal. Base.
 * &lt;exponent&gt; : 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
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
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
Mathematical signum function. Returns a unit value that extracts the sign of a value see: Signum function

Parameters
 * &lt;x&gt; : 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
Mathematical square root function.

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

Infinities and NaNs
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.