Are hyperbolic trigonometric functions used to calculate hyperbolic orbits?

Question

The following comment is really intriguing!

I would say, "Yes, The equations for going from Mean Anomaly to Eccentric Anomaly to True Anomaly are indeed different for hyperbolic orbits than for elliptical ones, if that's part of your process." The biggest differences are sign-flipping on some of the terms, and the use of hyperbolic trigonometric functions rather than the circular trig functions.

Question: Are hyperbolic trigonometric functions used in calculating hyperbolic orbits? If so, how?

Update: I just found this answer that I wrote a while ago, which was triggered by this answer

Answer 1

The equations for the position in a hyperbolic trajectory contain the hyperbolic sine, cosine and tangent.

A hyperbola is defined by the equation:

$$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$

It can be described by several parametric equations:

Using the hyperbolic sine and cosine functions, (1), plot cyan: $$ \boxed{x = \pm a \cosh(t) \\y = b \sinh(t) \\ t\in\mathbb{R} }$$

Using the complex exponential function, (2), plot magenta: $$ \boxed{z = c e^t + \overline{c} e^{-t} \\ c = \frac{a + i b}{2} \\ \overline{c} = \frac{a - i b}{2} \\ t\in\mathbb{R} }$$

Solving the definition for x, (3), plot blue: $$ \boxed{x = a \sqrt{\frac{y^2}{b^2} + 1} \\ y\in\mathbb{R} }$$

Solving the definition for y, (4), plot green: $$ \boxed{y = b \sqrt{\frac{x^2}{a^2} - 1} \\ x \geq a , x \leq -a }$$

Using cosine and tangent, (5), plot yellow: $$ \boxed{x = \frac{a} {\cos(t)} = a \sec(t) \\y = b \tan(t) \\ 0 \leq t \leq 2\pi \\ t \neq \frac{\pi}{2} , t \neq \frac{3\pi}{2} }$$

Using a rational parametric equation, (6), plot red: $$ \boxed{x = \pm a \frac{t^2 + 1}{2t} \\y = b \frac{t^2 - 1}{2t} \\ t\in\mathbb{R}, t > 0 }$$

Using sine and cosine with complex arguments, (7), plot grey: $$ \boxed{z = a \cos(it) + b \sin(it) \\ t\in\mathbb{R} }$$

I found no documentation about complex arguments for the Python Numpy sin and cos functions but it simply works perfect.

The equation (7) looks similar to: $$ \boxed{z = a \cos(t) + ib \sin(t) \\ 0 \leq t \leq 2\pi }$$ used to calculate an ellipse or circle.

import matplotlib.pyplot as plt
import numpy as np
import math as math
#
def check(x,y,a,b,eps):
    a2 = np.square(a)
    b2 = np.square(b)
    res = np.square(x)/a2 - np.square(y)/b2
    test = True
    lowlim = 1.0-eps
    highlim = 1.0+eps
    for i in range(len(res)):
        if res[i] < lowlim or res[i] > highlim : test = False
    return test
#
omega = np.pi*0.5
steps = 15
#
# 1: using hyperbolic sine and cosine, plot cyan
a = 1.0
b = 1.0
eps = 1E-13
t1 = np.linspace(-omega, omega, steps)
x1 = a*np.cosh(t1)
y1 = b*np.sinh(t1)
plt.plot(x1, y1, color='c', marker="x")   
print('cosh sinh check ', check(x1, y1, a, b, eps))
#
# 2: using complex exponential function, plot magenta
a = 1.2
c = (a + b*1j)*0.5
ck = (a - b*1j)*0.5
z2 = c*np.exp(t1) + ck*np.exp(-t1)
plt.plot(np.real(z2), np.imag(z2), color='m', marker="x")   
print('complex exp check ', check(np.real(z2), np.imag(z2), a, b, eps))
#
# 3: solving equation for x, plot blue
ymin = min(y1)
ymax = max(y1)
a = 1.4
a2 = np.square(a)
b2 = np.square(b)
y3 = np.linspace(ymin, ymax, steps)
x3 = a*np.sqrt(np.square(y3)/b2 + 1.0)
plt.plot(x3, y3, color='b', marker="x")
print('normal form y check ', check(x3, y3, a, b, eps))

# 4: solving equation for y, plot green
a = 1.6
a2 = np.square(a)
xmin = a
xmax = a*np.sqrt(np.square(ymax)/b2 + 1.0)
x4 = np.linspace(xmin, xmax, steps//2)
y4 = b*np.sqrt(np.square(x4)/a2 - 1.0)
x4 = np.concatenate((np.flip(x4, 0), x4), axis=None)
y4 = np.concatenate((np.flip(-y4, 0), y4), axis=None)
plt.plot(x4, y4, color='g', marker="x")
print('normal form x check ', check(x4, y4, a, b, eps))

# 5: using cosine and tangent functions, plot yellow
a = 1.8
tmax = np.arctan(ymax/b)
t5 = np.linspace(-tmax, tmax, steps)
x5 = a/np.cos(t5)
y5 = b*np.tan(t5)
plt.plot(x5, y5, color='y', marker="x")   
print('cos tan check ', check(x5, y5, a, b, eps))

# 6: using parametric equation, plot red
a = 2.0
tmin = ymax/b + np.sqrt(np.square(ymax/b) + 1.0)
#t6 = np.geomspace(tmin, 1.0, steps//2)
t6 = np.linspace(tmin, 1.0, steps//2)
x6 = a*(np.square(t6) + 1.0)/(2.0*t6)
xmax = max(x6)
y6 = b*(np.square(t6) - 1.0)/(2.0*t6)
x6 = np.concatenate((x6, np.flip(x6, 0)), axis=None)
y6 = np.concatenate((y6, np.flip(-y6, 0)), axis=None)
plt.plot(x6, y6, color='r', marker="x")   
print('t square check ', check(x6, y6, a, b, eps))

# 7: using sine and cosine with complex arguments, plot grey
a = 2.2
t7 = np.linspace(-omega*1j, omega*1j, steps)
z7 = a*np.cos(t7) + b*np.sin(t7)
plt.plot(np.real(z7), np.imag(z7), color='grey', marker="x")   
print('cos sin check ', check(np.real(z7), np.imag(z7), a, b, eps))

plt.grid(b=None, which='both', axis='both')
plt.axis('scaled')
plt.xlim(0.0,  math.ceil(xmax+0.5))
plt.ylim(math.floor(ymin), math.ceil(ymax))
plt.show()