// AsinH function returns the inverse hyperbolic sine. Therefore: ASINH( SINH( z ) ) = z
func AsinH(number float64) (float64, error) {
if math.IsNaN(number) {
return 0.0, errors.New("#VALUE! - Occurred because the supplied number argument is non-numeric")
}
return math.Asinh(number), nil
}
func TestMathAsinh(t *testing.T) {
// This is just an interface to Go's function, so just a quick simple test
ctx := runtime.NewCtx(nil, nil)
mm := new(MathMod)
mm.SetCtx(ctx)
val := 0.12
ret := mm.math_Asinh(runtime.Number(val))
exp := math.Asinh(val)
if ret.Float() != exp {
t.Errorf("expected %f, got %f", exp, ret.Float())
}
}
// Asinh returns the inverse hyperbolic sine of x.
func Asinh(x complex128) complex128 {
// TODO check range
if imag(x) == 0 {
if math.Fabs(real(x)) > 1 {
return cmplx(math.Pi/2, 0) // DOMAIN error
}
return cmplx(math.Asinh(real(x)), 0)
}
xx := x * x
x1 := cmplx(1+real(xx), imag(xx)) // 1 + x*x
return Log(x + Sqrt(x1)) // log(x + sqrt(1 + x*x))
}
// Chebyshev biquad (2-poles) recursive coefficients
// Adapted from The Scientist and Engineer's Guide to Digital Signal Processing, Steven W. Smith
// poleIndex = [0, poleCount)
// percentRipple in the pass band can range from 0 for a butterworth to about 0.29. Something like 0.005 is a good trade-off.
func chebyshevBiquad(freq, percentRipple float64, poleIndex, poleCount int, highpass bool) (stageAs, stageBs []float64) {
// We start off by designing a low-pass filter with unity cut-off frequency
// Location of pole on unit circle, real and imaginary parts
// The maximally flat butterworth filter positions the poles so that
// they form a semi-circle on the left side of the s-plane (sigma < 0)
// The half offset keeps the poles evenly spaced and off of the sigma=0 line
// s-plane s = sigma + i * omega = poleR + i * poleI
poleI, poleR := math.Sincos((float64(poleIndex) + 0.5) * math.Pi / float64(poleCount))
poleR = -poleR
// The chebyshev filter uses an ellipse to move all of the poles closer to the sigma=0 line
// This causes pass-band ripple and sharpens the drop off
// Warp coordinates from being on a circle to an ellipse
if percentRipple != 0.0 {
e := math.Sqrt(1/((1-percentRipple)*(1-percentRipple)) - 1)
v := math.Asinh(1/e) / float64(poleCount)
k := math.Acosh(1/e) / float64(poleCount)
k = math.Cosh(k)
poleR = poleR * math.Sinh(v) / k
poleI = poleI * math.Cosh(v) / k
}
// bilinear s-domain to z-domain transformation
t := 2 * math.Tan(0.5)
w := 2 * math.Pi * freq
m := poleR*poleR + poleI*poleI
d := 4 - 4*poleR*t + m*t*t
x0 := t * t / d
x1 := 2 * t * t / d
x2 := t * t / d
y1 := (8 - 2*m*t*t) / d
y2 := (-4 - 4*poleR*t - m*t*t) / d
// We now have the coefficients of a low-pass filter with a cutoff frequency of 1 (2 times the nyquist)...
// We must now apply our desired frequency and convert to a high-pass filter if necessary
// as with the bilinear tranform, these are the results of a substitution in the transfer function...
var k float64
if highpass {
k = -math.Cos(w/2+0.5) / math.Cos(w/2-0.5)
} else {
k = math.Sin(0.5-w/2) / math.Sin(0.5+w/2)
}
d = 1 + (y1*k - y2*k*k)
a0 := (x0 - x1*k + x2*k*k) / d
a1 := (-2*x0*k + x1 + (x1*k*k - 2*x2*k)) / d
a2 := (x0*k*k - x1*k + x2) / d
b1 := (2*k + y1 + y1*k*k - 2*y2*k) / d
b2 := (-k*k - y1*k + y2) / d
if highpass {
a1, b1 = -a1, -b1
}
// we now have the desired coefficients of our low/high pass filter with the desired cutoff frequency
// however, the gain has not been normalized, if that is desired...
stageAs = []float64{a0, a1, a2}
stageBs = []float64{0, b1, b2}
return
}
func Gps2webmerc(lng, lat float64) (float64, float64) {
lat_rad := lat * math.Pi / 180
return lng, math.Asinh(math.Tan(lat_rad)) * 180 / math.Pi
}
// Inverse transverse mercator projection: Projection of an cylinder onto the surface of
// of an ellipsoid. Also known as reverse Gauss-Krüger projection. Input parameters:
//
// pt *GeoPoint: Easting(Y) and Northing(X) of map point to be projected; in meters
// latO, longO: Shifted origin of latitude and longitude in decimal degrees
// fe, fn: False easting and northing respectively in meters
// scale: Projection scaling; Dimensionless, typically 1 or little bellow
//
// This algorithm uses the algorithm described by Redfearn
// http://en.wikipedia.org/wiki/Transverse_Mercator:_Redfearn_series
//
// Taken from "OGP Publication 373-7-2 – Surveying and Positioning Guidance Note number 7, part 2 – November 2010",
// pp. 48 - 51
//
// More accurate, iterative but slower algorithmic implementation
func InverseTransverseMercator(pt *GeoPoint, latO, longO, scale, fe, fn float64) *PolarCoord {
var gc PolarCoord
el := pt.El
latOrad := degtorad(latO)
longOrad := degtorad(longO)
f := 1 - el.b/el.a
esq := math.Sqrt(2.0*f - f*f)
n := f / (2.0 - f)
B := (el.a / (1 + n)) * (1 + n*n/4.0 + n*n*n*n/64.0)
var SO float64
if latOrad != 0.0 {
h1 := n/2.0 - (2.0/3.0)*n*n + (5.0/16.0)*n*n*n + (41.0/180.0)*n*n*n*n
h2 := (13.0/48.0)*n*n - (3.0/5.0)*n*n*n + (557.0/1440.0)*n*n*n*n
h3 := (61.0/240.0)*n*n*n - (103.0/140.0)*n*n*n*n
h4 := (49561.0 / 161280.0) * n * n * n * n
QO := math.Asinh(math.Tan(latOrad)) - (esq * math.Atanh(esq*math.Sin(latOrad)))
bO := math.Atan(math.Sinh(QO))
xiO0 := bO // math.Asin(math.Sin(bO))
xiO1 := h1 * math.Sin(2.0*xiO0)
xiO2 := h2 * math.Sin(4.0*xiO0)
xiO3 := h3 * math.Sin(6.0*xiO0)
xiO4 := h4 * math.Sin(8.0*xiO0)
xiO := xiO0 + xiO1 + xiO2 + xiO3 + xiO4
SO = B * xiO
}
h1i := n/2.0 - (2.0/3.0)*n*n + (37.0/96.0)*n*n*n - (1.0/360.0)*n*n*n*n
h2i := (1.0/48.0)*n*n + (1.0/15.0)*n*n*n - (437.0/1440.0)*n*n*n*n
h3i := (17.0/480.0)*n*n*n - (37.0/840.0)*n*n*n*n
h4i := (4397.0 / 161280.0) * n * n * n * n
etai := (pt.X - fe) / (B * scale)
xii := ((pt.Y - fn) + scale*SO) / (B * scale)
xi1i := h1i * math.Sin(2*xii) * math.Cosh(2*etai)
xi2i := h2i * math.Sin(4*xii) * math.Cosh(4*etai)
xi3i := h3i * math.Sin(6*xii) * math.Cosh(6*etai)
xi4i := h4i * math.Sin(8*xii) * math.Cosh(8*etai)
eta1i := h1i * math.Cos(2*xii) * math.Sinh(2*etai)
eta2i := h2i * math.Cos(4*xii) * math.Sinh(4*etai)
eta3i := h3i * math.Cos(6*xii) * math.Sinh(6*etai)
eta4i := h4i * math.Cos(8*xii) * math.Sinh(8*etai)
xi0i := xii - (xi1i + xi2i + xi3i + xi4i)
eta0i := etai - (eta1i + eta2i + eta3i + eta4i)
bi := math.Asin(math.Sin(xi0i) / math.Cosh(eta0i))
Qi := math.Asinh(math.Tan(bi))
Qiiold := Qi + (esq * math.Atanh(esq*math.Tanh(Qi)))
Qii := Qi + (esq * math.Atanh(esq*math.Tanh(Qiiold)))
for math.Abs(Qiiold-Qii) > 1e-12 {
Qiiold = Qii
Qii = Qi + (esq * math.Atanh(esq*math.Tanh(Qiiold)))
}
gc.Latitude = radtodeg(math.Atan(math.Sinh(Qii)))
gc.Longitude = radtodeg(longOrad + math.Asin(math.Tanh(eta0i)/math.Cos(bi)))
gc.El = el
return &gc
}
// Direct transverse mercator projection: Projection of an ellipsoid onto the surface of
// of a cylinder. Also known as Gauss-Krüger projection. Input parameters:
//
// gc *PolarCoord: Latitude and Longitude or point to be projected; in decimal degrees
// latO, longO: Shifted origin of latitude and longitude in decimal degrees
// fe, fn: False easting and northing respectively in meters
// scale: Projection scaling; Dimensionless, typically 1 or little bellow
//
// This algorithm uses the algorithm described by Redfearn
// http://en.wikipedia.org/wiki/Transverse_Mercator:_Redfearn_series
//
// Taken from "OGP Publication 373-7-2 – Surveying and Positioning Guidance Note number 7, part 2 – November 2010",
// pp. 48 - 51
func DirectTransverseMercator(gc *PolarCoord, latO, longO, scale, fe, fn float64) *GeoPoint {
var pt GeoPoint
el := gc.El
latOrad := degtorad(latO)
longOrad := degtorad(longO)
latrad := degtorad(gc.Latitude)
longrad := degtorad(gc.Longitude)
f := 1 - el.b/el.a
esq := math.Sqrt(2.0*f - f*f)
n := f / (2.0 - f)
B := (el.a / (1 + n)) * (1 + n*n/4.0 + n*n*n*n/64.0)
h1 := n/2.0 - (2.0/3.0)*(n*n) + (5.0/16.0)*(n*n*n) + (41.0/180.0)*(n*n*n*n)
h2 := (13.0/48.0)*(n*n) - (3.0/5.0)*(n*n*n) + (557.0/1440.0)*(n*n*n*n)
h3 := (61.0/240.0)*(n*n*n) - (103.0/140.0)*(n*n*n*n)
h4 := (49561.0 / 161280.0) * (n * n * n * n)
var SO float64
if latOrad != 0.0 {
QO := math.Asinh(math.Tan(latOrad)) - (esq * math.Atanh(esq*math.Sin(latOrad)))
bO := math.Atan(math.Sinh(QO))
xiO0 := bO // math.Asin(math.Sin(bO))
xiO1 := h1 * math.Sin(2.0*xiO0)
xiO2 := h2 * math.Sin(4.0*xiO0)
xiO3 := h3 * math.Sin(6.0*xiO0)
xiO4 := h4 * math.Sin(8.0*xiO0)
xiO := xiO0 + xiO1 + xiO2 + xiO3 + xiO4
SO = B * xiO
}
Q := math.Asinh(math.Tan(latrad)) - (esq * math.Atanh(esq*math.Sin(latrad)))
b := math.Atan(math.Sinh(Q))
eta0 := math.Atanh(math.Cos(b) * math.Sin(longrad-longOrad))
xi0 := math.Asin(math.Sin(b) * math.Cosh(eta0))
xi1 := h1 * math.Sin(2*xi0) * math.Cosh(2*eta0)
xi2 := h2 * math.Sin(4*xi0) * math.Cosh(4*eta0)
xi3 := h3 * math.Sin(6*xi0) * math.Cosh(6*eta0)
xi4 := h4 * math.Sin(8*xi0) * math.Cosh(8*eta0)
xi := xi0 + xi1 + xi2 + xi3 + xi4
eta1 := h1 * math.Cos(2*xi0) * math.Sinh(2*eta0)
eta2 := h2 * math.Cos(4*xi0) * math.Sinh(4*eta0)
eta3 := h3 * math.Cos(6*xi0) * math.Sinh(6*eta0)
eta4 := h4 * math.Cos(8*xi0) * math.Sinh(8*eta0)
eta := eta0 + eta1 + eta2 + eta3 + eta4
pt.X = fe + scale*B*eta
pt.Y = fn + scale*(B*xi-SO)
pt.El = el
return &pt
}
// Asinh returns the inverse hyperbolic sine of x.
//
// Special cases are:
// Asinh(±0) = ±0
// Asinh(±Inf) = ±Inf
// Asinh(NaN) = NaN
func Asinh(x float32) float32 {
return float32(math.Asinh(float64(x)))
}
func (m *MathMod) math_Asinh(args ...runtime.Val) runtime.Val {
runtime.ExpectAtLeastNArgs(1, args)
return runtime.Number(math.Asinh(args[0].Float()))
}
func FunctionVal(f Function, x float64) float64 {
//
switch f {
case Floor:
i, _ := math.Modf(x)
return i
case Fract:
_, f := math.Modf(x)
return f
case Chs:
return -x
case Rec:
if x == 0 { // TODO
return math.NaN()
}
return 1 / x
case Sqr:
return x * x
case Sqrt:
return math.Sqrt(x)
case Exp:
return math.Exp(x)
case Exp10:
return math.Exp(x * math.Ln10)
case Exp2:
return math.Exp(x * math.Ln2)
case Log:
return math.Log(x)
case Lg:
return math.Log10(x)
case Ld:
return math.Log2(x)
case Sin:
return math.Sin(x)
case Cos:
return math.Cos(x)
case Tan:
return math.Tan(x)
case Cot:
return 1 / math.Tan(x)
case Arcsin:
return math.Asin(x)
case Arccos:
return math.Acos(x)
case Arctan:
return math.Atan(x)
case Arccot:
return math.Atan(x)
case Sinh:
return math.Sinh(x)
case Cosh:
return (math.Exp(x) + math.Exp(-x)) / 2
case Tanh:
return math.Tanh(x)
case Coth:
return (math.Exp(x) + math.Exp(-x)) / (math.Exp(x) - math.Exp(-x))
case Arsinh:
return math.Asinh(x)
case Arcosh:
return math.Log(x + math.Sqrt(x*x-1))
case Artanh:
return math.Atanh(x)
case Arcoth:
return math.Log((x+1)/(x-1)) / 2
case Gamma:
return math.Gamma(x)
}
return math.NaN()
}