|
Elliptic 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 b ² – a ² y ² Collecting terms : b ² x ² = a ² ( b ² – y ² ) Dividing both sides of the equation by ( b ² – y ² ) : a ² = b ² x ² / ( b ² – y ² ) Taking the square root of both sides of the equation : a = bx / Ö ( b ² – y 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 negative value of the discriminant is discarded; the result is illustrated below. |
