Blog Entry © Monday, September 14, 2026, by James Pate Williams, Jr. Cardioid Curve Graph

// Cardioid.cpp : Defines the entry point for the application.
// Copyright (c) Sunday, September 13, 2026
// by James Pate Williams, Jr. BA, BS, MSwE, PhD
// https://en.wikipedia.org/wiki/Cardioid

#include "framework.h"
#include "Cardioid.h"
#include <float.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
std::vector<Point2d> points;

// 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 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_CARDIOID, szWindowClass, MAX_LOADSTRING);
    MyRegisterClass(hInstance);

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

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

    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_CARDIOID));
    wcex.hCursor        = LoadCursor(nullptr, IDC_ARROW);
    wcex.hbrBackground  = (HBRUSH)(COLOR_WINDOW+1);
    wcex.lpszMenuName   = MAKEINTRESOURCEW(IDC_CARDIOID);
    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 < points.size(); i++)
    {
        Point2d pt = points[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);
}

static void CreateCurve(double a, int n)
{
    double pi = 4.0 * atan(1.0), phi = 0.0;

    while (phi < 2.0 * pi)
    {
        double phi1 = 1.0 - cos(phi);
        double y = 2.0 * a * phi1 * cos(phi);
        double x = 2.0 * a * phi1 * sin(phi);
        Point2d pt = { 0, 0 };

        pt.x = x;
        pt.y = y;
        points.push_back(pt);
        phi += 0.001;
    }
}

//
//  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_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:
    {
        CreateCurve(1.0, 256);
        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, "%5.4lf", x);
            SIZE size = { };
            GetTextExtentPoint32A(
                hdc,
                buffer,
                (int)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, "%+5.3lf", y);
                SIZE size = { };
                GetTextExtentPoint32A(
                    hdc,
                    buffer,
                    (int)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 bPenNew = NULL;
        HGDIOBJ hPenOld = NULL;

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

        px = (float)points[0].x;
        py = (float)points[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 < points.size(); j++)
        {
            px = (float)points[j].x;
            py = (float)points[j].y;
            sx = xSlope * px + xInter;
            sy = ySlope * py + yInter;
            LineTo(hdc, (int)sx, (int)sy);
        }

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

        return (INT_PTR)FALSE;
    }
    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;
}

Blog Entry © Monday – Thursday, September 7 – 10, 2026, by James Pate Williams, Jr. Solutions of Three Nonhomogeneous Second Order Linear Ordinary Differential Equation Boundary Value Problems and a Series Solution of the Bessel Function of the First Kind of Integer Eigenvalue (Order) of Zero

Blog Entry Monday September 7 2026 (AutoRecovered)Download
// FiniteDifference.cpp : This file contains the 'main' function.
// Program execution begins and ends there.
// Copyright (c) Wednesday, August 9, 2026
// by James Pate Williams, Jr., BA, BS, MSwE, PhD
// Reference: "Elementary Numerical Analysis an Algorithmic Approach
// Third Edition" © 1980 by S. D. Conte and Carl de Boor 

#include <math.h>
#include <stdio.h>

#define MAX_ROWS 8192

double ac[MAX_ROWS];
double fv[MAX_ROWS];
double gv[MAX_ROWS];
double qv[MAX_ROWS];
double xv[MAX_ROWS];
double yv[MAX_ROWS];

double ta[MAX_ROWS];
double tb[MAX_ROWS];
double td[MAX_ROWS];
double tc[MAX_ROWS];
double tx[MAX_ROWS];

int SolveTridiagonal(int n)
{
    for (int k = 2; k <= n; k++)
    {
        if (td[k - 1] == 0)
            return 0;

        double m = ta[k] / td[k - 1];
        td[k] -= m * tc[k - 1];
        tb[k] -= m * tb[k - 1];
    }

    if (td[n] == 0)
        return 0;

    tx[n] = tb[n] / td[n];

    for (int k = n - 1; k >= 1; k--)
        tx[k] = (tb[k] - tc[k] * tx[k + 1]) / td[k];

    return 1;
}

double f1(double x)
{
	return 0.0;
}

double g1(double x)
{
	return -1.0;
}

double q1(double x)
{
	return 0.0;
}

double f2(double x)
{
	return 0.0;
}

double g2(double x)
{
	return 1.0;
}

double q2(double x)
{
	return 0.0;
}

double f3(double x)
{
	return x;
}

double g3(double x)
{
	return 1.0;
}

double q3(double x)
{
	return 2.0 * x;
}

double f4(double x)
{
	return 2.0;
}

double g4(double x)
{
	return 1.0;
}

double q4(double x)
{
	return x;
}

double f5(double x)
{
	return 1.0 / x;
}

double g5(double x)
{
	return 1.0;
}

double q5(double x)
{
	return 0.0;
}

double FD_Solution(
	double a,
	double b,
	double ya,
	double yb,
	double (*f)(double),
	double (*g)(double),
	double (*q)(double),
	int N)
{
	double h = (b - a) / N;

	for (int i = 1; i <= N - 1; i++)
	{
		xv[i] = a + i * h;
		fv[i] = f(xv[i]);
		gv[i] = g(xv[i]);
		qv[i] = q(xv[i]);
	}

	tb[1] = h * h * qv[1] - (1.0 - 0.5 * h * fv[1]) * ya;

	for (int i = 2; i <= N - 1; i++)
		tb[i] = h * h * qv[i];

	for (int i = 1; i <= N - 1; i++)
		td[i] = -2.0 + h * h * gv[i];

	for (int i = 1; i <= N - 2; i++)
		tc[i] = 1.0 + 0.5 * h * fv[i];

	for (int i = 2; i <= N - 1; i++)
		ta[i] = 1.0 - 0.5 * h * fv[i];

	tb[N - 1] = h * h * qv[N - 1] - (1.0 + 0.5 * h * fv[N - 1]) * yb;

	return SolveTridiagonal(N - 1);
}

double IS_Solution(
	double a,
	double b,
	double ya,
	double yb,
	double x)
{
	double A[13] = { 0 };

	A[0] = 1.0;
	A[1] = 0.0;
	A[2] = -0.25;
	A[3] = 0.0;
	A[4] = A[0] / 64.0;
	A[5] = 0.0;
	A[6] = -A[0] / (36.0 * 64.0);
	A[7] = 0.0;
	A[8] = A[0] / (64.0 * 36.0 * 64.0);
	A[9] = 0.0;
	A[10] = -A[8] / 100.0;
	A[11] = 0.0;
	A[12] = -A[10] / 144.0;

	double s = A[12];

	for (int i = 11; i >= 0; i--)
		s = s * x + A[i];

	return s;
}

int main()
{
	double h = 0.05, x0 = 0.0, x1 = 1.0, y0 = 0, y1 = 1.0;
	int N = (int)((x1 - x0) / h);
	double fd1 = FD_Solution(
		x0, x1, y0, y1, f1, g1, q1, N);
	FILE* file = 0;
	int errno = fopen_s(&file, "Chapter9.txt", "w");

	if (errno != 0)
		return -1;

	fprintf_s(file, "Example 9.1\r\n");
	fprintf_s(file, " x\tapproximate\texact\t\tpercent error\r\n");

	for (int i = 1; i < N; i++)
	{
		double exact = sinh(xv[i]) / sinh(1.0);
		double error = 100.0 * (fabs(tx[i] - exact) / fabs(exact));
		fprintf_s(file, "%3.2lf\t%11.10lf\t%11.10lf\t%11.10lf\r\n",
			xv[i], tx[i], exact, error);

	}

	h = 0.25;
	x0 = 0.0, x1 = 1.0, y0 = 0, y1 = 1.0;
	N = (int)((x1 - x0) / h);
	double fd2 = FD_Solution(
		x0, x1, y0, y1, f2, g2, q2, N);

	fprintf_s(file, "Exercise 9.1-1\r\n");
	fprintf_s(file, " x\tapprox\r\n");

	for (int i = 1; i < N; i++)
		fprintf_s(file, "%3.2lf\t%5.4lf\r\n", xv[i], tx[i]);

	h = 0.1;
	x0 = 0.0, x1 = 1.0, y0 = 1.0, y1 = 0.0;
	N = (int)((x1 - x0) / h);
	double fd3 = FD_Solution(
		x0, x1, y0, y1, f3, g3, q3, N);

	fprintf_s(file, "Exercise 9.1-3\r\n");
	fprintf_s(file, " x\t approximate\r\n");

	for (int i = 1; i < N; i++)
		fprintf_s(file, "%3.2lf\t%11.10lf\r\n", xv[i], tx[i]);

	h = 1.0 / 16.0;
	x0 = 0.0, x1 = 1.0, y0 = 0.0, y1 = 1.0;
	N = (int)((x1 - x0) / h);
	double fd4 = FD_Solution(
		x0, x1, y0, y1, f4, g4, q4, N);

	fprintf_s(file, "Exercise 9.1-4\r\n");
	fprintf_s(file, " x\t approximate\r\n");

	for (int i = 1; i < N; i++)
		fprintf_s(file, "%3.2lf\t%11.10lf\r\n", xv[i], tx[i]);

	h = 1.0 / 16.0;
	x0 = 0.0, x1 = 2.0, y0 = 1.0, y1 = 2.238907555e-01;
	N = (int)((x1 - x0) / h);
	double fd5 = FD_Solution(
		x0, x1, y0, y1, f5, g5, q5, N);

	fprintf_s(file, "Bessel Equation of Order Zero\r\n");
	fprintf_s(file, "Approximate Finite Difference Solution\r\n");

	fprintf_s(file, " x\t approximate\r\n");

	for (int i = 1; i < N; i++)
		fprintf_s(file, "%3.2lf\t%11.10lf\r\n", xv[i], tx[i]);

	fprintf_s(file, "Bessel Equation of Order Zero\r\n");
	fprintf_s(file, "Approximate Infinite Series Solution\r\n");

	fprintf_s(file, " x\t approximate\r\n");

	for (int i = 1; i < N; i++)
	{
		double x = x0 + i * h;
		fprintf_s(file, "%3.2lf\t%11.10lf\r\n", x,
			IS_Solution(x0, x1, y0, y1, x));
	}

    return 0;
}

Blog Entry © Friday, August 28, 2026, by James Pate Williams, Jr. New Cryptography Project Continued Basic RSA Functions

References: Guide to Elliptic Curve Cryptography © 2004 by Darrel Hankerson, Alfred Menezes, and Scott Vanstone, Handbook of Applied Cryptography © 1997 by A. Menezes, P. van Oorschot and S. Vanstone

Blog Entry Friday August 28 2026Download

Blog Entry © Monday, August 24, 2026, by James Pate Williams, Jr. New Cryptography Project

Blog Entry Monday August 24 2026Download

Blog Entry © Friday, August 21, 2026, by James Pate Williams, Jr. NIST FIPS 203 C/C++Implementations

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Blog Entry © Thursday, August 20, 2026, by James Pate Williams, Jr. A Tale of Two SecureHashing Algorithms (SHA-1 and SHA-3)

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Blog Entry © Tuesday, August 18, 2026, by James Pate Williams, Jr. Some Results from My Unoptimized and Modified Factoring with Cubic Integers a la J. M. Pollard

Blog Entry Tuesday August 18 2026Download

Some More C Multiple Quadratic Polynomial Sieve Factoring Results, C/C++ SHA-1 Results, and C# SHA-3 Results © Monday, August 17, 2026, by James Pate Williams, Jr.

Blog Entry Monday August 17 2026Download

Blog Entry © Wednesday, August 12, 2026, by James Pate Williams, Jr., Two Brute Force Elliptic Curve Point Counting Algorithms

Blog Entry Wednesday August 12 2026Download

Blog Entry © Monday, August 3, 2026, by James Pate Williams, Jr., BA, BS, MSwE, PhD Free Particle Schrödinger Equation in Confocal Parabolic Coordinates

Blog Entry Monday August 3 2026Download