Category: C++ Computer Applications

Blog Entry © Thursday, July 31, 2025, Numerically Solving Four Second Order Linear Ordinary Differential Equation Boundary Value Problems by James Pate Williams, Jr.

Blog Entry Thursday July 31 2025Download

Blog Entry © Sunday, July 20, 2025, by James Pate Williams, Jr. Example and Two Exercises from a Numerical Analysis Textbook

Happy 59th Anniversary of the historic moon landing by Neil Armstrong, Buzz Aldrin, and Michael Collins in Apolla 11.

Blog Entry July 20 2025Download

Newtonian Gravity and Cosmology © Friday, July 18, 2025, by James Pate Williams, Jr. in Conjunction with the Microsoft Copilot AI

Newtonian Gravity and CosmologyDownload

Some General Relativity Mathematics by James Pate Williams, Jr. (c) July 13, 2025

Some Relativity MathDownload
// SomeRelativityMath.cpp : Defines the entry point for the application.
// https://arxiv.org/pdf/2506.09946

#include "pch.h"
#include "framework.h"
#include "SomeRelativityMath.h"
#include <stdio.h>
#include <vector>

#define MAX_LOADSTRING 100

typedef struct tagPoint2d
{
    double x, y;
} POINT2D, * PPOINT2D;

// Global Variables:
HINSTANCE hInst;                                // current instance
WCHAR szTitle[MAX_LOADSTRING];                  // The title bar text
WCHAR szWindowClass[MAX_LOADSTRING];            // the main window class name
char text[8192];
std::vector<POINT2D> points1, points2;
std::vector<POINT2D> points3, points4;

// 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_PTR CALLBACK    DrawGraph(HWND, UINT, WPARAM, LPARAM);

int APIENTRY wWinMain(_In_ HINSTANCE hInstance,
                     _In_opt_ HINSTANCE hPrevInstance,
                     _In_ LPWSTR    lpCmdLine,
                     _In_ int       nCmdShow)
{
    UNREFERENCED_PARAMETER(hPrevInstance);
    UNREFERENCED_PARAMETER(lpCmdLine);

    // TODO: Place code here.

    // Initialize global strings
    LoadStringW(hInstance, IDS_APP_TITLE, szTitle, MAX_LOADSTRING);
    LoadStringW(hInstance, IDC_SOMERELATIVITYMATH, szWindowClass, MAX_LOADSTRING);
    MyRegisterClass(hInstance);

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

    HACCEL hAccelTable = LoadAccelerators(hInstance, MAKEINTRESOURCE(IDC_SOMERELATIVITYMATH));

    MSG msg;

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

    return (int) msg.wParam;
}

//
//  FUNCTION: MyRegisterClass()
//
//  PURPOSE: Registers the window class.
//
ATOM MyRegisterClass(HINSTANCE hInstance)
{
    WNDCLASSEXW wcex = { 0 };

    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_SOMERELATIVITYMATH));
    wcex.hCursor        = LoadCursor(nullptr, IDC_ARROW);
    wcex.hbrBackground  = (HBRUSH)(COLOR_WINDOW+1);
    wcex.lpszMenuName   = MAKEINTRESOURCEW(IDC_SOMERELATIVITYMATH);
    wcex.lpszClassName  = szWindowClass;
    wcex.hIconSm        = LoadIcon(wcex.hInstance, MAKEINTRESOURCE(IDI_SMALL));

    return RegisterClassExW(&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)
{
   hInst = hInstance; // Store instance handle in our global variable

   HWND hWnd = CreateWindowW(szWindowClass, szTitle, WS_OVERLAPPEDWINDOW,
      CW_USEDEFAULT, 0, CW_USEDEFAULT, 0, nullptr, nullptr, hInstance, nullptr);

   if (!hWnd)
   {
      return FALSE;
   }

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

   return TRUE;
}

static void FindMinMax(
    double& xMin, double& xMax,
    double& yMin, double& yMax)
{
    // uses global 2D double points structure

    xMin = yMin = DBL_MAX;
    xMax = yMax = DBL_MIN;

    for (size_t i = 0; i < points1.size(); i++)
    {
        POINT2D pt = points1[i];
        double x = pt.x;
        double y = pt.y;

        if (x < xMin)
            xMin = x;
        if (x > xMax)
            xMax = x;
        if (y < yMin)
            yMin = y;
        if (y > yMax)
            yMax = y;
    }

    for (size_t i = 0; i < points2.size(); i++)
    {
        POINT2D pt = points2[i];
        double x = pt.x;
        double y = pt.y;

        if (x < xMin)
            xMin = x;
        if (x > xMax)
            xMax = x;
        if (y < yMin)
            yMin = y;
        if (y > yMax)
            yMax = y;
    }

    for (size_t i = 0; i < points3.size(); i++)
    {
        POINT2D pt = points3[i];
        double x = pt.x;
        double y = pt.y;

        if (x < xMin)
            xMin = x;
        if (x > xMax)
            xMax = x;
        if (y < yMin)
            yMin = y;
        if (y > yMax)
            yMax = y;
    }

    for (size_t i = 0; i < points4.size(); i++)
    {
        POINT2D pt = points4[i];
        double x = pt.x;
        double y = pt.y;

        if (x < xMin)
            xMin = x;
        if (x > xMax)
            xMax = x;
        if (y < yMin)
            yMin = y;
        if (y > yMax)
            yMax = y;
    }
}

static void DrawFormattedText(HDC hdc, char text[], RECT rect)
{
    // Draw the text with formatting options
    DrawTextA(hdc, text, -1, &rect, DT_SINGLELINE | DT_NOCLIP);
}

//
//  FUNCTION: WndProc(HWND, UINT, WPARAM, LPARAM)
//
//  PURPOSE: Processes messages for the main window.
//
//  WM_COMMAND  - process the application menu
//  WM_PAINT    - Paint the main window
//  WM_DESTROY  - post a quit message and return
//
//
LRESULT CALLBACK WndProc(HWND hWnd, UINT message, WPARAM wParam, LPARAM lParam)
{
    switch (message)
    {
    case WM_CREATE:
    {
        double P1 = 0, P2 = 0, P3 = 0, P4 = 0;
        int numberPts = 1024;
        double x0 = 1, h = 0.5 / numberPts, x = x0;

        points1.resize(numberPts);
        points2.resize(numberPts);
        points3.resize(numberPts);
        points4.resize(numberPts);

        for (int i = 0; i < numberPts; i++)
        {
            P1 = x + 1.0;
            P2 = 3.0 * x * x + 4.0 * x + 2.0;
            P3 = 15.0 * x * x * x + 16.0 * x * x + 6.0 * x;
            P4 = 105.0 * x * x * x * x + 156.0 * x * x * x +
                94.0 * x * x + 24.0 * x;
            POINT2D pt = { 0 };
            pt.x = x;
            pt.y = log(P1);
            points1[i] = pt;
            pt.y = log(P2);
            points2[i] = pt;
            pt.y = log(P3);
            points3[i] = pt;
            pt.y = log(P4);
            points4[i] = pt;
            x += h;
        }
        break;
    }
    case WM_COMMAND:
        {
            int wmId = LOWORD(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:
    {
        double h = 0, pi = 0, plm = 0, theta = 0;
        double xMax = 0, xMin = 0, yMax = 0, yMin = 0;
        FindMinMax(xMin, xMax, yMin, yMax);
        float xSpan = (float)(xMax - xMin);
        float ySpan = (float)(yMax - yMin);
        RECT rect = { };
        GetClientRect(hWnd, &rect);
        float width = (float)(rect.right - rect.left + 1);
        float height = (float)(rect.bottom - rect.top - 32 + 1);
        float sx0 = 2.0f * width / 16.0f;
        float sx1 = 14.0f * width / 16.0f;
        float sy0 = 2.0f * height / 16.0f;
        float sy1 = 14.0f * height / 16.0f;
        float deltaX = xSpan / 8.0f;
        float deltaY = ySpan / 8.0f;
        float xSlope = (sx1 - sx0) / xSpan;
        float xInter = (float)(sx0 - xSlope * xMin);
        float ySlope = (sy0 - sy1) / ySpan;
        float yInter = (float)(sy0 - ySlope * yMax);
        float px = 0, py = 0, sx = 0, sy = 0;
        PAINTSTRUCT ps;
        POINT wPt = { };
        HDC hdc = BeginPaint(hWnd, &ps);
        int i = 0;
        float x = (float)xMin;
        float y = (float)yMax;
        px = x;
        py = y;
        sx = xSlope * px + xInter;
        sy = ySlope * py + yInter;
        MoveToEx(hdc, (int)sx, (int)sy0, &wPt);
        char buffer[128] = { };

        while (i <= 8)
        {
            sx = xSlope * x + xInter;
            wPt.x = wPt.y = 0;
            MoveToEx(hdc, (int)sx, (int)sy0, &wPt);
            LineTo(hdc, (int)sx, (int)sy1);

            sprintf_s(buffer, "%3.2f", x);
            SIZE size = { };
            GetTextExtentPoint32A(
                hdc,
                buffer,
                strlen(buffer),
                &size);
            RECT textRect = { };
            textRect.left = (long)(sx - size.cx / 2.0f);
            textRect.right = (long)(sx + size.cx / 2.0f);
            textRect.top = (long)sy1;
            textRect.bottom = (long)(sy1 + size.cy / 2.0f);
            DrawFormattedText(hdc, buffer, textRect);
            x += deltaX;
            i++;
        }

        i = 0;
        y = (float)yMin;

        while (i <= 8)
        {
            sy = ySlope * y + yInter;
            wPt.x = wPt.y = 0;
            MoveToEx(hdc, (int)sx0, (int)sy, &wPt);
            LineTo(hdc, (int)sx, (int)sy);

            if (i != 0)
            {
                sprintf_s(buffer, "%3.1f", y);
                SIZE size = { };
                GetTextExtentPoint32A(
                    hdc,
                    buffer,
                    strlen(buffer),
                    &size);
                RECT textRect = { };
                textRect.left = (long)(sx0 - size.cx - size.cx / 5.0f);
                textRect.right = (long)(sx0 - size.cx / 2.0f);
                textRect.top = (long)(sy - size.cy / 2.0f);
                textRect.bottom = (long)(sy + size.cy / 2.0f);
                DrawFormattedText(hdc, buffer, textRect);
            }

            y += deltaY;
            i++;
        }

        HGDIOBJ rPenNew = NULL;
        HGDIOBJ hPenOld = NULL;
        rPenNew = CreatePen(PS_SOLID, 2, RGB(255, 0, 0));
        hPenOld = SelectObject(hdc, rPenNew);

        px = (float)points1[0].x;
        py = (float)points1[0].y;
        sx = xSlope * px + xInter;
        sy = ySlope * py + yInter;
        wPt.x = wPt.y = 0;
        MoveToEx(hdc, (int)sx, (int)sy, &wPt);

        for (size_t j = 1; j < points1.size(); j++)
        {
            px = (float)points1[j].x;
            py = (float)points1[j].y;
            sx = xSlope * px + xInter;
            sy = ySlope * py + yInter;
            LineTo(hdc, (int)sx, (int)sy);
        }

        SelectObject(hdc, hPenOld);
        DeleteObject(rPenNew);

        HGDIOBJ gPenNew = CreatePen(PS_SOLID, 2, RGB(0, 255, 0));
        hPenOld = SelectObject(hdc, gPenNew);

        px = (float)points2[0].x;
        py = (float)points2[0].y;
        sx = xSlope * px + xInter;
        sy = ySlope * py + yInter;
        wPt.x = wPt.y = 0;
        MoveToEx(hdc, (int)sx, (int)sy, &wPt);

        for (size_t j = 1; j < points2.size(); j++)
        {
            px = (float)points2[j].x;
            py = (float)points2[j].y;
            sx = xSlope * px + xInter;
            sy = ySlope * py + yInter;
            LineTo(hdc, (int)sx, (int)sy);
        }

        SelectObject(hdc, hPenOld);
        DeleteObject(gPenNew);

        HGDIOBJ bPenNew = CreatePen(PS_SOLID, 2, RGB(0, 0, 255));
        hPenOld = SelectObject(hdc, bPenNew);

        px = (float)points3[0].x;
        py = (float)points3[0].y;
        sx = xSlope * px + xInter;
        sy = ySlope * py + yInter;
        wPt.x = wPt.y = 0;
        MoveToEx(hdc, (int)sx, (int)sy, &wPt);

        for (size_t j = 1; j < points3.size(); j++)
        {
            px = (float)points3[j].x;
            py = (float)points3[j].y;
            sx = xSlope * px + xInter;
            sy = ySlope * py + yInter;
            LineTo(hdc, (int)sx, (int)sy);
        }

        SelectObject(hdc, hPenOld);
        DeleteObject(bPenNew);

        HGDIOBJ pPenNew = CreatePen(PS_SOLID, 2, RGB(128, 0, 128));
        hPenOld = SelectObject(hdc, pPenNew);

        px = (float)points4[0].x;
        py = (float)points4[0].y;
        sx = xSlope * px + xInter;
        sy = ySlope * py + yInter;
        wPt.x = wPt.y = 0;
        MoveToEx(hdc, (int)sx, (int)sy, &wPt);

        for (size_t j = 1; j < points4.size(); j++)
        {
            px = (float)points4[j].x;
            py = (float)points4[j].y;
            sx = xSlope * px + xInter;
            sy = ySlope * py + yInter;
            LineTo(hdc, (int)sx, (int)sy);
        }

        SelectObject(hdc, hPenOld);
        DeleteObject(pPenNew);

        EndPaint(hWnd, &ps);
    }
    break;
    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;
}

static INT_PTR CALLBACK DrawGraph(
    HWND hDlg, UINT message, WPARAM wParam, LPARAM lParam)
{
    UNREFERENCED_PARAMETER(lParam);
    char line[256] = { };
    switch (message)
    {
    case WM_INITDIALOG:
        GetWindowTextA(hDlg, line, 256);
        strcat_s(text, 8192, " ");
        strcat_s(text, 8192, line);
        SetWindowTextA(hDlg, text);
        return (INT_PTR)TRUE;

    case WM_COMMAND:
        if (LOWORD(wParam) == IDOK || LOWORD(wParam) == IDCANCEL)
        {
            EndDialog(hDlg, LOWORD(wParam));
            return (INT_PTR)TRUE;
        }
        break;
    case WM_PAINT:
        double h = 0, pi = 0, plm = 0, theta = 0;
        double xMax = 0, xMin = 0, yMax = 0, yMin = 0;
        FindMinMax(xMin, xMax, yMin, yMax);
        float xSpan = (float)(xMax - xMin);
        float ySpan = (float)(yMax - yMin);
        RECT rect = { };
        GetClientRect(hDlg, &rect);
        float width = (float)(rect.right - rect.left + 1);
        float height = (float)(rect.bottom - rect.top - 32 + 1);
        float sx0 = 2.0f * width / 16.0f;
        float sx1 = 14.0f * width / 16.0f;
        float sy0 = 2.0f * height / 16.0f;
        float sy1 = 14.0f * height / 16.0f;
        float deltaX = xSpan / 8.0f;
        float deltaY = ySpan / 8.0f;
        float xSlope = (sx1 - sx0) / xSpan;
        float xInter = (float)(sx0 - xSlope * xMin);
        float ySlope = (sy0 - sy1) / ySpan;
        float yInter = (float)(sy0 - ySlope * yMax);
        float px = 0, py = 0, sx = 0, sy = 0;
        PAINTSTRUCT ps;
        POINT wPt = { };
        HDC hdc = BeginPaint(hDlg, &ps);
        int i = 0;
        float x = (float)xMin;
        float y = (float)yMax;
        px = x;
        py = y;
        sx = xSlope * px + xInter;
        sy = ySlope * py + yInter;
        MoveToEx(hdc, (int)sx, (int)sy0, &wPt);
        char buffer[128] = { };

        while (i <= 8)
        {
            sx = xSlope * x + xInter;
            wPt.x = wPt.y = 0;
            MoveToEx(hdc, (int)sx, (int)sy0, &wPt);
            LineTo(hdc, (int)sx, (int)sy1);

            sprintf_s(buffer, "%3.0f", x);
            SIZE size = { };
            GetTextExtentPoint32A(
                hdc,
                buffer,
                strlen(buffer),
                &size);
            RECT textRect = { };
            textRect.left = (long)(sx - size.cx / 2.0f);
            textRect.right = (long)(sx + size.cx / 2.0f);
            textRect.top = (long)sy1;
            textRect.bottom = (long)(sy1 + size.cy / 2.0f);
            DrawFormattedText(hdc, buffer, textRect);
            x += deltaX;
            i++;
        }

        i = 0;
        y = (float)yMin;

        while (i <= 8)
        {
            sy = ySlope * y + yInter;
            wPt.x = wPt.y = 0;
            MoveToEx(hdc, (int)sx0, (int)sy, &wPt);
            LineTo(hdc, (int)sx, (int)sy);

            if (i != 0)
            {
                sprintf_s(buffer, "%3.0f", y);
                SIZE size = { };
                GetTextExtentPoint32A(
                    hdc,
                    buffer,
                    strlen(buffer),
                    &size);
                RECT textRect = { };
                textRect.left = (long)(sx0 - size.cx - size.cx / 5.0f);
                textRect.right = (long)(sx0 - size.cx / 2.0f);
                textRect.top = (long)(sy - size.cy / 2.0f);
                textRect.bottom = (long)(sy + size.cy / 2.0f);
                DrawFormattedText(hdc, buffer, textRect);
            }

            y += deltaY;
            i++;
        }

        HGDIOBJ rPenNew = NULL;
        HGDIOBJ hPenOld = NULL;
        rPenNew = CreatePen(PS_SOLID, 2, RGB(255, 0, 0));
        hPenOld = SelectObject(hdc, rPenNew);

        px = (float)points1[1].x;
        py = (float)points1[1].y;
        sx = xSlope * px + xInter;
        sy = ySlope * py + yInter;
        wPt.x = wPt.y = 0;
        MoveToEx(hdc, (int)sx, (int)sy, &wPt);

        for (size_t j = 1; j < points1.size(); j++)
        {
            px = (float)points1[j].x;
            py = (float)points1[j].y;
            sx = xSlope * px + xInter;
            sy = ySlope * py + yInter;
            LineTo(hdc, (int)sx, (int)sy);
        }

        SelectObject(hdc, hPenOld);
        DeleteObject(rPenNew);

        px = (float)points2[1].x;
        py = (float)points2[1].y;
        sx = xSlope * px + xInter;
        sy = ySlope * py + yInter;
        wPt.x = wPt.y = 0;
        MoveToEx(hdc, (int)sx, (int)sy, &wPt);

        HGDIOBJ gPenNew = CreatePen(PS_SOLID, 2, RGB(0, 255, 0));
        hPenOld = SelectObject(hdc, gPenNew);
        px = (float)points2[1].x;
        py = (float)points2[1].y;
        sx = xSlope * px + xInter;
        sy = ySlope * py + yInter;
        wPt.x = wPt.y = 0;
        MoveToEx(hdc, (int)sx, (int)sy, &wPt);

        for (size_t j = 1; j < points2.size(); j++)
        {
            px = (float)points2[j].x;
            py = (float)points2[j].y;
            sx = xSlope * px + xInter;
            sy = ySlope * py + yInter;
            LineTo(hdc, (int)sx, (int)sy);
        }

        SelectObject(hdc, hPenOld);
        DeleteObject(gPenNew);

        px = (float)points3[1].x;
        py = (float)points3[1].y;
        sx = xSlope * px + xInter;
        sy = ySlope * py + yInter;
        wPt.x = wPt.y = 0;
        MoveToEx(hdc, (int)sx, (int)sy, &wPt);

        HGDIOBJ bPenNew = CreatePen(PS_SOLID, 2, RGB(0, 0, 255));
        hPenOld = SelectObject(hdc, bPenNew);
        px = (float)points3[1].x;
        py = (float)points3[1].y;
        sx = xSlope * px + xInter;
        sy = ySlope * py + yInter;
        wPt.x = wPt.y = 0;
        MoveToEx(hdc, (int)sx, (int)sy, &wPt);

        for (size_t j = 1; j < points3.size(); j++)
        {
            px = (float)points3[j].x;
            py = (float)points3[j].y;
            sx = xSlope * px + xInter;
            sy = ySlope * py + yInter;
            LineTo(hdc, (int)sx, (int)sy);
        }

        SelectObject(hdc, hPenOld);
        DeleteObject(bPenNew);

        px = (float)points4[1].x;
        py = (float)points4[1].y;
        sx = xSlope * px + xInter;
        sy = ySlope * py + yInter;
        wPt.x = wPt.y = 0;
        MoveToEx(hdc, (int)sx, (int)sy, &wPt);

        HGDIOBJ tPenNew = CreatePen(PS_SOLID, 2, RGB(128, 0, 128));
        hPenOld = SelectObject(hdc, tPenNew);
        px = (float)points4[1].x;
        py = (float)points4[1].y;
        sx = xSlope * px + xInter;
        sy = ySlope * py + yInter;
        wPt.x = wPt.y = 0;
        MoveToEx(hdc, (int)sx, (int)sy, &wPt);

        for (size_t j = 1; j < points4.size(); j++)
        {
            px = (float)points4[j].x;
            py = (float)points4[j].y;
            sx = xSlope * px + xInter;
            sy = ySlope * py + yInter;
            LineTo(hdc, (int)sx, (int)sy);
        }

        SelectObject(hdc, hPenOld);
        DeleteObject(tPenNew);

        EndPaint(hDlg, &ps);
        return (INT_PTR)FALSE;
        break;
    }

    return (INT_PTR)FALSE;
}

Historical Attendance Data for the First United Methodist Church (FUMC) of LaGrange, Georgia for the Year 2006 and Modeling the Data Using Various Polynomials and Least Squares Fitting by James Pate Williams, Jr., BA, BS, Master of Software Engineering and Doctor of Philosophy in Computer Science

Church Data Polynomial Fitting 2006Download

Historical Attendance Data for the Four Services Offered by the First United Methodist Church of LaGrange, Georgia, Compiled and Analyzed by James Pate Williams, Jr.

Church Attendance BPNN 2006 to 2008Download

Blog Entry © Thursday, June 5, 2025, by James Pate Williams, Jr., Analytic, Numeric, and Siacci’s Method for Solving Ballistic Trajectory Problems (Point-Mass Projectile Motion)

Blog Entry Thursday June 5 2025Download