OpenSCAD User Manual/Mathematical Functions

__FORCETOC__ Trigonometric functions cos Mathematical cosine function of degrees. See Cosine Parameters <degrees> Number. Angle in degrees. Usage example: for(i=[0:36]) translate([i*10,0,0]) cylinder(r=5,h=cos(i*10)*50+60);thumb|OpenSCAD Cos Function‎ sin Mathematical sine function. See Sine Parameters <degrees> Number. 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);thumb|OpenSCAD Sin Function tan Mathematical tangent function. See Tangent Parameters <degrees> Number. 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 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. atan 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 between the x axis and the vector(x,y) expressed in degrees, in the range . 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 abs Mathematical absolute value function. Returns the value, with its sign changed to positive. 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 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 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 2×2 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 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 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 <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); } thumb|OpenSCAD Lookup Function 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) <n> Two or more numbers <vector> Single vector of numbers 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 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 numbers <vector> Single vector of numbers 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 Does not exist as a function; 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: This returns the actual numeric length while len() returns the number of elements in the vector or array. Usage examples: a=[1,2,3,4,5,6]; b="abcd"; c=[]; d=""; e=[[1,2,3,4],[1,2,3],[1,2],[1]]; echo(norm(a)); //9.53939 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: 9.53939 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 <base> Number. Base. <exponent> Number. 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]. The random numbers generated are a "half open interval"; each is greater than or equal to the minimum, and less than the maximum. 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] // Get a vector of integers between 1 and 10 inclusive. // Note that rands(1,10,...) only spans 9 numbers and so it is difficult to get it to yield equal // probabilities for 1..10 inclusive. We widen the range by 1 so that we have the right number // of intervals. function irands(minimum, maximum, n) = let(floats = rands(minimum, maximum+1, n)) [ for (f = floats) floor(f) ]; echo(irands(1, 10, 5)); // ECHO: [9, 6, 2, 4, 1] 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 <x> Number. 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. Usage example translate([sqrt(100),0,0])sphere(100); 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. 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