Question

I’m making just a basic application that just writes pixels along a curve in C#.

I came across this website with a formula that looks promising. I believe this website is also talking about the same thing here.

What I don’t really understand is how to implement it. I tried looking at the JavaScript code on the first link but I can’t really tell what data I need to supply. The things involving the PVC, PVI, or PVT are the things I don’t understand.

The example situation I’m going to set up is just both of the grades (vertical incline/decline) is just 5 and -5. Let’s say point 1 is at 0, 0 and point 2 is 100, 100.

Can someone explain some of the obscure variables in the formula and how would I use the formula to draw the curve?

Answer 1

Generally, to draw a curve in 2D you vary one parameter, and then collect x,y point pairs, and plot the pairs. In your case it will work to just vary the horizontal distance (x), and then collect the corresponding y-values, and then you can plot these.

As for the formula, it is very unclear. Basically it’s just a parabola with a bunch of (poorly defined) jargon around it. To graph this, you want to vary x from 0 to L (this isn’t obvious, btw, I had to work out the math, i.e., how to vary x so that the slopes would be as they suggest in the figure, anyway, it’s 0 to L, and they should have said so).

I don’t have C# running now, but hopefully you can translate this Python code:

from matplotlib.pyplot import plot, show
from numpy import arange

G1 = .1 # an initial slope (grade) of 10%  (note that one never uses percentages directly in calculations, it's always %/100)
G2 = -.02 # a final slope (grade) of 2%
c = 0    # elevation (value of curve when x=0, that is, y at PVC
L = 10.   # the length of the curve in whatever unit you want to use (ft, m, mi, etc), but this basically sets your unit system
N = 1000  # I'm going to calculate and plot 100 points to illustrate this curve

x = arange(0, L, float(L)/N)  # an array of N x values from 0 to (almost) L

# calculate the curve
a = (G2-G1)/(2*L)
b = G1
y = a*x*x + b*x + c  # this is shorthand for a loop y[0]=a*x[0]*x[0] + b*...

plot(x, y)    
show()

print (y[1]-y[0])/(x[1]-x[0]), (y[-1]-y[-2])/(x[-1]-x[-2])

The final line prints the initial and final slopes as a check (in Python neg indexing counts from the back of the array), and this match what I specified for G1 and G2. The plot looks like:

As for your requests: “The example situation I’m going to set up is just both of the grades (vertical incline/decline) is just 5 and -5. Let’s say point 1 is at 0, 0 and point 2 is 100, 100.”, in a parabola you basically get three free parameters (corresponding to a, b, and c), and here, I think, you over-specified it.

What are PVC, PVT, and PVI? PVC: the starting point, so Y_PVC is the height of the starting point. PVT: the ending point. PVI: if you draw a line from PVC at the initial slope G1 (ie the tangent to the curve on the left), and similarly from PVT, the point where they intersect is called PVI (though why someone would ever care about this point is beyond me).