Hyperbolic Arch Calculator
(x/a) ² – (y/b) ² = – 1
2x ... Arch Width =
at center ... Arch Height =

=
– ( b + h ) ... y =
on Major Axis ... Foci =
Arch Length =

Hyperbolic Arch: Major Axis of Hyperbola on X-Axis
Hyperbolic Arch: Major Axis of Hyperbola on Y-Axis

Hyperbolic Axis Calculations
(x/a) ² – (y/b) ² = – 1
Arch Height = h       Arch Width = w

Given : Semi-Axis on the y-axis = b
The equation of the Ellipse is written as:
b ² x ² – a ² y ² = – a ² b ²   , where x = w / 2   and   y = b + h
Therefore : b ² x ² = a ² y ² – a ² b ²
Collecting terms : b ² x ² = a ² ( y ² – b ² )
Dividing both sides of the equation by ( y ² – b ² ) :
a ² = b ² x ² / ( y ² – b ² )
Taking the square root of both sides of the equation :
a = bx / Ö ( y ² – b ² )   , and the axis on the x-axis = 2a

Given : Semi-Axis on the x-axis = a
The equation of the Ellipse is written as:
b ² x ² – a ² y ² = – a ² b ²   , where x = w / 2   and   b = y – h
Transposing the term from the right side :
a ² y ² – b ² x ² – a ² b ² = 0
Collecting like terms :
a ² y ² – b ² ( a ² + x ² ) = 0
Dividing by ( a ² + x ² ) :
y ² a ² / ( a ² + x ² ) – b ² = 0
Substituting for b to express the equation
in terms of y and h :
y ² a ² / ( a ² + x ² ) – ( y – h ) ² = 0
Expanding the term ( y – h ) ² :
y ² a ² / ( a ² + x ² ) – ( y ² – 2 y h + h ² ) = 0
Removing the parentheses :
y ² a ² / ( a ² + x ² ) – y ² + 2 y h – h ² = 0
Collecting terms :
y ² [ a ² / ( a ² + x ² ) – 1 ] + y 2 h – h ² = 0
The equation is quadratic in y, where :
A = a ² / ( a ² + x ² ) – 1     B = 2 h     C = – h ²
Substituting in the General Quadratic Equation :
[ – B ± Ö ( B ² – 4 A C ) ] / 2A returns the value of y
Therefore b = y – h , and the axis on the y-axis = 2b

Although only one value for b is returned by the calculator,
due to the ± sign in the General Quadratic Equation there
are two possible solutions.
The calculator returns values based on the negative value of the discriminant.

Given : Equal Semi-Axes
The equation of the Ellipse is written as:
a ² x ² – a ² y ² = – a ² a ²   , where x = w / 2   and   y = a + h
Therefore : a ² = y ² – x ²
Substituting for x and y :
a ² = ( a + h ) ² – (w / 2 ) ²
Expanding the terms :
a ² = a ² + 2 a h + h ² – w ² / 4
Transposing terms and multiplying all terms by 4 :
4 a h = w ² – 4 h ²
Dividing both sides by 4 h :
a = ( w ² – 4 h ² ) / 4 h

Joe Bartok