////////////////////////////////////////////////////////////////////// // LibFile: math.scad // Assorted math functions, including linear interpolation, list operations (sums, mean, products), // convolution, quantization, log2, hyperbolic trig functions, random numbers, derivatives, // polynomials, and root finding. // Includes: // include <BOSL2/std.scad> // FileGroup: Math // FileSummary: Math on lists, special functions, quantization, random numbers, calculus, root finding // // FileFootnotes: STD=Included in std.scad ////////////////////////////////////////////////////////////////////// // Section: Math Constants // Constant: PHI // 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 PHI = (1+sqrt(5))/2; // Constant: EPSILON // Synopsis: A tiny value to compare floating point values. `1e-9` // Topics: Constants, Math // See Also: PHI, EPSILON, INF, NAN // Description: A really small value useful in comparing floating point numbers. ie: abs(a-b)<EPSILON `1e-9` EPSILON = 1e-9; // Constant: INF // Synopsis: The floating point value for Infinite. // Topics: Constants, Math // See Also: PHI, EPSILON, INF, NAN // Description: The value `inf`, useful for comparisons. INF = 1/0; // Constant: NAN // Synopsis: The floating point value for Not a Number. // Topics: Constants, Math // See Also: PHI, EPSILON, INF, NAN // Description: The value `nan`, useful for comparisons. NAN = acos(2); // Section: Interpolation and Counting // Function: count() // 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: // n = The length of the list of numbers to create, or a list to match the length of // s = The starting value of the list of numbers. // step = The amount to increment successive numbers in the list. // reverse = Reverse the list. Default: false. // Example: // 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] function count(n,s=0,step=1,reverse=false) = let(n=is_list(n) ? len(n) : n) reverse? [for (i=[n-1:-1:0]) s+i*step] : [for (i=[0:1:n-1]) s+i*step]; // Function: lerp() // 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: // a = First value or vector. // b = Second value or vector. // u = The proportion from `a` to `b` to calculate. Standard range is 0.0 to 1.0, inclusive. If given as a list or range of values, returns a list of results. // Example: // 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(2D): // 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); function lerp(a,b,u) = assert(same_shape(a,b), "Bad or inconsistent inputs to lerp") is_finite(u)? (1-u)*a + u*b : assert(is_finite(u) || is_vector(u) || valid_range(u), "Input u to lerp must be a number, vector, or valid range.") [for (v = u) (1-v)*a + v*b ]; // Function: lerpn() // 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: // a = First value or vector. // b = Second value or vector. // n = The number of values to return. // endpoint = If true, the last value will be exactly `b`. If false, the last value will be one step less. // Example: // 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] function lerpn(a,b,n,endpoint=true) = assert(same_shape(a,b), "Bad or inconsistent inputs to lerpn") assert(is_int(n)) assert(is_bool(endpoint)) let( d = n - (endpoint? 1 : 0) ) [for (i=[0:1:n-1]) let(u=i/d) (1-u)*a + u*b]; // Section: Miscellaneous Functions // Function: sqr() // 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: // sqr(3); // Returns: 9 // sqr(-4); // Returns: 16 // sqr([2,3,4]); // Returns: 29 // sqr([[1,2],[3,4]]); // Returns [[7,10],[15,22]] function sqr(x) = assert(is_finite(x) || is_vector(x) || is_matrix(x), "Input is not a number nor a list of numbers.") x*x; // Function: log2() // 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: // log2(0.125); // Returns: -3 // log2(16); // Returns: 4 // log2(256); // Returns: 8 function log2(x) = assert( is_finite(x), "Input is not a number.") ln(x)/ln(2); // this may return NAN or INF; should it check x>0 ? // Function: hypot() // Synopsis: Returns the hypotenuse length of a 2D or 3D triangle. // Topics: Math // See Also: hypot(), sqr(), log2() // Usage: // l = hypot(x, y, [z]); // Description: // Calculate hypotenuse length of a 2D or 3D triangle. // Arguments: // x = Length on the X axis. // y = Length on the Y axis. // z = Length on the Z axis. Optional. // Example: // l = hypot(3,4); // Returns: 5 // l = hypot(3,4,5); // Returns: ~7.0710678119 function hypot(x,y,z=0) = assert( is_vector([x,y,z]), "Improper number(s).") norm([x,y,z]); // Function: factorial() // 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: // n = The integer number to get the factorial of. (n!) // d = If given, the returned value will be (n! / d!) // Example: // x = factorial(4); // Returns: 24 // y = factorial(6); // Returns: 720 // z = factorial(9); // Returns: 362880 function factorial(n,d=0) = assert(is_int(n) && is_int(d) && n>=0 && d>=0, "Factorial is defined only for non negative integers") assert(d<=n, "d cannot be larger than n") product([1,for (i=[n:-1:d+1]) i]); // Function: binomial() // 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: // n = The integer to get the binomial coefficients of // Example: // 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] function binomial(n) = assert( is_int(n) && n>0, "Input is not an integer greater than 0.") [for( c = 1, i = 0; i<=n; c = c*(n-i)/(i+1), i = i+1 ) c ] ; // Function: binomial_coefficient() // 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: // n = The integer to get the binomial coefficient of // k = The binomial coefficient index // Example: // x = binomial_coefficient(3,2); // Returns: 3 // y = binomial_coefficient(10,6); // Returns: 210 function binomial_coefficient(n,k) = assert( is_int(n) && is_int(k), "Some input is not a number.") k < 0 || k > n ? 0 : k ==0 || k ==n ? 1 : let( k = min(k, n-k), b = [for( c = 1, i = 0; i<=k; c = c*(n-i)/(i+1), i = i+1 ) c] ) b[len(b)-1]; // Function: gcd() // Synopsis: Returns the Greatest Common Divisor/Factor of two integers. // Topics: Math // See Also: hypot(), sqr(), log2(), factorial(), binomial(), gcd(), lcm() // Usage: // x = gcd(a,b) // Description: // Computes the Greatest Common Divisor/Factor of `a` and `b`. function gcd(a,b) = assert(is_int(a) && is_int(b),"Arguments to gcd must be integers") b==0 ? abs(a) : gcd(b,a % b); // Computes lcm for two integers function _lcm(a,b) = assert(is_int(a) && is_int(b), "Invalid non-integer parameters to lcm") assert(a!=0 && b!=0, "Arguments to lcm should not be zero") abs(a*b) / gcd(a,b); // Computes lcm for a list of values function _lcmlist(a) = len(a)==1 ? a[0] : _lcmlist(concat(lcm(a[0],a[1]),list_tail(a,2))); // Function: lcm() // Synopsis: Returns the Least Common Multiple of two or more integers. // Topics: Math // See Also: hypot(), sqr(), log2(), factorial(), binomial(), gcd(), lcm() // 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. function lcm(a,b=[]) = !is_list(a) && !is_list(b) ? _lcm(a,b) : let( arglist = concat(force_list(a),force_list(b)) ) assert(len(arglist)>0, "Invalid call to lcm with empty list(s)") _lcmlist(arglist); // Function rational_approx() // Usage: // pq = rational_approx(x, maxq); // Description: // Finds the best rational approximation p/q to the number x so that q<=maxq. Returns // the result as `[p,q]`. If the input is zero, then returns `[0,1]`. // Example: // pq1 = rational_approx(PI,10); // Returns: [22,7] // pq2 = rational_approx(PI,10000); // Returns: [355, 113] // pq3 = rational_approx(221/323,500); // Returns: [13,19] // pq4 = rational_approx(0,50); // Returns: [0,1] function rational_approx(x, maxq, cfrac=[], p, q) = let( next = floor(x), fracpart = x-next, cfrac = [each cfrac, next], pq = _cfrac_to_pq(cfrac) ) approx(fracpart,0) ? pq : pq[1]>maxq ? [p,q] : rational_approx(1/fracpart,maxq,cfrac, pq[0], pq[1]); // Converts a continued fraction given as a list with leading integer term // into a fraction in the form p / q, returning [p,q]. function _cfrac_to_pq(cfrac,p=0,q=1,ind) = is_undef(ind) ? _cfrac_to_pq(cfrac,p,q,len(cfrac)-1) : ind==0 ? [p+q*cfrac[0], q] : _cfrac_to_pq(cfrac, q, cfrac[ind]*q+p, ind-1); // Section: Hyperbolic Trigonometry // Function: sinh() // Synopsis: Returns the hyperbolic sine of the given value. // Topics: Math, Trigonometry // See Also: sinh(), cosh(), tanh(), asinh(), acosh(), atanh() // Usage: // a = sinh(x); // Description: Takes a value `x`, and returns the hyperbolic sine of it. function sinh(x) = assert(is_finite(x), "The input must be a finite number.") (exp(x)-exp(-x))/2; // Function: cosh() // Synopsis: Returns the hyperbolic cosine of the given value. // Topics: Math, Trigonometry // See Also: sinh(), cosh(), tanh(), asinh(), acosh(), atanh() // Usage: // a = cosh(x); // Description: Takes a value `x`, and returns the hyperbolic cosine of it. function cosh(x) = assert(is_finite(x), "The input must be a finite number.") (exp(x)+exp(-x))/2; // Function: tanh() // Synopsis: Returns the hyperbolic tangent of the given value. // Topics: Math, Trigonometry // See Also: sinh(), cosh(), tanh(), asinh(), acosh(), atanh() // Usage: // a = tanh(x); // Description: Takes a value `x`, and returns the hyperbolic tangent of it. function tanh(x) = assert(is_finite(x), "The input must be a finite number.") sinh(x)/cosh(x); // Function: asinh() // Synopsis: Returns the hyperbolic arc-sine of the given value. // Topics: Math, Trigonometry // See Also: sinh(), cosh(), tanh(), asinh(), acosh(), atanh() // Usage: // a = asinh(x); // Description: Takes a value `x`, and returns the inverse hyperbolic sine of it. function asinh(x) = assert(is_finite(x), "The input must be a finite number.") ln(x+sqrt(x*x+1)); // Function: acosh() // Synopsis: Returns the hyperbolic arc-cosine of the given value. // Topics: Math, Trigonometry // See Also: sinh(), cosh(), tanh(), asinh(), acosh(), atanh() // Usage: // a = acosh(x); // Description: Takes a value `x`, and returns the inverse hyperbolic cosine of it. function acosh(x) = assert(is_finite(x), "The input must be a finite number.") ln(x+sqrt(x*x-1)); // Function: atanh() // Synopsis: Returns the hyperbolic arc-tangent of the given value. // Topics: Math, Trigonometry // See Also: sinh(), cosh(), tanh(), asinh(), acosh(), atanh() // Usage: // a = atanh(x); // Description: Takes a value `x`, and returns the inverse hyperbolic tangent of it. function atanh(x) = assert(is_finite(x), "The input must be a finite number.") ln((1+x)/(1-x))/2; // Section: Quantization // Function: quant() // Synopsis: Returns `x` quantized to the nearest integer multiple of `y`. // Topics: Math, Quantization // See Also: quant(), 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: // x = The value or list to quantize. // y = Positive quantum to quantize to // Example: // 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]] function quant(x,y) = assert( is_finite(y) && y>0, "The quantum `y` must be a positive value.") is_num(x) ? round(x/y)*y : _roundall(x/y)*y; function _roundall(data) = [for(x=data) is_list(x) ? _roundall(x) : round(x)]; // Function: quantdn() // Synopsis: Returns `x` quantized down to an integer multiple of `y`. // Topics: Math, Quantization // See Also: quant(), quantdn(), 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: // x = The value or list to quantize. // y = Postive quantum to quantize to. // Example: // 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]] function quantdn(x,y) = assert( is_finite(y) && y>0, "The quantum `y` must be a positive value.") is_num(x) ? floor(x/y)*y : _floorall(x/y)*y; function _floorall(data) = [for(x=data) is_list(x) ? _floorall(x) : floor(x)]; // Function: quantup() // Synopsis: Returns `x` quantized uo to an integer multiple of `y`. // Topics: Math, Quantization // See Also: quant(), quantdn(), quantup() // 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: // x = The value or list to quantize. // y = Positive quantum to quantize to. // Example: // 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]] function quantup(x,y) = assert( is_finite(y) && y>0, "The quantum `y` must be a positive value.") is_num(x) ? ceil(x/y)*y : _ceilall(x/y)*y; function _ceilall(data) = [for(x=data) is_list(x) ? _ceilall(x) : ceil(x)]; // Section: Constraints and Modulos // Function: constrain() // 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: // v = value to constrain. // minval = minimum value to return, if out of range. // maxval = maximum value to return, if out of range. // Example: // 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 function constrain(v, minval, maxval) = assert( is_finite(v+minval+maxval), "Input must be finite number(s).") min(maxval, max(minval, v)); // Function: posmod() // Synopsis: Returns the positive modulo of a value. // Topics: Math // See Also: constrain(), posmod(), 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: // x = The value to constrain. // m = Modulo value. // Example: // 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 function posmod(x,m) = assert( is_finite(x) && is_finite(m) && !approx(m,0) , "Input must be finite numbers. The divisor cannot be zero.") (x%m+m)%m; // Function: modang() // Synopsis: Returns an angle normalized to between -180º and 180º. // Topics: Math // See Also: constrain(), posmod(), modang() // Usage: // ang = modang(x); // Description: // Takes an angle in degrees and normalizes it to an equivalent angle value between -180 and 180. // Example: // 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 function modang(x) = assert( is_finite(x), "Input must be a finite number.") let(xx = posmod(x,360)) xx<180? xx : xx-360; // Section: Operations on Lists (Sums, Mean, Products) // Function: sum() // 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: // v = The list to get the sum of. // dflt = The default value to return if `v` is an empty list. Default: 0 // Example: // sum([1,2,3]); // returns 6. // sum([[1,2,3], [3,4,5], [5,6,7]]); // returns [9, 12, 15] function sum(v, dflt=0) = v==[]? dflt : assert(is_consistent(v), "Input to sum is non-numeric or inconsistent") is_finite(v[0]) || is_vector(v[0]) ? [for(i=v) 1]*v : _sum(v,v[0]*0); function _sum(v,_total,_i=0) = _i>=len(v) ? _total : _sum(v,_total+v[_i], _i+1); // Function: mean() // Synopsis: Returns the mean value of a list of values. // Topics: Math, Statistics // See Also: sum(), mean(), 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: // v = The list of values to get the mean of. // Example: // mean([2,3,4]); // returns 3. // mean([[1,2,3], [3,4,5], [5,6,7]]); // returns [3, 4, 5] function mean(v) = assert(is_list(v) && len(v)>0, "Invalid list.") sum(v)/len(v); // Function: median() // Synopsis: Returns the median value of a list of values. // Topics: Math, Statistics // See Also: sum(), mean(), median(), product() // Usage: // middle = median(v) // Description: // Returns the median of the given vector. function median(v) = assert(is_vector(v), "Input to median must be a vector") len(v)%2 ? max( list_smallest(v, ceil(len(v)/2)) ) : let( lowest = list_smallest(v, len(v)/2 + 1), max = max(lowest), imax = search(max,lowest,1), max2 = max([for(i=idx(lowest)) if(i!=imax[0]) lowest[i] ]) ) (max+max2)/2; // Function: deltas() // 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: // v = The list to get the deltas of. // wrap = If true, wrap back to the start from the end. ie: return the difference between the last and first items as the last delta. Default: false // Example: // 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]] function deltas(v, wrap=false) = assert( is_consistent(v) && len(v)>1 , "Inconsistent list or with length<=1.") [for (p=pair(v,wrap)) p[1]-p[0]] ; // Function: cumsum() // 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: // v = The list to get the sum of. // Example: // 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]] function cumsum(v) = v==[] ? [] : assert(is_consistent(v), "The input is not consistent." ) [for (a = v[0], i = 1 ; i <= len(v) ; a = i<len(v) ? a+v[i] : a, i = i+1) a]; // Function: product() // Synopsis: Returns the multiplicative product of a list of values. // Topics: Math, Statistics // See Also: sum(), mean(), median(), product(), cumsum() // Usage: // x = product(v); // Description: // Returns the product of all entries in the given list. // If passed a list of vectors of same length, returns a vector of the component-wise products of the input. // If passed a list of square matrices, returns the resulting product matrix. Matrices are multiplied in the order they appear in the list. // Arguments: // v = The list to get the product of. // Example: // product([2,3,4]); // returns 24. // product([[1,2,3], [3,4,5], [5,6,7]]); // returns [15, 48, 105] function product(list,right=true) = list==[] ? [] : is_matrix(list) ? [for (a = list[0], i = 1 ; i <= len(list) ; a = i<len(list) ? v_mul(a,list[i]) : 0, i = i+1) if (i==len(list)) a][0] : assert(is_vector(list) || (is_matrix(list[0],square=true) && is_consistent(list)), "Input must be a vector, a list of vectors, or a list of matrices.") [for (a = list[0], i = 1 ; i <= len(list) ; a = i<len(list) ? a*list[i] : 0, i = i+1) if (i==len(list)) a][0]; // Function: cumprod() // 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: // list = The list to get the cumulative product of. // right = if true multiply matrices on the right // Example: // 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]] function cumprod(list,right=false) = list==[] ? [] : is_matrix(list) ? [for (a = list[0], i = 1 ; i <= len(list) ; a = i<len(list) ? v_mul(a,list[i]) : 0, i = i+1) a] : assert(is_vector(list) || (is_matrix(list[0],square=true) && is_consistent(list)), "Input must be a listector, a list of listectors, or a list of matrices.") [for (a = list[0], i = 1 ; i <= len(list) ; a = i<len(list) ? (right ? a*list[i] : list[i]*a) : 0, i = i+1) a]; // Function: convolve() // 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: // p = The first vector or path. // q = The second vector or path. // Example: // 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] function convolve(p,q) = p==[] || q==[] ? [] : assert( (is_vector(p) || is_matrix(p)) && ( is_vector(q) || (is_matrix(q) && ( !is_vector(p[0]) || (len(p[0])==len(q[0])) ) ) ) , "The inputs should be vectors or paths all of the same dimension.") let( n = len(p), m = len(q)) [for(i=[0:n+m-2], k1 = max(0,i-n+1), k2 = min(i,m-1) ) sum([for(j=[k1:k2]) p[i-j]*q[j] ]) ]; // Function: sum_of_sines() // 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: // a = Angle to get the value for. // sines = List of [amplitude, frequency, phase_angle] items, where the frequency is the number of times the cycle repeats around the circle. // Example: // v = sum_of_sines(30, [[10,3,0], [5,5.5,60]]); function sum_of_sines(a, sines) = assert( is_finite(a) && is_matrix(sines,undef,3), "Invalid input.") sum([ for (s = sines) let( ss=point3d(s), v=ss[0]*sin(a*ss[1]+ss[2]) ) v ]); // Section: Random Number Generation // Function: rand_int() // Synopsis: Returns a random integer. // Topics: Random // See Also: rand_int(), 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: // minval = Minimum integer value to return. // maxval = Maximum integer value to return. // N = Number of random integers to return. // seed = If given, sets the random number seed. // Example: // ints = rand_int(0,100,3); // int = rand_int(-10,10,1)[0]; function rand_int(minval, maxval, n, seed=undef) = assert( is_finite(minval+maxval+n) && (is_undef(seed) || is_finite(seed) ), "Input must be finite numbers.") assert(maxval >= minval, "Max value cannot be smaller than minval") let (rvect = is_def(seed) ? rands(minval,maxval+1,n,seed) : rands(minval,maxval+1,n)) [for(entry = rvect) floor(entry)]; // Function: random_points() // Synopsis: Returns a list of random points. // Topics: Random, Points // See Also: rand_int(), random_points(), 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: // n = number of points to generate. Default: 1 // dim = dimension of the points. Default: 2 // scale = the scale of the point coordinates. Default: 1 // seed = an optional seed for the random generation. function random_points(n, dim, scale=1, seed) = assert( is_int(n) && n>=0, "The number of points should be a non-negative integer.") assert( is_int(dim) && dim>=1, "The point dimensions should be an integer greater than 1.") assert( is_finite(scale) || is_vector(scale,dim), "The scale should be a number or a vector with length equal to d.") let( rnds = is_undef(seed) ? rands(-1,1,n*dim) : rands(-1,1,n*dim, seed) ) is_num(scale) ? scale*[for(i=[0:1:n-1]) [for(j=[0:dim-1]) rnds[i*dim+j] ] ] : [for(i=[0:1:n-1]) [for(j=[0:dim-1]) scale[j]*rnds[i*dim+j] ] ]; // Function: gaussian_rands() // Synopsis: Returns a list of random numbers with a gaussian distribution. // Topics: Random, Statistics // See Also: rand_int(), random_points(), gaussian_rands(), 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: // n = the number of points to return. Default: 1 // mean = The average of the random value (a number or vector). Default: 0 // cov = covariance matrix of the random numbers, or variance in the 1D case. Default: 1 // seed = If given, sets the random number seed. function gaussian_rands(n=1, mean=0, cov=1, seed=undef) = assert(is_num(mean) || is_vector(mean)) let( dim = is_num(mean) ? 1 : len(mean) ) assert((dim==1 && is_num(cov)) || is_matrix(cov,dim,dim),"mean and covariance matrix not compatible") assert(is_undef(seed) || is_finite(seed)) let( nums = is_undef(seed)? rands(0,1,dim*n*2) : rands(0,1,dim*n*2,seed), rdata = [for (i = count(dim*n,0,2)) sqrt(-2*ln(nums[i]))*cos(360*nums[i+1])] ) dim==1 ? add_scalar(sqrt(cov)*rdata,mean) : assert(is_matrix_symmetric(cov),"Supplied covariance matrix is not symmetric") let( L = cholesky(cov) ) assert(is_def(L), "Supplied covariance matrix is not positive definite") move(mean,list_to_matrix(rdata,dim)*transpose(L)); // Function: exponential_rands() // 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: // n = number of points to return. Default: 1 // lambda = distribution parameter. The mean will be 1/lambda. Default: 1 function exponential_rands(n=1, lambda=1, seed) = assert( is_int(n) && n>=1, "The number of points should be an integer greater than zero.") assert( is_num(lambda) && lambda>0, "The lambda parameter must be a positive number.") let( unif = is_def(seed) ? rands(0,1,n,seed=seed) : rands(0,1,n) ) -(1/lambda) * [for(x=unif) x==1 ? 708.3964185322641 : ln(1-x)]; // Use ln(min_float) when x is 1 // Function: spherical_random_points() // 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(), spherical_random_points() // 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: // n = number of points to generate. Default: 1 // radius = the sphere radius. Default: 1 // seed = an optional seed for the random generation. // See https://mathworld.wolfram.com/SpherePointPicking.html function spherical_random_points(n=1, radius=1, seed) = assert( is_int(n) && n>=1, "The number of points should be an integer greater than zero.") assert( is_num(radius) && radius>0, "The radius should be a non-negative number.") let( theta = is_undef(seed) ? rands(0,360,n) : rands(0,360,n, seed), cosphi = rands(-1,1,n)) [for(i=[0:1:n-1]) let( sin_phi=sqrt(1-cosphi[i]*cosphi[i]) ) radius*[sin_phi*cos(theta[i]),sin_phi*sin(theta[i]), cosphi[i]]]; // Function: random_polygon() // 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: // n = number of vertices of the polygon. Default: 3 // size = the radius of a circle centered at the origin containing the polygon. Default: 1 // seed = an optional seed for the random generation. function random_polygon(n=3,size=1, seed) = assert( is_int(n) && n>2, "Improper number of polygon vertices.") assert( is_num(size) && size>0, "Improper size.") let( seed = is_undef(seed) ? rands(0,1,1)[0] : seed, cumm = cumsum(rands(0.1,10,n+1,seed)), angs = 360*cumm/cumm[n-1], rads = rands(.01,size,n,seed+cumm[0]) ) [for(i=count(n)) rads[i]*[cos(angs[i]), sin(angs[i])] ]; // Section: Calculus // Function: deriv() // Synopsis: Returns the first derivative estimate of a list of data. // Topics: Math, Calculus // See Also: deriv(), 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: // data = the list of the elements to compute the derivative of. // h = the parametric sampling of the data. // closed = boolean to indicate if the data set should be wrapped around from the end to the start. function deriv(data, h=1, closed=false) = assert( is_consistent(data) , "Input list is not consistent or not numerical.") assert( len(data)>=2, "Input `data` should have at least 2 elements.") assert( is_finite(h) || is_vector(h), "The sampling `h` must be a number or a list of numbers." ) assert( is_num(h) || len(h) == len(data)-(closed?0:1), str("Vector valued `h` must have length ",len(data)-(closed?0:1))) is_vector(h) ? _deriv_nonuniform(data, h, closed=closed) : let( L = len(data) ) closed ? [ for(i=[0:1:L-1]) (data[(i+1)%L]-data[(L+i-1)%L])/2/h ] : let( first = L<3 ? data[1]-data[0] : 3*(data[1]-data[0]) - (data[2]-data[1]), last = L<3 ? data[L-1]-data[L-2]: (data[L-3]-data[L-2])-3*(data[L-2]-data[L-1]) ) [ first/2/h, for(i=[1:1:L-2]) (data[i+1]-data[i-1])/2/h, last/2/h ]; function _dnu_calc(f1,fc,f2,h1,h2) = let( f1 = h2<h1 ? lerp(fc,f1,h2/h1) : f1 , f2 = h1<h2 ? lerp(fc,f2,h1/h2) : f2 ) (f2-f1) / 2 / min(h1,h2); function _deriv_nonuniform(data, h, closed) = let( L = len(data) ) closed ? [for(i=[0:1:L-1]) _dnu_calc(data[(L+i-1)%L], data[i], data[(i+1)%L], select(h,i-1), h[i]) ] : [ (data[1]-data[0])/h[0], for(i=[1:1:L-2]) _dnu_calc(data[i-1],data[i],data[i+1], h[i-1],h[i]), (data[L-1]-data[L-2])/h[L-2] ]; // Function: deriv2() // Synopsis: Returns the second derivative estimate of a list of data. // Topics: Math, Calculus // See Also: deriv(), deriv2(), 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)-2*f(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) = (2*f(t) - 5*f(t+h) + 4*f(t+2*h) - f(t+3*h))/h^2 or // f''(t) = (35*f(t) - 104*f(t+h) + 114*f(t+2*h) - 56*f(t+3*h) + 11*f(t+4*h)) / 12h^2 // if five points are available. // Arguments: // data = the list of the elements to compute the derivative of. // h = the constant parametric sampling of the data. // closed = boolean to indicate if the data set should be wrapped around from the end to the start. function deriv2(data, h=1, closed=false) = assert( is_consistent(data) , "Input list is not consistent or not numerical.") assert( is_finite(h), "The sampling `h` must be a number." ) let( L = len(data) ) assert( L>=3, "Input list has less than 3 elements.") closed ? [ for(i=[0:1:L-1]) (data[(i+1)%L]-2*data[i]+data[(L+i-1)%L])/h/h ] : let( first = L==3? data[0] - 2*data[1] + data[2] : L==4? 2*data[0] - 5*data[1] + 4*data[2] - data[3] : (35*data[0] - 104*data[1] + 114*data[2] - 56*data[3] + 11*data[4])/12, last = L==3? data[L-1] - 2*data[L-2] + data[L-3] : L==4? -2*data[L-1] + 5*data[L-2] - 4*data[L-3] + data[L-4] : (35*data[L-1] - 104*data[L-2] + 114*data[L-3] - 56*data[L-4] + 11*data[L-5])/12 ) [ first/h/h, for(i=[1:1:L-2]) (data[i+1]-2*data[i]+data[i-1])/h/h, last/h/h ]; // Function: deriv3() // Synopsis: Returns the third derivative estimate of a list of data. // Topics: Math, Calculus // See Also: deriv(), deriv2(), deriv3() // 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-2*h)+2*f(t-h)-2*f(t+h)+f(t+2*h)) / 2h^3. At the first and second points from the end // the estimates are f'''(t) = (-5*f(t)+18*f(t+h)-24*f(t+2*h)+14*f(t+3*h)-3*f(t+4*h)) / 2h^3 and // f'''(t) = (-3*f(t-h)+10*f(t)-12*f(t+h)+6*f(t+2*h)-f(t+3*h)) / 2h^3. // Arguments: // data = the list of the elements to compute the derivative of. // h = the constant parametric sampling of the data. // closed = boolean to indicate if the data set should be wrapped around from the end to the start. function deriv3(data, h=1, closed=false) = assert( is_consistent(data) , "Input list is not consistent or not numerical.") assert( len(data)>=5, "Input list has less than 5 elements.") assert( is_finite(h), "The sampling `h` must be a number." ) let( L = len(data), h3 = h*h*h ) closed? [ for(i=[0:1:L-1]) (-data[(L+i-2)%L]+2*data[(L+i-1)%L]-2*data[(i+1)%L]+data[(i+2)%L])/2/h3 ] : let( first=(-5*data[0]+18*data[1]-24*data[2]+14*data[3]-3*data[4])/2, second=(-3*data[0]+10*data[1]-12*data[2]+6*data[3]-data[4])/2, last=(5*data[L-1]-18*data[L-2]+24*data[L-3]-14*data[L-4]+3*data[L-5])/2, prelast=(3*data[L-1]-10*data[L-2]+12*data[L-3]-6*data[L-4]+data[L-5])/2 ) [ first/h3, second/h3, for(i=[2:1:L-3]) (-data[i-2]+2*data[i-1]-2*data[i+1]+data[i+2])/2/h3, prelast/h3, last/h3 ]; // Section: Complex Numbers // Function: complex() // 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. function complex(list) = is_num(list) ? [list,0] : [for(entry=list) is_num(entry) ? [entry,0] : complex(entry)]; // Function: c_mul() // Synopsis: Multiplies two complex numbers. // Topics: Math, Complex Numbers // See Also: complex(), c_mul(), 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: // z1 = First complex number, vector or matrix // z2 = Second complex number, vector or matrix function c_mul(z1,z2) = is_matrix([z1,z2],2,2) ? _c_mul(z1,z2) : _combine_complex(_c_mul(_split_complex(z1), _split_complex(z2))); function _split_complex(data) = is_vector(data,2) ? data : is_num(data[0][0]) ? [data*[1,0], data*[0,1]] : [ [for(vec=data) vec * [1,0]], [for(vec=data) vec * [0,1]] ]; function _combine_complex(data) = is_vector(data,2) ? data : is_num(data[0][0]) ? [for(i=[0:len(data[0])-1]) [data[0][i],data[1][i]]] : [for(i=[0:1:len(data[0])-1]) [for(j=[0:1:len(data[0][0])-1]) [data[0][i][j], data[1][i][j]]]]; function _c_mul(z1,z2) = [ z1.x*z2.x - z1.y*z2.y, z1.x*z2.y + z1.y*z2.x ]; // Function: c_div() // Synopsis: Divides two complex numbers. // Topics: Math, Complex Numbers // See Also: complex(), c_mul(), c_div(), 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: // z1 = First complex number, given as a 2D vector [REAL, IMAGINARY] // z2 = Second complex number, given as a 2D vector [REAL, IMAGINARY] function c_div(z1,z2) = assert( is_vector(z1,2) && is_vector(z2), "Complex numbers should be represented by 2D vectors." ) assert( !approx(z2,0), "The divisor `z2` cannot be zero." ) let(den = z2.x*z2.x + z2.y*z2.y) [(z1.x*z2.x + z1.y*z2.y)/den, (z1.y*z2.x - z1.x*z2.y)/den]; // Function: c_conj() // Synopsis: Returns the complex conjugate of the input. // Topics: Math, Complex Numbers // See Also: complex(), c_mul(), c_div(), c_conj(), 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. function c_conj(z) = is_vector(z,2) ? [z.x,-z.y] : [for(entry=z) c_conj(entry)]; // Function: c_real() // 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_real(), c_imag(), c_ident(), c_norm() // Usage: // x = c_real(z) // Description: // Returns real part of a complex number, vector or matrix. function c_real(z) = is_vector(z,2) ? z.x : is_num(z[0][0]) ? z*[1,0] : [for(vec=z) vec * [1,0]]; // Function: c_imag() // 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_imag(), c_ident(), c_norm() // Usage: // x = c_imag(z) // Description: // Returns imaginary part of a complex number, vector or matrix. function c_imag(z) = is_vector(z,2) ? z.y : is_num(z[0][0]) ? z*[0,1] : [for(vec=z) vec * [0,1]]; // Function: c_ident() // 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_ident(), c_norm() // Usage: // I = c_ident(n) // Description: // Produce an n by n complex identity matrix function c_ident(n) = [for (i = [0:1:n-1]) [for (j = [0:1:n-1]) (i==j)?[1,0]:[0,0]]]; // Function: c_norm() // 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(), c_norm() // Usage: // n = c_norm(z) // Description: // Compute the norm of a complex number or vector. function c_norm(z) = norm_fro(z); // Section: Polynomials // Function: quadratic_roots() // Synopsis: Computes roots for the quadratic equation. // Topics: Math, Geometry, Complex Numbers // See Also: quadratic_roots(), polynomial(), poly_mult(), poly_div(), poly_add() // Usage: // roots = quadratic_roots(a, b, c, [real]) // Description: // Computes roots of the quadratic equation a*x^2+b*x+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 function quadratic_roots(a,b,c,real=false) = real ? [for(root = quadratic_roots(a,b,c,real=false)) if (root.y==0) root.x] : is_undef(b) && is_undef(c) && is_vector(a,3) ? quadratic_roots(a[0],a[1],a[2]) : assert(is_num(a) && is_num(b) && is_num(c)) assert(a!=0 || b!=0 || c!=0, "Quadratic must have a nonzero coefficient") a==0 && b==0 ? [] : // No solutions a==0 ? [[-c/b,0]] : let( descrim = b*b-4*a*c, sqrt_des = sqrt(abs(descrim)) ) descrim < 0 ? // Complex case [[-b, sqrt_des], [-b, -sqrt_des]]/2/a : b<0 ? // b positive [[2*c/(-b+sqrt_des),0], [(-b+sqrt_des)/a/2,0]] : // b negative [[(-b-sqrt_des)/2/a, 0], [2*c/(-b-sqrt_des),0]]; // Function: polynomial() // Synopsis: Evaluate a polynomial at a real or complex value. // Topics: Math, Complex Numbers // See Also: quadratic_roots(), polynomial(), 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. function polynomial(p,z,k,total) = is_undef(k) ? assert( is_vector(p) , "Input polynomial coefficients must be a vector." ) assert( is_finite(z) || is_vector(z,2), "The value of `z` must be a real or a complex number." ) polynomial( _poly_trim(p), z, 0, is_num(z) ? 0 : [0,0]) : k==len(p) ? total : polynomial(p,z,k+1, is_num(z) ? total*z+p[k] : c_mul(total,z)+[p[k],0]); // Function: poly_mult() // Synopsis: Compute product of two polynomials, returning a polynomial. // Topics: Math // See Also: quadratic_roots(), polynomial(), poly_mult(), 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. function poly_mult(p,q) = is_undef(q) ? len(p)==2 ? poly_mult(p[0],p[1]) : poly_mult(p[0], poly_mult(list_tail(p))) : assert( is_vector(p) && is_vector(q),"Invalid arguments to poly_mult") p*p==0 || q*q==0 ? [0] : _poly_trim(convolve(p,q)); // Function: poly_div() // Synopsis: Returns the polynomial quotient and remainder results of dividing two polynomials. // Topics: Math // See Also: quadratic_roots(), polynomial(), poly_mult(), poly_div(), 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]. function poly_div(n,d) = assert( is_vector(n) && is_vector(d) , "Invalid polynomials." ) let( d = _poly_trim(d), n = _poly_trim(n) ) assert( d!=[0] , "Denominator cannot be a zero polynomial." ) n==[0] ? [[0],[0]] : _poly_div(n,d,q=[]); function _poly_div(n,d,q) = len(n)<len(d) ? [q,_poly_trim(n)] : let( t = n[0] / d[0], newq = concat(q,[t]), newn = [for(i=[1:1:len(n)-1]) i<len(d) ? n[i] - t*d[i] : n[i]] ) _poly_div(newn,d,newq); /// Internal Function: _poly_trim() /// Usage: /// _poly_trim(p, [eps]) /// Description: /// Removes leading zero terms of a polynomial. By default zeros must be exact, /// or give epsilon for approximate zeros. Returns [0] for a zero polynomial. function _poly_trim(p,eps=0) = let( nz = [for(i=[0:1:len(p)-1]) if ( !approx(p[i],0,eps)) i]) len(nz)==0 ? [0] : list_tail(p,nz[0]); // Function: poly_add() // Synopsis: Returns the polynomial sum of adding two polynomials. // Topics: Math // See Also: quadratic_roots(), polynomial(), poly_mult(), poly_div(), poly_add(), poly_roots() // Usage: // sum = poly_add(p,q) // Description: // Computes the sum of two polynomials. function poly_add(p,q) = assert( is_vector(p) && is_vector(q), "Invalid input polynomial(s)." ) let( plen = len(p), qlen = len(q), long = plen>qlen ? p : q, short = plen>qlen ? q : p ) _poly_trim(long + concat(repeat(0,len(long)-len(short)),short)); // Function: poly_roots() // Synopsis: Returns all complex valued 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 = 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: // p = polynomial coefficients with higest power coefficient first // tol = tolerance for iteration. Default: 1e-14 // Uses the Aberth method https://en.wikipedia.org/wiki/Aberth_method // // Dario Bini. "Numerical computation of polynomial zeros by means of Aberth's Method", Numerical Algorithms, Feb 1996. // https://www.researchgate.net/publication/225654837_Numerical_computation_of_polynomial_zeros_by_means_of_Aberth's_method function poly_roots(p,tol=1e-14,error_bound=false) = assert( is_vector(p), "Invalid polynomial." ) let( p = _poly_trim(p,eps=0) ) assert( p!=[0], "Input polynomial cannot be zero." ) p[len(p)-1] == 0 ? // Strip trailing zero coefficients let( solutions = poly_roots(list_head(p),tol=tol, error_bound=error_bound)) (error_bound ? [ [[0,0], each solutions[0]], [0, each solutions[1]]] : [[0,0], each solutions]) : len(p)==1 ? (error_bound ? [[],[]] : []) : // Nonzero constant case has no solutions len(p)==2 ? let( solution = [[-p[1]/p[0],0]]) // Linear case needs special handling (error_bound ? [solution,[0]] : solution) : let( n = len(p)-1, // polynomial degree pderiv = [for(i=[0:n-1]) p[i]*(n-i)], s = [for(i=[0:1:n]) abs(p[i])*(4*(n-i)+1)], // Error bound polynomial from Bini // Using method from: http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/0915-24.pdf beta = -p[1]/p[0]/n, r = 1+pow(abs(polynomial(p,beta)/p[0]),1/n), init = [for(i=[0:1:n-1]) // Initial guess for roots let(angle = 360*i/n+270/n/PI) [beta,0]+r*[cos(angle),sin(angle)] ], roots = _poly_roots(p,pderiv,s,init,tol=tol), error = error_bound ? [for(xi=roots) n * (norm(polynomial(p,xi))+tol*polynomial(s,norm(xi))) / abs(norm(polynomial(pderiv,xi))-tol*polynomial(s,norm(xi)))] : 0 ) error_bound ? [roots, error] : roots; // Internal function // p = polynomial // pderiv = derivative polynomial of p // z = current guess for the roots // tol = root tolerance // i=iteration counter function _poly_roots(p, pderiv, s, z, tol, i=0) = assert(i<45, str("Polyroot exceeded iteration limit. Current solution:", z)) let( n = len(z), svals = [for(zk=z) tol*polynomial(s,norm(zk))], p_of_z = [for(zk=z) polynomial(p,zk)], done = [for(k=[0:n-1]) norm(p_of_z[k])<=svals[k]], newton = [for(k=[0:n-1]) c_div(p_of_z[k], polynomial(pderiv,z[k]))], zdiff = [for(k=[0:n-1]) sum([for(j=[0:n-1]) if (j!=k) c_div([1,0], z[k]-z[j])])], w = [for(k=[0:n-1]) done[k] ? [0,0] : c_div( newton[k], [1,0] - c_mul(newton[k], zdiff[k]))] ) all(done) ? z : _poly_roots(p,pderiv,s,z-w,tol,i+1); // Function: real_roots() // 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 error can make their imaginary parts // large enough to appear non-zero. // Arguments: // p = polynomial to solve as coefficient list, highest power term first // eps = used to determine whether imaginary parts of roots are zero // tol = tolerance for the complex polynomial root finder function real_roots(p,eps=undef,tol=1e-14) = assert( is_vector(p), "Invalid polynomial." ) let( p = _poly_trim(p,eps=0) ) assert( p!=[0], "Input polynomial cannot be zero." ) let( roots_err = poly_roots(p,error_bound=true), roots = roots_err[0], err = roots_err[1] ) is_def(eps) ? [for(z=roots) if (abs(z.y)/(1+norm(z))<eps) z.x] : [for(i=idx(roots)) if (abs(roots[i].y)<=err[i]) roots[i].x]; // Section: Operations on Functions // Function: root_find() // 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: // f = function literal for a scalar-valued single variable function // x0 = endpoint of interval to search for root // x1 = second endpoint of interval to search for root // tol = tolerance for solution. Default: 1e-15 // Example(2D): Solve x*sin(x)=4 // f = function (x) x*sin(x)-4; // root=root_find(f, 0,25); // root = 15.2284 // // Graphical verification // stroke([for(x=[0:25]) [x,f(x)]],width=.2); // color("red")move([root,f(root)]) // circle(r=.25,$fn=16); // The algorithm is based on Brent's method and is a combination of // bisection and inverse quadratic approximation, where bisection occurs // at every step, with refinement using inverse quadratic approximation // only when that approximation gives a good result. The detail // of how to decide when to use the quadratic came from an article // by Crenshaw on "The World's Best Root Finder". // https://www.embedded.com/worlds-best-root-finder/ function root_find(f,x0,x1,tol=1e-15) = let( y0 = f(x0), y1 = f(x1), yrange = y0<y1 ? [y0,y1] : [y1,y0] ) // Check endpoints y0==0 || _rfcheck(x0, y0,yrange,tol) ? x0 : y1==0 || _rfcheck(x1, y1,yrange,tol) ? x1 : assert(y0*y1<0, "Sign of function must be different at the interval endpoints") _rootfind(f,[x0,x1],[y0,y1],yrange,tol); function _rfcheck(x,y,range,tol) = assert(is_finite(y), str("Function not finite at ",x)) abs(y) < tol*(range[1]-range[0]); // xpts and ypts are arrays whose first two entries contain the // interval bracketing the root. Extra entries are ignored. // yrange is the total observed range of y values (used for the // tolerance test). function _rootfind(f, xpts, ypts, yrange, tol, i=0) = assert(i<100, "root_find did not converge to a solution") let( xmid = (xpts[0]+xpts[1])/2, ymid = f(xmid), yrange = [min(ymid, yrange[0]), max(ymid, yrange[1])] ) _rfcheck(xmid, ymid, yrange, tol) ? xmid : let( // Force root to be between x0 and midpoint y = ymid * ypts[0] < 0 ? [ypts[0], ymid, ypts[1]] : [ypts[1], ymid, ypts[0]], x = ymid * ypts[0] < 0 ? [xpts[0], xmid, xpts[1]] : [xpts[1], xmid, xpts[0]], v = y[2]*(y[2]-y[0]) - 2*y[1]*(y[1]-y[0]) ) v <= 0 ? _rootfind(f,x,y,yrange,tol,i+1) // Root is between first two points, extra 3rd point doesn't hurt : let( // Do quadratic approximation B = (x[1]-x[0]) / (y[1]-y[0]), C = y*[-1,2,-1] / (y[2]-y[1]) / (y[2]-y[0]), newx = x[0] - B * y[0] *(1-C*y[1]), newy = f(newx), new_yrange = [min(yrange[0],newy), max(yrange[1], newy)], // select interval that contains the root by checking sign yinterval = newy*y[0] < 0 ? [y[0],newy] : [newy,y[1]], xinterval = newy*y[0] < 0 ? [x[0],newx] : [newx,x[1]] ) _rfcheck(newx, newy, new_yrange, tol) ? newx : _rootfind(f, xinterval, yinterval, new_yrange, tol, i+1); // vim: expandtab tabstop=4 shiftwidth=4 softtabstop=4 nowrap