Slope-Polarized Distance Parametric Graph System

[ganondorfchampin]

In case you don't know Polar is parametric graphing system based around angle (θ) and radius (r) and is represented by the equation r= f(θ). Polar graphs each point the distance away from the origin, called the pole, for each angle. To convert a polar equation to parametric form you use the formula x = rcosθ and y = sinθ. θ will then act as the parameter is the system of parametric equations, and it will graph the polar equation.

The only other commonly used parametric system is Rectangular, represented by y=f(x) and a point is graphed at the proper height for each x value, and the parametric form is simply x = t and y = f(t), where t is the parameter.

I have created a third parametric graphing system similar to Polar, only instead of being based around angle and radius it is based around slope (m) and polarized distance (P) and is represented by P = f(m). Normally distance is only non-negative, so polarized distance is polarized in that it can be any real number, and the sign is determined by whether the point is to the left, where it would be negative, or right of the origin. If the point is on the y-axis the sign is determined whether it is above, where it would be positive, or below the origin. I have derived the parametric form to be x = (Psqrt(1+m^2))/(1+m^2) and y = (sgn(m)Psqrt(1+(1/m^2)))/(1+(1/m^2)), where sqrt is the square root and sgn is the sign function (sgn(x)=|x|/x). If you graph P = m correctly you should get what looks like a parabola, but it is bounded between x = -1 and x = 1.

[Fringe Pioneer]

Of what is m the slope? In Cartesian coordinates, x is the horizontal distance from the origin and y is the vertical distance from the origin. In polar coordinates, θ is the angle from the vector i (which is commonly expressed as <1, 0> in two dimensions or <1, 0, 0> in three dimensions) and r is the distance from the point to the origin.

How do you define m? Do you presuppose that there already exists a function from which to get the slope, or do you define it relative to an origin, or do you define m in another way?

[ganondorfchampin]

m is the slope of the line connecting the point to the origin, so it is y/x. m is the input value, like how θ and x are the input values.

[Fringe Pioneer]

Ok, so it looks like you have something of a hybrid between Cartesian and polar coordinates, where the dependent variable P is essentially equal to r and the independent variable m is equal to tan(θ).

I do have one question: how would you specify the point represented in the Cartesian coordinate system as (0, y) or in the polar coordinate system as (π/2, r)? In both cases, m = y/x = tan(θ) is undefined...

[ganondorfchampin]

I guess you just use some sort of symbol to denote 1/0 or something. You can't really denote those points using just real numbers, you either need to use an extension, or a function.