Blog Entry © Tuesday, May 27, 2025, Translation of a Siacci’s Method C# App to C/C++ by James Pate Williams, Jr., BA, BS, MSWE, PhD

Blog Entry © Friday, May 23, 2025, NASA Geopotential Altitude and the Variation of the Gravitational Acceleration by James Pate Williams, Jr., BA, BS, Master of Software Engineering, PhD Computer Science

More source code is available upon request.
Reference: 19770009539.pdf
Win32 C/C++ application output to a Win32 window:
H Z1 Z2 Z3 g
80000 79005.7 79005.7 79005.7 9.56735
80500 79493.3 79493.3 79493.3 9.56590
81000 79980.9 79980.9 79980.9 9.56446
81500 80468.3 80468.3 80468.3 9.56301
82000 80955.7 80955.7 80955.7 9.56156
82500 81443.0 81443.0 81443.0 9.56011
83000 81930.2 81930.2 81930.2 9.55867
83500 82417.4 82417.4 82417.4 9.55722
84000 82904.5 82904.5 82904.5 9.55577
84500 83391.5 83391.5 83391.5 9.55433
85000 83878.4 83878.4 83878.4 9.55288
85500 84365.3 84365.3 84365.3 9.55144
H Geopotential Altitude
Z1 Geopotential Gauss-Legendre 128 Steps
Z2 Geopotential Simpson's Rule 256 Steps
Z3 Geopotential Trapezoidal Rule 512 Steps
g gravitational acceleration in m/s/s
https://ntrs.nasa.gov/api/citations/19770009539/downloads/19770009539.pdf
See Table 8. Page 9
// NASAGeopotentialAltitude.cpp : Defines the entry point for the application.
// Copyright (c) Friday, May 23, 2025 by James Pate Williams, Jr.
// BA, BS, MSWE, PhD All Applicable Rights Reserved

#include "stdafx.h"
#include "GeopotentialAltitude.h"
#include "Integration1d.h"
#include "NASAGeopotentialAltitude.h"
#include <math.h>
#include <stdio.h>
#include <tchar.h>
#include <string.h>
#include <vector>
using namespace std;

#define MAX_LOADSTRING 100

// Global Variables:
HINSTANCE hInst;								// current instance
TCHAR szTitle[MAX_LOADSTRING];					// The title bar text
TCHAR szWindowClass[MAX_LOADSTRING];			// the main window class name

// Forward declarations of functions included in this code module:
ATOM				MyRegisterClass(HINSTANCE hInstance);
BOOL				InitInstance(HINSTANCE, int);
LRESULT CALLBACK	WndProc(HWND, UINT, WPARAM, LPARAM);
INT_PTR CALLBACK	About(HWND, UINT, WPARAM, LPARAM);

int APIENTRY _tWinMain(HINSTANCE hInstance,
                     HINSTANCE hPrevInstance,
                     LPTSTR    lpCmdLine,
                     int       nCmdShow)
{
	UNREFERENCED_PARAMETER(hPrevInstance);
	UNREFERENCED_PARAMETER(lpCmdLine);

 	// TODO: Place code here.
	MSG msg;
	HACCEL hAccelTable;

	// Initialize global strings
	LoadString(hInstance, IDS_APP_TITLE, szTitle, MAX_LOADSTRING);
	LoadString(hInstance, IDC_NASAGEOPOTENTIALALTITUDE, szWindowClass, MAX_LOADSTRING);
	MyRegisterClass(hInstance);

	// Perform application initialization:
	if (!InitInstance (hInstance, nCmdShow))
	{
		return FALSE;
	}

	hAccelTable = LoadAccelerators(hInstance, MAKEINTRESOURCE(IDC_NASAGEOPOTENTIALALTITUDE));

	// Main message loop:
	while (GetMessage(&msg, NULL, 0, 0))
	{
		if (!TranslateAccelerator(msg.hwnd, hAccelTable, &msg))
		{
			TranslateMessage(&msg);
			DispatchMessage(&msg);
		}
	}

	return (int) msg.wParam;
}

//
//  FUNCTION: MyRegisterClass()
//
//  PURPOSE: Registers the window class.
//
//  COMMENTS:
//
//    This function and its usage are only necessary if you want this code
//    to be compatible with Win32 systems prior to the 'RegisterClassEx'
//    function that was added to Windows 95. It is important to call this function
//    so that the application will get 'well formed' small icons associated
//    with it.
//
ATOM MyRegisterClass(HINSTANCE hInstance)
{
	WNDCLASSEX wcex;

	wcex.cbSize = sizeof(WNDCLASSEX);

	wcex.style			= CS_HREDRAW | CS_VREDRAW;
	wcex.lpfnWndProc	= WndProc;
	wcex.cbClsExtra		= 0;
	wcex.cbWndExtra		= 0;
	wcex.hInstance		= hInstance;
	wcex.hIcon			= LoadIcon(hInstance, MAKEINTRESOURCE(IDI_NASAGEOPOTENTIALALTITUDE));
	wcex.hCursor		= LoadCursor(NULL, IDC_ARROW);
	wcex.hbrBackground	= (HBRUSH)(COLOR_WINDOW+1);
	wcex.lpszMenuName	= MAKEINTRESOURCE(IDC_NASAGEOPOTENTIALALTITUDE);
	wcex.lpszClassName	= szWindowClass;
	wcex.hIconSm		= LoadIcon(wcex.hInstance, MAKEINTRESOURCE(IDI_SMALL));

	return RegisterClassEx(&wcex);
}

//
//   FUNCTION: InitInstance(HINSTANCE, int)
//
//   PURPOSE: Saves instance handle and creates main window
//
//   COMMENTS:
//
//        In this function, we save the instance handle in a global variable and
//        create and display the main program window.
//
BOOL InitInstance(HINSTANCE hInstance, int nCmdShow)
{
   HWND hWnd;

   hInst = hInstance; // Store instance handle in our global variable

   hWnd = CreateWindow(szWindowClass, szTitle, WS_OVERLAPPEDWINDOW,
      CW_USEDEFAULT, 0, CW_USEDEFAULT, 0, NULL, NULL, hInstance, NULL);

   if (!hWnd)
   {
      return FALSE;
   }

   ShowWindow(hWnd, nCmdShow);
   UpdateWindow(hWnd);

   return TRUE;
}

//
//  FUNCTION: WndProc(HWND, UINT, WPARAM, LPARAM)
//
//  PURPOSE:  Processes messages for the main window.
//
//  WM_CREATE   - creates a multiple line edit control
//  WM_COMMAND	- process the application menu
//  WM_PAINT	- Paint the main window
//  WM_DESTROY	- post a quit message and return
//
//

#define ID_EDITCHILD 100
char asciiData[16384];
char asciiLine[16384];
TCHAR textData[16384];

void ConvertCharToTChar(const char* charArray, TCHAR* tcharArray, size_t tcharSize) {
#ifdef _UNICODE
	MultiByteToWideChar(CP_ACP, 0, charArray, -1, tcharArray, (int)tcharSize);
#else
    strncpy(tcharArray, charArray, tcharSize - 1);
    tcharArray[tcharSize - 1] = '\0'; // Ensure null-termination
#endif
}

LRESULT CALLBACK WndProc(HWND hWnd, UINT message, WPARAM wParam, LPARAM lParam)
{
	int wmId, wmEvent;
	PAINTSTRUCT ps;
	HDC hdc;
	static HWND hwndEdit;
	static double zArray[] = {
		80000, 80500, 81000, 81500, 82000, 82500,
		83000, 83500, 84000, 84500, 85000, 85500 };
	static vector<double> z(12);
	static vector<double> potential1(12);
	static vector<double> potential2(12);
	static vector<double> potential3(12);
	static vector<double> potential4(12);
	size_t numberPoints = z.size();

	for (size_t i = 0; i < z.size(); i++)
	{
		z[i] = zArray[i];
	}

	switch (message)
	{
	case WM_CREATE:
	GeopotentialAltitude::GaussianLegendre(
		numberPoints, z, potential1);
	GeopotentialAltitude::SimpsonsRule(
		numberPoints, z, potential2);
	GeopotentialAltitude::TrapezoidalRule(
		numberPoints, z, potential3);

	strcpy_s(asciiData, 16384, "H\tZ1\tZ2\tZ3\tg\r\n");

	for (size_t i = 0; i < z.size(); i++)
	{
		double zi = z[i];
		double p1 = potential1[i];
		double p2 = potential2[i];
		double p3 = potential3[i];
		double g = g0 * pow(r0 / (r0 + p1), 2.0);

		sprintf_s(
			asciiLine, 16384,
			"%5.0lf\t%5.1lf\t%5.1lf\t%5.1lf\t%7.5lf\r\n",
			zi, p1, p2, p3, g);
		strcat_s(asciiData, 16384, asciiLine);
	}

	strcat_s(asciiData, "H Geopotential Altitude\r\n");
	strcat_s(asciiData, "Z1 Geopotential Gauss-Legendre 128 Steps\r\n");
	strcat_s(asciiData, "Z2 Geopotential Simpson's Rule 256 Steps\r\n");
	strcat_s(asciiData, "Z3 Geopotential Trapezoidal Rule 512 Steps\r\n");
	strcat_s(asciiData, "g gravitational acceleration in m/s/s\r\n");
	strcat_s(asciiData, "https://ntrs.nasa.gov/api/citations/19770009539/downloads/19770009539.pdf\r\n");
	strcat_s(asciiData, "See Table 8. Page 9\r\n");

	ConvertCharToTChar(asciiData, textData, sizeof(textData));

	// https://learn.microsoft.com/en-us/windows/win32/controls/use-a-multiline-edit-control

	hwndEdit = CreateWindowEx(
		0, L"EDIT",   // predefined class 
        NULL,         // no window title 
        WS_CHILD | WS_VISIBLE | WS_VSCROLL | 
        ES_LEFT | ES_MULTILINE | ES_AUTOVSCROLL, 
        0, 0, 0, 0,   // set size in WM_SIZE message 
        hWnd,         // parent window 
        (HMENU) ID_EDITCHILD,   // edit control ID 
		hInst,
        /*(HINSTANCE) GetWindowLongPtr(hWnd, GWLP_HINSTANCE),*/ 
		NULL);        // pointer not needed 
    // Add text to the window. 
	SendMessage(hwndEdit, WM_SETTEXT, 0, (LPARAM) textData); 
    return 0; 
	case WM_COMMAND:
		wmId    = LOWORD(wParam);
		wmEvent = HIWORD(wParam);
		// Parse the menu selections:
		switch (wmId)
		{
		case IDM_ABOUT:
			DialogBox(hInst, MAKEINTRESOURCE(IDD_ABOUTBOX), hWnd, About);
			break;
		case IDM_EXIT:
			DestroyWindow(hWnd);
			break;
		default:
			return DefWindowProc(hWnd, message, wParam, lParam);
		}
		break;
	case WM_PAINT:
		hdc = BeginPaint(hWnd, &ps);
		// TODO: Add any drawing code here...
		EndPaint(hWnd, &ps);
		break;
	case WM_SIZE: 
            // Make the edit control the size of the window's client area. 
		MoveWindow(hwndEdit, 
			0, 0,                  // starting x- and y-coordinates 
            LOWORD(lParam),        // width of client area 
            HIWORD(lParam),        // height of client area 
            TRUE);                 // repaint window 
        return 0; 
	case WM_DESTROY:
		PostQuitMessage(0);
		break;
	default:
		return DefWindowProc(hWnd, message, wParam, lParam);
	}
	return 0;
}

// Message handler for about box.
INT_PTR CALLBACK About(HWND hDlg, UINT message, WPARAM wParam, LPARAM lParam)
{
	UNREFERENCED_PARAMETER(lParam);
	switch (message)
	{
	case WM_INITDIALOG:
		return (INT_PTR)TRUE;

	case WM_COMMAND:
		if (LOWORD(wParam) == IDOK || LOWORD(wParam) == IDCANCEL)
		{
			EndDialog(hDlg, LOWORD(wParam));
			return (INT_PTR)TRUE;
		}
		break;
	}
	return (INT_PTR)FALSE;
}

Blog Entry © Thursday, May 22, 2025, Baseball Ballistics with Simple Drag Computation by James Pate Williams, Jr.

// https://phys.libretexts.org/Bookshelves/University_Physics/Physics_(Boundless)/3%3A_Two-Dimensional_Kinematics/3.3%3A_Projectile_Motion

#pragma once

class ClassicalBallistics
{
public:
	ClassicalBallistics(void);
	~ClassicalBallistics(void);
	// analytical formula for maximum height
	double H(double V0, double theta0, double g);
	// analytical formula for flight time
	double T(double H, double g);
	// analytical formula for velocity at apex
	double Va(double V0, double theta0);
	// analytical formula for the range
	double L(double Va, double T);
	// analytical formula for time to apex
	double ta(double T);
	// analytical formula for range to apex
	double xa(double L);
	double xa(double H, double L, double theta0);
	double Theta1(double theta0);
	double Theta1(double H, double L, double xa);
	double ClassicalBallistics::V1(double V0);
};

#include "StdAfx.h"
#include "ClassicalBallistics.h"
#include <math.h>

ClassicalBallistics::ClassicalBallistics(void)
{
}
ClassicalBallistics::~ClassicalBallistics(void)
{
}
// analytical formula for maximum height
double ClassicalBallistics::H(double V0, double theta0, double g)
{
	double V02 = V0 * V0, sinTheta0 = sin(theta0);
	double sinTheta02 = sinTheta0 * sinTheta0;

	return V02 * sinTheta02 / (g * 2.0);
}
// analytical formula for flight time
double ClassicalBallistics::T(double H, double g)
{
	return 2.0 * sqrt(2.0 * H / g);
}
// analytical formula for velocity at apex
double ClassicalBallistics::Va(double V0, double theta0)
{
	double cosTheta0 = cos(theta0);
	double V02 = V0 * V0;

	return V0 * cosTheta0;
}
// analytical formula for the range
double ClassicalBallistics::L(double Va, double T)
{
	return Va * T;
}
// analytical formula for time to apex
double ClassicalBallistics::ta(double T)
{
	return 0.5 * T;
}
// analytical formula for range to apex
double ClassicalBallistics::xa(double L)
{
	return 0.5 * L;
}
double ClassicalBallistics::xa(double H, double L, double theta0)
{
	return sqrt(L * H / tan(theta0));
}
double ClassicalBallistics::Theta1(double theta0)
{
	return -theta0;
}
double ClassicalBallistics::Theta1(double H, double L, double xa)
{
	return atan(L * H / pow(L - xa, 2.0));
}
double ClassicalBallistics::V1(double V0)
{ 
	return V0;
}

// http://www.lajpe.org/sep13/04-LAJPE-782_Chudinov.pdf

#pragma once

const double g = 9.81;
// accleration due to gravity in Metric (SI) Units
const double k = 0.000625;
// k is typical for a baseball

class DragBallistics
{

public:

	DragBallistics(void);
	~DragBallistics(void);
    double f(double theta);
    // analytical formula for maximum height
    double H(double V0, double theta0, double g, double k);
    // analytical formula for flight time
    double T(double H, double g);
    // analytical formula for velocity at apex
    double Va(double V0, double theta0, double k);
    // analytical formula for the range
	double L(double Va, double T);
    double ta(double H, double T, double Va, double k);
    double xa(double H, double L, double theta0);
    double Theta1(double H, double L, double xa);
    double V1(double V0, double theta0, double theta1, double k);
};

#include "StdAfx.h"
#include "DragBallistics.h"
#include <math.h>

DragBallistics::DragBallistics(void)
{
}
DragBallistics::~DragBallistics(void)
{
}
double DragBallistics::f(double theta)
{
	double pi = 4.0 * atan(1.0);
	double sinTheta = sin(theta), cosTheta = cos(theta),
		cosTheta2 = cosTheta * cosTheta;
    return (sinTheta / cosTheta2) + log(tan(0.5 * theta + pi / 4.0));
}
// analytical formula for maximum height
double DragBallistics::H(double V0, double theta0, double g, double k)
{
	double V02 = V0 * V0, sinTheta0 = sin(theta0),
		sinTheta02 = sinTheta0 * sinTheta0;
    return V02 * sinTheta02 / (g * (2.0 + k * V02 * sinTheta0));
}
// analytical formula for flight time
double DragBallistics::T(double H, double g)
{
	return 2.0 * sqrt(2.0 * H / g);
}
// analytical formula for velocity at apex
double DragBallistics::Va(double V0, double theta0, double k)
{
	double pi = 4.0 * atan(1.0);
	double cosTheta0 = cos(theta0), cosTheta02 = cosTheta0 * cosTheta0;
    double V02 = V0 * V0, sinTheta0 = sin(theta0);
    return V0 * cosTheta0 / sqrt(1.0 + k * V02 * (sinTheta0 + 
		cosTheta02 * log(tan(0.5 * theta0 + pi / 4.0))));
}
// analytical formula for the range
double DragBallistics::L(double Va, double T)
{
	return Va * T;
}
double DragBallistics::ta(double H, double T, double Va, double k)
{
	return 0.5 * (T - k * H * Va);
}
double DragBallistics::xa(double H, double L, double theta0)
{
	return sqrt(L * H / tan(theta0));
}
double DragBallistics::Theta1(double H, double L, double xa)
{
	return -atan(L * H / pow(L - xa, 2.0));
}
double DragBallistics::V1(double V0, double theta0, double theta1, double k)
{
	double V02 = V0 * V0;
    double sinTheta0 = sin(theta0), cosTheta0 = cos(theta0),
		cosTheta02 = cosTheta0 * cosTheta0;
    return V0 * cosTheta0 / (cos(theta1) * sqrt(1.0 + k * 
		V02 * cosTheta02 * (f(theta0) - f(theta1))));
}

// BaseballBallisticsWin32Console.cpp
// Translated from August 2017 C# application
// May 21, 2025 (c) James Pate Williams, Jr.

#include "stdafx.h"
#include "ClassicalBallistics.h"
#include "DragBallistics.h"
#include <iomanip>
#include <iostream>
#include <math.h>

void PrintResults(char title[],
				  double H, double T, double Va,
				  double L, double Ta, double xa,
				  double Theta1, double V1)
{
	std::cout << title << std::endl;
	std::cout << std::fixed << std::setprecision(2);
	std::cout << "H      = " << H << std::endl;
	std::cout << "T      = " << T << std::endl;
	std::cout << "Va     = " << Va << std::endl;
	std::cout << "L      = " << L << std::endl;
	std::cout << "Ta     = " << Ta << std::endl;
	std::cout << "xa     = " << xa << std::endl;
	std::cout << "theta1 = " << Theta1 << std::endl;
	std::cout << "V1     = " << V1 << std::endl;
}

int _tmain(int argc, _TCHAR* argv[])
{
	while (true)
	{
		char line[128] = { };
		std::cout << "V0 (m / s) or 0 to quit: ";
		std::cin.getline(line, 128);
		double V0 = atof(line);
		if (V0 == 0)
		{
			break;
		}
		std::cout << "Enter angle in degrees: ";
		std::cin.getline(line, 128);
		double theta0 = atof(line);
		double pi = 4.0 * atan(1.0);
		theta0 *= pi / 180.0;
		ClassicalBallistics cBall;
		DragBallistics dBall;
		double cH = cBall.H(V0, theta0, g);
        double dH = dBall.H(V0, theta0, g, k);
        double cT = cBall.T(cH, g);
        double dT = dBall.T(dH, g);
        double cVa = cBall.Va(V0, theta0);
        double dVa = dBall.Va(V0, theta0, k);
        double cL = cBall.L(cVa, cT);
        double dL = dBall.L(dVa, dT);
        double cta = cBall.ta(cT);
        double dta = dBall.ta(dH, dT, dVa, k);
        double cxa = cBall.xa(cL);
        double dxa = dBall.xa(dH, dL, theta0);
        double cTheta1 = 180.0 * cBall.Theta1(theta0) / pi;
        double dTheta1 = 180.0 * dBall.Theta1(dH, dL, dxa) / pi;
        double cV1 = cBall.V1(V0);
        double dV1 = dBall.V1(V0, theta0, pi * dTheta1 / 180.0, k);
		PrintResults("Classical Ballistics",
			cH, cT, cVa, cL, cta, cxa, cTheta1, cV1);
		PrintResults("Drag Ballistics",
			dH, dT, dVa, dL, dta, dxa, dTheta1, dV1);
	}
	return 0;
}

Blog Entry © Wednesday, May 21, 2025, Backpropagation Artificial Neural Network Experiments by James Pate Williams, Jr.

Live Person-to-Person Tutoring

Blog Entry (c) Thursday, November 21, 2024, by James Pate Williams, Jr. Comparison of Homegrown Fifth Order Runge-Kutta Method Versus a Limited Number of Predictor-Corrector Algorithm

The first table is based on Conte-de Boor Fourth Runge-Kutta formulas that I converted to Fifth Order Runge-Kutta. Initial values: V = 2600 feet per second angle of elevation 30 degrees diameter 16 inches coefficient of form 0.61 density ratio 1.00 are from LCDR Ernest Edward Herrmann’s Exterior ballistics, 1935 My results are given first then LCDR Herrmann’s results:

x deg min sec time v vx vy y
0 30 0 0 0.00 2600 2252 1300 0
563 29 50 45 0.25 2578 2236 1283 325
1122 29 41 24 0.50 2556 2221 1266 646
1677 29 31 57 0.75 2535 2206 1250 962
2229 29 22 25 1.00 2515 2191 1233 1275
2776 29 12 47 1.25 2494 2177 1217 1583
3321 29 3 4 1.50 2474 2163 1201 1887
3861 28 53 15 1.75 2455 2149 1186 2188
x deg min sec time v vx vy y
0 30 0 0 0.00 2600 2252 1300 0
561 29 50 7 0.25 2582 2259 1285 323
1120 29 41 4 0.50 2564 2227 1270 642
1675 29 32 2 0.75 2546 2216 1255 958
2228 29 22 5 1.00 2529 2204 1241 1270

It is amazing how accurate Herrmann’s results were based on only a couple iterations using the Mayevski seven zone velocity retardation formulas.

Blog Entry © Sunday, November 17, 2024, by James Pate Williams, Jr. Three Methods of Solving the Exterior Ballistics Problem for the Fast Battleship USS Iowa (BB-61) 16-Inch/50 Caliber Rifles

Blog Entry (c) Saturday, September 28, 2024, by James Pate Williams, Jr. New Iowa Ballistics Table

I created the following fast battleship Iowa class ballistics table using a single coefficient of form with the value 0.531. The ballistic coefficient is 19.86229.

For comparison use the actual October 1941 Iowa class ballistics tables:

https://eugeneleeslover.com/USN-GUNS-AND-RANGE-TABLES/OP-770-1.html