Smooth movement to ascend through the atmosphere

Question

I\'m moving through the atmosphere with Microsoft Virtual Earth 3D and I can descend smoothly, but I don\'t know the math to ascend smoothly.

I\'m descending like this:<

Answer 1

An even better solution might be to use a function like the Logistic function.

Double minAlt = 0.0;
Double maxAlt = 500000.0;

Int32 numberSteps = 1000;

Double boundary = +6.0;

for (Int32 step = 0; step < numberSteps; step++)
{
   Double t = -boundary + 2.0 * boundary * step / (numberSteps - 1);
   Double correction = 1.0 / (1.0 + Math.Exp(Math.Abs(boundary)));
   Double value = 1.0 / (1.0 + Math.Exp(-t));
   Double correctedValue = (value - correction) / (1.0 - 2.0 * correction);
   Double curAlt = correctedValue * (maxAlt - minAlt) + minAlt;
}

Because the current altitude is explicitly calculated you do not have to rely on a iterative calculation introducing all sorts of precision releated errors.

See the sample code for how to tune the function shape.

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Here is a sample console application that displays the function. You can play a bit with the parameters to get a feeling for the behavior.

using System;

namespace LogisticFunction
{
    class Program
    {
        static void Main(string[] args)
        {
            Double minAlt = 5.0;
            Double maxAlt = 95.0;

            Int32 numberSteps = 60;

            // Keep maxAlt and numberSteps small if you don't want a giant console window.
            Console.SetWindowSize((Int32)maxAlt + 12, numberSteps + 1);

            // Positive values produce ascending functions.
            // Negative values produce descending functions.
            // Values with smaller magnitude produce more linear functions.
            // Values with larger magnitude produce more step like functions.
            // Zero causes an error.
            // Try for example +1.0, +6.0, +20.0 and -1.0, -6.0, -20.0
            Double boundary = +6.0;

            for (Int32 step = 0; step < numberSteps; step++)
            {
                Double t = -boundary + 2.0 * boundary * step / (numberSteps - 1);
                Double correction = 1.0 / (1.0 + Math.Exp(Math.Abs(boundary)));
                Double value = 1.0 / (1.0 + Math.Exp(-t));
                Double correctedValue = (value - correction) / (1.0 - 2.0 * correction);
                Double curAlt = correctedValue * (maxAlt - minAlt) + minAlt;

                Console.WriteLine(String.Format("{0, 10:N4} {1}", curAlt, new String('#', (Int32)Math.Round(curAlt))));
            }

            Console.ReadLine();
        }
    }
}