About this parabola solver whatchamacallit
This is a shaggy hunk of Ruby code I wrote for playing with basic quadratic equations.
parabola.rb
This contains a Ruby class, Parabola, that encapsulates the features of a basic quadratic equation a*x^2 + b*x + c. Objects can be instantiated by providing the a, b, and c coefficients directly:
p = Parabola.new(2, 3, 4)
They can also be instantiated by providing a set of three Cartesian coordinates all at once. For example, to provide the coordinates [1, 1], [2, 3], and [7, 4]:
p = Parabola.from_3_points(1, 1, 2, 3, 7, 4)
# => #<Parabola:0x007fb71a91c020 @a=-0.3, @b=2.9, @c=-1.6>
Once instantiated, the object can be used to find any point on the curve given x or y:
p.y_for_x(2) # => 2.9999999999999996
p.x_for_y(4) # => [2.6666666666666665, 7.000000000000001]
p.x_for_y(6) # => raises Math::DomainError
Note that #x_for_y returns an array of two values for x, which will always
be in increasing order.
This is what happens when you give two or three coordinates with the same x value:
p = Parabola.from_3_points(1, 1, 2, 8, 1, 3)
# => #<Parabola:0x007fb719c50328 @a=-Infinity, @b=Infinity, @c=NaN>
parabola_solver.rb
This is just spaghetti code that I used to work out and check my work on the
math in parabola.rb. It has little practical use, but I kept it. It
generates random output that looks like this:
x = 10
y = -3
a*x^2 + b*x + c = y
(1) a*100 + b*10 + c = -3
x = -3
y = -3
a*x^2 + b*x + c = y
(2) a*9 + b*-3 + c = -3
x = 3
y = -7
a*x^2 + b*x + c = y
(3) a*9 + b*3 + c = -7
Subtracting (2) from (1) and (3) from (2) creates two new equations
that eliminate c...
(4) a*91.0 + b*13.0 = 0.0
(5) a*0.0 + b*-6.0 = 4.0
Solving for b in (4)...
a*7.0 + b*1.0 = 0.0
(6) b = 0.0 + a*-7.0
Then replacing b in (5)...
a*0.0 + (0.0 + a*-7.0) * -6.0 = 4.0
a*0.0 + -0.0 + a*42.0 = 4.0
a*0.0 + a*42.0 = 4.0
a*42.0 = 4.0
a = 4.0 / 42.0
(7) a = 0.09523809523809523
Use (6) to find b...
b = 0.0 + 0.09523809523809523 * -7.0
(8) b = -0.6666666666666666
Use (1) to find c...
0.09523809523809523*100.0 + -0.6666666666666666*10.0 + c = -3.0
9.523809523809524 + -6.666666666666666 + c = -3.0
c = -3.0 - (9.523809523809524 + -6.666666666666666)
(9) c = -5.857142857142858
Apologia
If this looks like the work of someone with a shaky grasp of basic algebra, that's because it is. The last time I remember playing with quadratic equations was roughly three decades ago in my high school's "College Algebra" course.
Author
Lawrence Leonard Gilbert (2015-08-15)