Assorted math functions, including linear interpolation, list operations (sums, mean, products), convolution, quantization, log2, hyperbolic trig functions, random numbers, derivatives, polynomials, and root finding.

To use, add the following lines to the beginning of your file:

include <BOSL2/std.scad>

1. Section: Math Constants

   • PHI – The golden ratio φ (phi). Approximately 1.6180339887
   • EPSILON – A tiny value to compare floating point values. 1e-9
   • INF – The floating point value for Infinite.
   • NAN – The floating point value for Not a Number.

2. Section: Interpolation and Counting

   • count() – Creates a list of incrementing numbers.
   • lerp() – Linearly interpolates between two values.
   • lerpn() – Returns exactly n values, linearly interpolated between a and b.

3. Section: Miscellaneous Functions

   • sqr() – Returns the square of the given value.
   • log2() – Returns the log base 2 of the given value.
   • hypot() – Returns the hypotenuse length of a 2D or 3D triangle.
   • factorial() – Returns the factorial of the given integer.
   • binomial() – Returns the binomial coefficients of the integer n.
   • binomial_coefficient() – Returns the k-th binomial coefficient of the integer n.
   • gcd() – Returns the Greatest Common Divisor/Factor of two integers.
   • lcm() – Returns the Least Common Multiple of two or more integers.

4. Section: Hyperbolic Trigonometry

   • sinh() – Returns the hyperbolic sine of the given value.
   • cosh() – Returns the hyperbolic cosine of the given value.
   • tanh() – Returns the hyperbolic tangent of the given value.
   • asinh() – Returns the hyperbolic arc-sine of the given value.
   • acosh() – Returns the hyperbolic arc-cosine of the given value.
   • atanh() – Returns the hyperbolic arc-tangent of the given value.

5. Section: Quantization

   • quant() – Returns x quantized to the nearest integer multiple of y.
   • quantdn() – Returns x quantized down to an integer multiple of y.
   • quantup() – Returns x quantized uo to an integer multiple of y.

6. Section: Constraints and Modulos

   • constrain() – Returns a value constrained between minval and maxval, inclusive.
   • posmod() – Returns the positive modulo of a value.
   • modang() – Returns an angle normalized to between -180º and 180º.

7. Section: Operations on Lists (Sums, Mean, Products)

   • sum() – Returns the sum of a list of values.
   • mean() – Returns the mean value of a list of values.
   • median() – Returns the median value of a list of values.
   • deltas() – Returns the deltas between a list of values.
   • cumsum() – Returns the running cumulative sum of a list of values.
   • product() – Returns the multiplicative product of a list of values.
   • cumprod() – Returns the running cumulative product of a list of values.
   • convolve() – Returns the convolution of p and q.
   • sum_of_sines() – Returns the sum of one or more sine waves at a given angle.

8. Section: Random Number Generation

   • rand_int() – Returns a random integer.
   • random_points() – Returns a list of random points.
   • gaussian_rands() – Returns a list of random numbers with a gaussian distribution.
   • exponential_rands() – Returns a list of random numbers with an exponential distribution.
   • spherical_random_points() – Returns a list of random points on the surface of a sphere.
   • random_polygon() – Returns the CCW path of a simple random polygon.

9. Section: Calculus

   • deriv() – Returns the first derivative estimate of a list of data.
   • deriv2() – Returns the second derivative estimate of a list of data.
   • deriv3() – Returns the third derivative estimate of a list of data.

10. Section: Complex Numbers

    • complex() – Replaces scalars in a list or matrix with complex number 2-vectors.
    • c_mul() – Multiplies two complex numbers.
    • c_div() – Divides two complex numbers.
    • c_conj() – Returns the complex conjugate of the input.
    • c_real() – Returns the real part of a complex number, vector or matrix..
    • c_imag() – Returns the imaginary part of a complex number, vector or matrix..
    • c_ident() – Returns an n by n complex identity matrix.
    • c_norm() – Returns the norm of a complex number or vector.

11. Section: Polynomials

    • quadratic_roots() – Computes roots for the quadratic equation.
    • polynomial() – Calculates a polynomial equation at a given value.
    • poly_mult() – Returns the polynomial result of multiplying two polynomial equations.
    • poly_div() – Returns the polynomial quotient and remainder results of dividing two polynomial equations.
    • poly_add() – Returns the polynomial sum of adding two polynomial equations.
    • poly_roots() – Returns all complex number roots of the given real polynomial.
    • real_roots() – Returns all real roots of the given real polynomial.

12. Section: Operations on Functions

    • root_find() – Finds a root of the given continuous function.

Synopsis: The golden ratio φ (phi). Approximately 1.6180339887

Topics: Constants, Math

See Also: EPSILON, INF, NAN

Description:

The golden ratio φ (phi). Approximately 1.6180339887

Synopsis: A tiny value to compare floating point values. 1e-9

Topics: Constants, Math

See Also: PHI, INF, NAN

Description:

A really small value useful in comparing floating point numbers. ie: abs(a-b)<EPSILON 1e-9

Synopsis: The floating point value for Infinite.

Topics: Constants, Math

See Also: PHI, EPSILON, NAN

Description:

The value inf, useful for comparisons.

Synopsis: The floating point value for Not a Number.

Topics: Constants, Math

See Also: PHI, EPSILON, INF

Description:

The value nan, useful for comparisons.

Synopsis: Creates a list of incrementing numbers.

Topics: Math, Indexing

See Also: idx()

Usage:

• list = count(n, [s], [step], [reverse]);

Description:

Creates a list of n numbers, starting at s, incrementing by step each time. You can also pass a list for n and then the length of the input list is used.

Arguments:

Example 1:

include <BOSL2/std.scad>
nl1 = count(5);  // Returns: [0,1,2,3,4]
nl2 = count(5,3);  // Returns: [3,4,5,6,7]
nl3 = count(4,3,2);  // Returns: [3,5,7,9]
nl4 = count(5,reverse=true);    // Returns: [4,3,2,1,0]
nl5 = count(5,3,reverse=true);  // Returns: [7,6,5,4,3]

Synopsis: Linearly interpolates between two values.

Topics: Interpolation, Math

See Also: v_lookup(), lerpn()

Usage:

• x = lerp(a, b, u);
• l = lerp(a, b, LIST);

Description:

Interpolate between two values or vectors. If u is given as a number, returns the single interpolated value. If u is 0.0, then the value of a is returned. If u is 1.0, then the value of b is returned. If u is a range, or list of numbers, returns a list of interpolated values. It is valid to use a u value outside the range 0 to 1. The result will be an extrapolation along the slope formed by a and b.

Arguments:

Example 1:

include <BOSL2/std.scad>
x = lerp(0,20,0.3);  // Returns: 6
x = lerp(0,20,0.8);  // Returns: 16
x = lerp(0,20,-0.1); // Returns: -2
x = lerp(0,20,1.1);  // Returns: 22
p = lerp([0,0],[20,10],0.25);  // Returns [5,2.5]
l = lerp(0,20,[0.4,0.6]);  // Returns: [8,12]
l = lerp(0,20,[0.25:0.25:0.75]);  // Returns: [5,10,15]

Example 2:

include <BOSL2/std.scad>
p1 = [-50,-20];  p2 = [50,30];
stroke([p1,p2]);
pts = lerp(p1, p2, [0:1/8:1]);
// Points colored in ROYGBIV order.
rainbow(pts) translate($item) circle(d=3,$fn=8);

Synopsis: Returns exactly n values, linearly interpolated between a and b.

Topics: Interpolation, Math

See Also: v_lookup(), lerp()

Usage:

• x = lerpn(a, b, n);
• x = lerpn(a, b, n, [endpoint]);

Description:

Returns exactly n values, linearly interpolated between a and b. If endpoint is true, then the last value will exactly equal b. If endpoint is false, then the last value will be a+(b-a)*(1-1/n).

Arguments:

Example 1:

include <BOSL2/std.scad>
l = lerpn(-4,4,9);        // Returns: [-4,-3,-2,-1,0,1,2,3,4]
l = lerpn(-4,4,8,false);  // Returns: [-4,-3,-2,-1,0,1,2,3]
l = lerpn(0,1,6);         // Returns: [0, 0.2, 0.4, 0.6, 0.8, 1]
l = lerpn(0,1,5,false);   // Returns: [0, 0.2, 0.4, 0.6, 0.8]

Synopsis: Returns the square of the given value.

Topics: Math

See Also: hypot(), log2()

Usage:

• x2 = sqr(x);

Description:

If given a number, returns the square of that number, If given a vector, returns the sum-of-squares/dot product of the vector elements. If given a matrix, returns the matrix multiplication of the matrix with itself.

Example 1:

include <BOSL2/std.scad>
sqr(3);     // Returns: 9
sqr(-4);    // Returns: 16
sqr([2,3,4]); // Returns: 29
sqr([[1,2],[3,4]]);  // Returns [[7,10],[15,22]]

Synopsis: Returns the log base 2 of the given value.

Topics: Math

See Also: hypot(), sqr()

Usage:

• val = log2(x);

Description:

Returns the logarithm base 2 of the value given.

Example 1:

include <BOSL2/std.scad>
log2(0.125);  // Returns: -3
log2(16);     // Returns: 4
log2(256);    // Returns: 8

Synopsis: Returns the hypotenuse length of a 2D or 3D triangle.

Topics: Math

See Also: sqr(), log2()

Usage:

• l = hypot(x, y, [z]);

Description:

Calculate hypotenuse length of a 2D or 3D triangle.

Arguments:

Example 1:

include <BOSL2/std.scad>
l = hypot(3,4);  // Returns: 5
l = hypot(3,4,5);  // Returns: ~7.0710678119

Synopsis: Returns the factorial of the given integer.

Topics: Math

See Also: hypot(), sqr(), log2()

Usage:

• x = factorial(n, [d]);

Description:

Returns the factorial of the given integer value, or n!/d! if d is given.

Arguments:

Example 1:

include <BOSL2/std.scad>
x = factorial(4);  // Returns: 24
y = factorial(6);  // Returns: 720
z = factorial(9);  // Returns: 362880

Synopsis: Returns the binomial coefficients of the integer n.

Topics: Math

See Also: hypot(), sqr(), log2(), factorial()

Usage:

• x = binomial(n);

Description:

Returns the binomial coefficients of the integer n.

Arguments:

Example 1:

include <BOSL2/std.scad>
x = binomial(3);  // Returns: [1,3,3,1]
y = binomial(4);  // Returns: [1,4,6,4,1]
z = binomial(6);  // Returns: [1,6,15,20,15,6,1]

Synopsis: Returns the k-th binomial coefficient of the integer n.

Topics: Math

See Also: hypot(), sqr(), log2(), factorial()

Usage:

• x = binomial_coefficient(n, k);

Description:

Returns the k-th binomial coefficient of the integer n.

Arguments:

Example 1:

include <BOSL2/std.scad>
x = binomial_coefficient(3,2);  // Returns: 3
y = binomial_coefficient(10,6); // Returns: 210

Synopsis: Returns the Greatest Common Divisor/Factor of two integers.

Topics: Math

See Also: hypot(), sqr(), log2(), factorial(), binomial(), lcm()

Usage:

• x = gcd(a,b)

Description:

Computes the Greatest Common Divisor/Factor of a and b.

Synopsis: Returns the Least Common Multiple of two or more integers.

Topics: Math

See Also: hypot(), sqr(), log2(), factorial(), binomial(), gcd()

Usage:

• div = lcm(a, b);
• divs = lcm(list);

Description:

Computes the Least Common Multiple of the two arguments or a list of arguments. Inputs should be non-zero integers. The output is always a positive integer. It is an error to pass zero as an argument.

Synopsis: Returns the hyperbolic sine of the given value.

Topics: Math, Trigonometry

See Also: cosh(), tanh(), asinh(), acosh(), atanh()

Usage:

• a = sinh(x);

Description:

Takes a value x, and returns the hyperbolic sine of it.

Synopsis: Returns the hyperbolic cosine of the given value.

Topics: Math, Trigonometry

See Also: sinh(), tanh(), asinh(), acosh(), atanh()

Usage:

• a = cosh(x);

Description:

Takes a value x, and returns the hyperbolic cosine of it.

Synopsis: Returns the hyperbolic tangent of the given value.

Topics: Math, Trigonometry

See Also: sinh(), cosh(), asinh(), acosh(), atanh()

Usage:

• a = tanh(x);

Description:

Takes a value x, and returns the hyperbolic tangent of it.

Synopsis: Returns the hyperbolic arc-sine of the given value.

Topics: Math, Trigonometry

See Also: sinh(), cosh(), tanh(), acosh(), atanh()

Usage:

• a = asinh(x);

Description:

Takes a value x, and returns the inverse hyperbolic sine of it.

Synopsis: Returns the hyperbolic arc-cosine of the given value.

Topics: Math, Trigonometry

See Also: sinh(), cosh(), tanh(), asinh(), atanh()

Usage:

• a = acosh(x);

Description:

Takes a value x, and returns the inverse hyperbolic cosine of it.

Synopsis: Returns the hyperbolic arc-tangent of the given value.

Topics: Math, Trigonometry

See Also: sinh(), cosh(), tanh(), asinh(), acosh()

Usage:

• a = atanh(x);

Description:

Takes a value x, and returns the inverse hyperbolic tangent of it.

Synopsis: Returns x quantized to the nearest integer multiple of y.

Topics: Math, Quantization

See Also: quantdn(), quantup()

Usage:

• num = quant(x, y);

Description:

Quantize a value x to an integer multiple of y, rounding to the nearest multiple. The value of y does NOT have to be an integer. If x is a list, then every item in that list will be recursively quantized.

Arguments:

Example 1:

include <BOSL2/std.scad>
a = quant(12,4);    // Returns: 12
b = quant(13,4);    // Returns: 12
c = quant(13.1,4);  // Returns: 12
d = quant(14,4);    // Returns: 16
e = quant(14.1,4);  // Returns: 16
f = quant(15,4);    // Returns: 16
g = quant(16,4);    // Returns: 16
h = quant(9,3);     // Returns: 9
i = quant(10,3);    // Returns: 9
j = quant(10.4,3);  // Returns: 9
k = quant(10.5,3);  // Returns: 12
l = quant(11,3);    // Returns: 12
m = quant(12,3);    // Returns: 12
n = quant(11,2.5);  // Returns: 10
o = quant(12,2.5);  // Returns: 12.5
p = quant([12,13,13.1,14,14.1,15,16],4);  // Returns: [12,12,12,16,16,16,16]
q = quant([9,10,10.4,10.5,11,12],3);      // Returns: [9,9,9,12,12,12]
r = quant([[9,10,10.4],[10.5,11,12]],3);  // Returns: [[9,9,9],[12,12,12]]

Synopsis: Returns x quantized down to an integer multiple of y.

Topics: Math, Quantization

See Also: quant(), quantup()

Usage:

• num = quantdn(x, y);

Description:

Quantize a value x to an integer multiple of y, rounding down to the previous multiple. The value of y does NOT have to be an integer. If x is a list, then every item in that list will be recursively quantized down.

Arguments:

Example 1:

include <BOSL2/std.scad>
a = quantdn(12,4);    // Returns: 12
b = quantdn(13,4);    // Returns: 12
c = quantdn(13.1,4);  // Returns: 12
d = quantdn(14,4);    // Returns: 12
e = quantdn(14.1,4);  // Returns: 12
f = quantdn(15,4);    // Returns: 12
g = quantdn(16,4);    // Returns: 16
h = quantdn(9,3);     // Returns: 9
i = quantdn(10,3);    // Returns: 9
j = quantdn(10.4,3);  // Returns: 9
k = quantdn(10.5,3);  // Returns: 9
l = quantdn(11,3);    // Returns: 9
m = quantdn(12,3);    // Returns: 12
n = quantdn(11,2.5);  // Returns: 10
o = quantdn(12,2.5);  // Returns: 10
p = quantdn([12,13,13.1,14,14.1,15,16],4);  // Returns: [12,12,12,12,12,12,16]
q = quantdn([9,10,10.4,10.5,11,12],3);      // Returns: [9,9,9,9,9,12]
r = quantdn([[9,10,10.4],[10.5,11,12]],3);  // Returns: [[9,9,9],[9,9,12]]

Synopsis: Returns x quantized uo to an integer multiple of y.

Topics: Math, Quantization

See Also: quant(), quantdn()

Usage:

• num = quantup(x, y);

Description:

Quantize a value x to an integer multiple of y, rounding up to the next multiple. The value of y does NOT have to be an integer. If x is a list, then every item in that list will be recursively quantized up.

Arguments:

Example 1:

include <BOSL2/std.scad>
a = quantup(12,4);    // Returns: 12
b = quantup(13,4);    // Returns: 16
c = quantup(13.1,4);  // Returns: 16
d = quantup(14,4);    // Returns: 16
e = quantup(14.1,4);  // Returns: 16
f = quantup(15,4);    // Returns: 16
g = quantup(16,4);    // Returns: 16
h = quantup(9,3);     // Returns: 9
i = quantup(10,3);    // Returns: 12
j = quantup(10.4,3);  // Returns: 12
k = quantup(10.5,3);  // Returns: 12
l = quantup(11,3);    // Returns: 12
m = quantup(12,3);    // Returns: 12
n = quantdn(11,2.5);  // Returns: 12.5
o = quantdn(12,2.5);  // Returns: 12.5
p = quantup([12,13,13.1,14,14.1,15,16],4);  // Returns: [12,16,16,16,16,16,16]
q = quantup([9,10,10.4,10.5,11,12],3);      // Returns: [9,12,12,12,12,12]
r = quantup([[9,10,10.4],[10.5,11,12]],3);  // Returns: [[9,12,12],[12,12,12]]

Synopsis: Returns a value constrained between minval and maxval, inclusive.

Topics: Math

See Also: posmod(), modang()

Usage:

• val = constrain(v, minval, maxval);

Description:

Constrains value to a range of values between minval and maxval, inclusive.

Arguments:

Example 1:

include <BOSL2/std.scad>
a = constrain(-5, -1, 1);   // Returns: -1
b = constrain(5, -1, 1);    // Returns: 1
c = constrain(0.3, -1, 1);  // Returns: 0.3
d = constrain(9.1, 0, 9);   // Returns: 9
e = constrain(-0.1, 0, 9);  // Returns: 0

Synopsis: Returns the positive modulo of a value.

Topics: Math

See Also: constrain(), modang()

Usage:

• mod = posmod(x, m)

Description:

Returns the positive modulo m of x. Value returned will be in the range 0 ... m-1.

Arguments:

Example 1:

include <BOSL2/std.scad>
a = posmod(-700,360);  // Returns: 340
b = posmod(-270,360);  // Returns: 90
c = posmod(-120,360);  // Returns: 240
d = posmod(120,360);   // Returns: 120
e = posmod(270,360);   // Returns: 270
f = posmod(700,360);   // Returns: 340
g = posmod(3,2.5);     // Returns: 0.5

Synopsis: Returns an angle normalized to between -180º and 180º.

Topics: Math

See Also: constrain(), posmod()

Usage:

• ang = modang(x);

Description:

Takes an angle in degrees and normalizes it to an equivalent angle value between -180 and 180.

Example 1:

include <BOSL2/std.scad>
a1 = modang(-700);  // Returns: 20
a2 = modang(-270);  // Returns: 90
a3 = modang(-120);  // Returns: -120
a4 = modang(120);   // Returns: 120
a5 = modang(270);   // Returns: -90
a6 = modang(700);   // Returns: -20

Synopsis: Returns the sum of a list of values.

Topics: Math

See Also: mean(), median(), product(), cumsum()

Usage:

• x = sum(v, [dflt]);

Description:

Returns the sum of all entries in the given consistent list. If passed an array of vectors, returns the sum the vectors. If passed an array of matrices, returns the sum of the matrices. If passed an empty list, the value of dflt will be returned.

Arguments:

Example 1:

include <BOSL2/std.scad>
sum([1,2,3]);  // returns 6.
sum([[1,2,3], [3,4,5], [5,6,7]]);  // returns [9, 12, 15]

Synopsis: Returns the mean value of a list of values.

Topics: Math, Statistics

See Also: sum(), median(), product()

Usage:

• x = mean(v);

Description:

Returns the arithmetic mean/average of all entries in the given array. If passed a list of vectors, returns a vector of the mean of each part.

Arguments:

Example 1:

include <BOSL2/std.scad>
mean([2,3,4]);  // returns 3.
mean([[1,2,3], [3,4,5], [5,6,7]]);  // returns [3, 4, 5]

Synopsis: Returns the median value of a list of values.

Topics: Math, Statistics

See Also: sum(), mean(), product()

Usage:

• middle = median(v)

Description:

Returns the median of the given vector.

Synopsis: Returns the deltas between a list of values.

Topics: Math, Statistics

See Also: sum(), mean(), median(), product()

Usage:

• delts = deltas(v,[wrap]);

Description:

Returns a list with the deltas of adjacent entries in the given list, optionally wrapping back to the front. The list should be a consistent list of numeric components (numbers, vectors, matrix, etc). Given [a,b,c,d], returns [b-a,c-b,d-c].

Arguments:

Example 1:

include <BOSL2/std.scad>
deltas([2,5,9,17]);  // returns [3,4,8].
deltas([[1,2,3], [3,6,8], [4,8,11]]);  // returns [[2,4,5], [1,2,3]]

Synopsis: Returns the running cumulative sum of a list of values.

Topics: Math, Statistics

See Also: sum(), mean(), median(), product()

Usage:

• sums = cumsum(v);

Description:

Returns a list where each item is the cumulative sum of all items up to and including the corresponding entry in the input list. If passed an array of vectors, returns a list of cumulative vectors sums.

Arguments:

Example 1:

include <BOSL2/std.scad>
cumsum([1,1,1]);  // returns [1,2,3]
cumsum([2,2,2]);  // returns [2,4,6]
cumsum([1,2,3]);  // returns [1,3,6]
cumsum([[1,2,3], [3,4,5], [5,6,7]]);  // returns [[1,2,3], [4,6,8], [9,12,15]]

Synopsis: Returns the multiplicative product of a list of values.

Topics: Math, Statistics

See Also: sum(), mean(), median(), cumsum()

Usage:

• x = product(v);

Description:

Returns the product of all entries in the given list. If passed a list of vectors of same dimension, returns a vector of products of each part. If passed a list of square matrices, returns the resulting product matrix.

Arguments:

Example 1:

include <BOSL2/std.scad>
product([2,3,4]);  // returns 24.
product([[1,2,3], [3,4,5], [5,6,7]]);  // returns [15, 48, 105]

Synopsis: Returns the running cumulative product of a list of values.

Topics: Math, Statistics

See Also: sum(), mean(), median(), product(), cumsum()

Usage:

• prod_list = cumprod(list, [right]);

Description:

Returns a list where each item is the cumulative product of all items up to and including the corresponding entry in the input list. If passed an array of vectors, returns a list of elementwise vector products. If passed a list of square matrices by default returns matrix products multiplying on the left, so a list [A,B,C] will produce the output [A,BA,CBA]. If you set right=true then it returns the product of multiplying on the right, so a list [A,B,C] will produce the output [A,AB,ABC] in that case.

Arguments:

Example 1:

include <BOSL2/std.scad>
cumprod([1,3,5]);  // returns [1,3,15]
cumprod([2,2,2]);  // returns [2,4,8]
cumprod([[1,2,3], [3,4,5], [5,6,7]]));  // returns [[1, 2, 3], [3, 8, 15], [15, 48, 105]]

Synopsis: Returns the convolution of p and q.

Topics: Math, Statistics

See Also: sum(), mean(), median(), product(), cumsum()

Usage:

• x = convolve(p,q);

Description:

Given two vectors, or one vector and a path or two paths of the same dimension, finds the convolution of them. If both parameter are vectors, returns the vector convolution. If one parameter is a vector and the other a path, convolves using products by scalars and returns a path. If both parameters are paths, convolve using scalar products and returns a vector. The returned vector or path has length len(p)+len(q)-1.

Arguments:

Example 1:

include <BOSL2/std.scad>
a = convolve([1,1],[1,2,1]); // Returns: [1,3,3,1]
b = convolve([1,2,3],[1,2,1])); // Returns: [1,4,8,8,3]
c = convolve([[1,1],[2,2],[3,1]],[1,2,1])); // Returns: [[1,1],[4,4],[8,6],[8,4],[3,1]]
d = convolve([[1,1],[2,2],[3,1]],[[1,2],[2,1]])); // Returns:  [3,9,11,7]

Synopsis: Returns the sum of one or more sine waves at a given angle.

Topics: Math, Statistics

See Also: sum(), mean(), median(), product(), cumsum()

Usage:

• sum_of_sines(a,sines)

Description:

Given a list of sine waves, returns the sum of the sines at the given angle. Each sine wave is given as an [AMPLITUDE, FREQUENCY, PHASE_ANGLE] triplet.

• AMPLITUDE is the height of the sine wave above (and below) 0.
• FREQUENCY is the number of times the sine wave repeats in 360º.
• PHASE_ANGLE is the offset in degrees of the sine wave.

Arguments:

Example 1:

include <BOSL2/std.scad>
v = sum_of_sines(30, [[10,3,0], [5,5.5,60]]);

Synopsis: Returns a random integer.

Topics: Random

See Also: random_points(), gaussian_rands(), random_polygon(), spherical_random_points(), exponential_rands()

Usage:

• rand_int(minval, maxval, n, [seed]);

Description:

Return a list of random integers in the range of minval to maxval, inclusive.

Arguments:

Example 1:

include <BOSL2/std.scad>
ints = rand_int(0,100,3);
int = rand_int(-10,10,1)[0];

Synopsis: Returns a list of random points.

Topics: Random, Points

See Also: rand_int(), random_polygon(), spherical_random_points()

Usage:

• points = random_points(n, dim, [scale], [seed]);

Description:

Generate n uniform random points of dimension dim with data ranging from -scale to +scale. The scale may be a number, in which case the random data lies in a cube, or a vector with dimension dim, in which case each dimension has its own scale.

Arguments:

Synopsis: Returns a list of random numbers with a gaussian distribution.

Topics: Random, Statistics

See Also: rand_int(), random_points(), random_polygon(), spherical_random_points(), exponential_rands()

Usage:

• arr = gaussian_rands([n],[mean], [cov], [seed]);

Description:

Returns a random number or vector with a Gaussian/normal distribution.

Arguments:

Synopsis: Returns a list of random numbers with an exponential distribution.

Topics: Random, Statistics

See Also: rand_int(), random_points(), gaussian_rands(), random_polygon(), spherical_random_points()

Usage:

• arr = exponential_rands([n], [lambda], [seed])

Description:

Returns random numbers with an exponential distribution with parameter lambda, and hence mean 1/lambda.

Arguments:

Synopsis: Returns a list of random points on the surface of a sphere.

Topics: Random, Points

See Also: rand_int(), random_points(), gaussian_rands(), random_polygon()

Usage:

• points = spherical_random_points([n], [radius], [seed]);

Description:

Generate n 3D uniformly distributed random points lying on a sphere centered at the origin with radius equal to radius.

Arguments:

Synopsis: Returns the CCW path of a simple random polygon.

Topics: Random, Polygon

See Also: random_points(), spherical_random_points()

Usage:

• points = random_polygon([n], [size], [seed]);

Description:

Generate the n vertices of a random counter-clockwise simple 2d polygon inside a circle centered at the origin with radius size.

Arguments:

Synopsis: Returns the first derivative estimate of a list of data.

Topics: Math, Calculus

See Also: deriv2(), deriv3()

Usage:

• x = deriv(data, [h], [closed])

Description:

Computes a numerical derivative estimate of the data, which may be scalar or vector valued. The h parameter gives the step size of your sampling so the derivative can be scaled correctly. If the closed parameter is true the data is assumed to be defined on a loop with data[0] adjacent to data[len(data)-1]. This function uses a symetric derivative approximation for internal points, f'(t) = (f(t+h)-f(t-h))/2h. For the endpoints (when closed=false) the algorithm uses a two point method if sufficient points are available: f'(t) = (3*(f(t+h)-f(t)) - (f(t+2*h)-f(t+h)))/2h.

If h is a vector then it is assumed to be nonuniform, with h[i] giving the sampling distance between data[i+1] and data[i], and the data values will be linearly resampled at each corner to produce a uniform spacing for the derivative estimate. At the endpoints a single point method is used: f'(t) = (f(t+h)-f(t))/h.

Arguments:

Synopsis: Returns the second derivative estimate of a list of data.

Topics: Math, Calculus

See Also: deriv(), deriv3()

Usage:

• x = deriv2(data, [h], [closed])

Description:

Computes a numerical estimate of the second derivative of the data, which may be scalar or vector valued. The h parameter gives the step size of your sampling so the derivative can be scaled correctly. If the closed parameter is true the data is assumed to be defined on a loop with data[0] adjacent to data[len(data)-1]. For internal points this function uses the approximation f''(t) = (f(t-h)-2f(t)+f(t+h))/h^2. For the endpoints (when closed=false), when sufficient points are available, the method is either the four point expression f''(t) = (2f(t) - 5f(t+h) + 4f(t+2h) - f(t+3h))/h^2 or f''(t) = (35f(t) - 104f(t+h) + 114f(t+2h) - 56f(t+3h) + 11f(t+4h)) / 12h^2 if five points are available.

Arguments:

Synopsis: Returns the third derivative estimate of a list of data.

Topics: Math, Calculus

See Also: deriv(), deriv2()

Usage:

• x = deriv3(data, [h], [closed])

Description:

Computes a numerical third derivative estimate of the data, which may be scalar or vector valued. The h parameter gives the step size of your sampling so the derivative can be scaled correctly. If the closed parameter is true the data is assumed to be defined on a loop with data[0] adjacent to data[len(data)-1]. This function uses a five point derivative estimate, so the input data must include at least five points: f'''(t) = (-f(t-2h)+2f(t-h)-2f(t+h)+f(t+2h)) / 2h^3. At the first and second points from the end the estimates are f'''(t) = (-5f(t)+18f(t+h)-24f(t+2h)+14f(t+3h)-3f(t+4h)) / 2h^3 and f'''(t) = (-3f(t-h)+10f(t)-12f(t+h)+6f(t+2h)-f(t+3h)) / 2h^3.

Arguments:

Synopsis: Replaces scalars in a list or matrix with complex number 2-vectors.

Topics: Math, Complex Numbers

See Also: c_mul(), c_div(), c_conj(), c_real(), c_imag(), c_ident(), c_norm()

Usage:

• z = complex(list)

Description:

Converts a real valued number, vector or matrix into its complex analog by replacing all entries with a 2-vector that has zero imaginary part.

Synopsis: Multiplies two complex numbers.

Topics: Math, Complex Numbers

See Also: complex(), c_div(), c_conj(), c_real(), c_imag(), c_ident(), c_norm()

Usage:

• c = c_mul(z1,z2)

Description:

Multiplies two complex numbers, vectors or matrices, where complex numbers or entries are represented as vectors: [REAL, IMAGINARY]. Note that all entries in both arguments must be complex.

Arguments:

Synopsis: Divides two complex numbers.

Topics: Math, Complex Numbers

See Also: complex(), c_mul(), c_conj(), c_real(), c_imag(), c_ident(), c_norm()

Usage:

• x = c_div(z1,z2)

Description:

Divides two complex numbers represented by 2D vectors. Returns a complex number as a 2D vector [REAL, IMAGINARY].

Arguments:

Synopsis: Returns the complex conjugate of the input.

Topics: Math, Complex Numbers

See Also: complex(), c_mul(), c_div(), c_real(), c_imag(), c_ident(), c_norm()

Usage:

• w = c_conj(z)

Description:

Computes the complex conjugate of the input, which can be a complex number, complex vector or complex matrix.

Synopsis: Returns the real part of a complex number, vector or matrix..

Topics: Math, Complex Numbers

See Also: complex(), c_mul(), c_div(), c_conj(), c_imag(), c_ident(), c_norm()

Usage:

• x = c_real(z)

Description:

Returns real part of a complex number, vector or matrix.

Synopsis: Returns the imaginary part of a complex number, vector or matrix..

Topics: Math, Complex Numbers

See Also: complex(), c_mul(), c_div(), c_conj(), c_real(), c_ident(), c_norm()

Usage:

• x = c_imag(z)

Description:

Returns imaginary part of a complex number, vector or matrix.

Synopsis: Returns an n by n complex identity matrix.

Topics: Math, Complex Numbers

See Also: complex(), c_mul(), c_div(), c_conj(), c_real(), c_imag(), c_norm()

Usage:

• I = c_ident(n)

Description:

Produce an n by n complex identity matrix

Synopsis: Returns the norm of a complex number or vector.

Topics: Math, Complex Numbers

See Also: complex(), c_mul(), c_div(), c_conj(), c_real(), c_imag(), c_ident()

Usage:

• n = c_norm(z)

Description:

Compute the norm of a complex number or vector.

Synopsis: Computes roots for the quadratic equation.

Topics: Math, Geometry, Complex Numbers

See Also: polynomial(), poly_mult(), poly_div(), poly_add()

Usage:

• roots = quadratic_roots(a, b, c, [real])

Description:

Computes roots of the quadratic equation ax^2+bx+c==0, where the coefficients are real numbers. If real is true then returns only the real roots. Otherwise returns a pair of complex values. This method may be more reliable than the general root finder at distinguishing real roots from complex roots. Algorithm from: https://people.csail.mit.edu/bkph/articles/Quadratics.pdf

Synopsis: Calculates a polynomial equation at a given value.

Topics: Math, Complex Numbers

See Also: quadratic_roots(), poly_mult(), poly_div(), poly_add(), poly_roots()

Usage:

• x = polynomial(p, z)

Description:

Evaluates specified real polynomial, p, at the complex or real input value, z. The polynomial is specified as p=[a_n, a_{n-1},...,a_1,a_0] where a_n is the z^n coefficient. Polynomial coefficients are real. The result is a number if z is a number and a complex number otherwise.

Synopsis: Returns the polynomial result of multiplying two polynomial equations.

Topics: Math

See Also: quadratic_roots(), polynomial(), poly_div(), poly_add(), poly_roots()

Usage:

• x = polymult(p,q)
• x = polymult([p1,p2,p3,...])

Description:

Given a list of polynomials represented as real algebraic coefficient lists, with the highest degree coefficient first, computes the coefficient list of the product polynomial.

Synopsis: Returns the polynomial quotient and remainder results of dividing two polynomial equations.

Topics: Math

See Also: quadratic_roots(), polynomial(), poly_mult(), poly_add(), poly_roots()

Usage:

• [quotient,remainder] = poly_div(n,d)

Description:

Computes division of the numerator polynomial by the denominator polynomial and returns a list of two polynomials, [quotient, remainder]. If the division has no remainder then the zero polynomial [0] is returned for the remainder. Similarly if the quotient is zero the returned quotient will be [0].

Synopsis: Returns the polynomial sum of adding two polynomial equations.

Topics: Math

See Also: quadratic_roots(), polynomial(), poly_mult(), poly_div(), poly_roots()

Usage:

• sum = poly_add(p,q)

Description:

Computes the sum of two polynomials.

Synopsis: Returns all complex number roots of the given real polynomial.

Topics: Math, Complex Numbers

See Also: quadratic_roots(), polynomial(), poly_mult(), poly_div(), poly_add()

Usage:

• roots = poly_roots(p, [tol]);

Description:

Returns all complex roots of the specified real polynomial p. The polynomial is specified as p=[a_n, a_{n-1},...,a_1,a_0] where a_n is the z^n coefficient. The tol parameter gives the stopping tolerance for the iteration. The polynomial must have at least one non-zero coefficient. Convergence is poor if the polynomial has any repeated roots other than zero.

Arguments:

Synopsis: Returns all real roots of the given real polynomial.

Topics: Math, Complex Numbers

See Also: quadratic_roots(), polynomial(), poly_mult(), poly_div(), poly_add(), poly_roots()

Usage:

• roots = real_roots(p, [eps], [tol])

Description:

Returns the real roots of the specified real polynomial p. The polynomial is specified as p=[a_n, a_{n-1},...,a_1,a_0] where a_n is the x^n coefficient. This function works by computing the complex roots and returning those roots where the imaginary part is closed to zero. By default it uses a computed error bound from the polynomial solver to decide whether imaginary parts are zero. You can specify eps, in which case the test is z.y/(1+norm(z)) < eps. Because of poor convergence and higher error for repeated roots, such roots may be missed by the algorithm because their imaginary part is large.

Arguments:

Synopsis: Finds a root of the given continuous function.

Topics: Math

See Also: quadratic_roots(), polynomial(), poly_mult(), poly_div(), poly_add(), poly_roots()

Usage:

• x = root_find(f, x0, x1, [tol])

Description:

Find a root of the continuous function f where the sign of f(x0) is different from the sign of f(x1). The function f is a function literal accepting one argument. You must have a version of OpenSCAD that supports function literals (2021.01 or newer). The tolerance (tol) specifies the accuracy of the solution: abs(f(x)) < tol * yrange, where yrange is the range of observed function values. This function can only find roots that cross the x axis: it cannot find the the root of x^2.

Arguments: