CmdPal: Fix window centering when moving to a display with different DPI (#45057)

## Summary of the Pull Request

This PR fixes centering of main window, when the window also moves
between move display with a different DPI.

- The centered position was calculated using the current window width
and height, but those values change after the window is moved to
accommodate the new display’s DPI.
- Calculations have been refactored out of main window to a helper
class.

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## PR Checklist

- [x] Closes: #44932
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issues) -->
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- [ ] **Tests:** Added/updated and all pass
- [ ] **Localization:** All end-user-facing strings can be localized
- [ ] **Dev docs:** Added/updated
- [ ] **New binaries:** Added on the required places
- [ ] [JSON for
signing](https://github.com/microsoft/PowerToys/blob/main/.pipelines/ESRPSigning_core.json)
for new binaries
- [ ] [WXS for
installer](https://github.com/microsoft/PowerToys/blob/main/installer/PowerToysSetup/Product.wxs)
for new binaries and localization folder
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pipeline](https://github.com/microsoft/PowerToys/blob/main/.pipelines/ci/templates/build-powertoys-steps.yml)
for new test projects
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pipeline](https://github.com/microsoft/PowerToys/blob/main/.pipelines/release.yml)
- [ ] **Documentation updated:** If checked, please file a pull request
on [our docs
repo](https://github.com/MicrosoftDocs/windows-uwp/tree/docs/hub/powertoys)
and link it here: #xxx

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## Detailed Description of the Pull Request / Additional comments

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## Validation Steps Performed

src/modules/cmdpal/Microsoft.CmdPal.UI/Helpers/WindowPositionHelper.cs

@@ -0,0 +1,152 @@
+// Copyright (c) Microsoft Corporation
+// The Microsoft Corporation licenses this file to you under the MIT license.
+// See the LICENSE file in the project root for more information.
+
+using Microsoft.UI;
+using Microsoft.UI.Windowing;
+using Windows.Graphics;
+using Windows.Win32;
+using Windows.Win32.Graphics.Gdi;
+using Windows.Win32.UI.HiDpi;
+
+namespace Microsoft.CmdPal.UI.Helpers;
+
+internal static class WindowPositionHelper
+{
+    private const int DefaultWidth = 800;
+    private const int DefaultHeight = 480;
+    private const int MinimumVisibleSize = 100;
+    private const int DefaultDpi = 96;
+
+    public static PointInt32? CalculateCenteredPosition(DisplayArea? displayArea, SizeInt32 windowSize, int windowDpi)
+    {
+        if (displayArea is null)
+        {
+            return null;
+        }
+
+        var workArea = displayArea.WorkArea;
+        if (workArea.Width <= 0 || workArea.Height <= 0)
+        {
+            return null;
+        }
+
+        var targetDpi = GetDpiForDisplay(displayArea);
+        var predictedSize = ScaleSize(windowSize, windowDpi, targetDpi);
+
+        // Clamp to work area
+        var width = Math.Min(predictedSize.Width, workArea.Width);
+        var height = Math.Min(predictedSize.Height, workArea.Height);
+
+        return new PointInt32(
+            workArea.X + ((workArea.Width - width) / 2),
+            workArea.Y + ((workArea.Height - height) / 2));
+    }
+
+    /// <summary>
+    /// Adjusts a saved window rect to ensure it's visible on the nearest display,
+    /// accounting for DPI changes and work area differences.
+    /// </summary>
+    ///
+    public static RectInt32 AdjustRectForVisibility(RectInt32 savedRect, SizeInt32 savedScreenSize, int savedDpi)
+    {
+        var displayArea = DisplayArea.GetFromRect(savedRect, DisplayAreaFallback.Nearest);
+        if (displayArea is null)
+        {
+            return savedRect;
+        }
+
+        var workArea = displayArea.WorkArea;
+        if (workArea.Width <= 0 || workArea.Height <= 0)
+        {
+            return savedRect;
+        }
+
+        var targetDpi = GetDpiForDisplay(displayArea);
+        if (savedDpi <= 0)
+        {
+            savedDpi = targetDpi;
+        }
+
+        var hasInvalidSize = savedRect.Width <= 0 || savedRect.Height <= 0;
+        if (hasInvalidSize)
+        {
+            savedRect = savedRect with { Width = DefaultWidth, Height = DefaultHeight };
+        }
+
+        if (targetDpi != savedDpi)
+        {
+            savedRect = ScaleRect(savedRect, savedDpi, targetDpi);
+        }
+
+        var clampedSize = ClampSize(savedRect.Width, savedRect.Height, workArea);
+
+        var shouldRecenter = hasInvalidSize ||
+                             IsOffscreen(savedRect, workArea) ||
+                             savedScreenSize.Width != workArea.Width ||
+                             savedScreenSize.Height != workArea.Height;
+
+        if (shouldRecenter)
+        {
+            return CenterRectInWorkArea(clampedSize, workArea);
+        }
+
+        return new RectInt32(savedRect.X, savedRect.Y, clampedSize.Width, clampedSize.Height);
+    }
+
+    private static int GetDpiForDisplay(DisplayArea displayArea)
+    {
+        var hMonitor = Win32Interop.GetMonitorFromDisplayId(displayArea.DisplayId);
+        if (hMonitor == IntPtr.Zero)
+        {
+            return DefaultDpi;
+        }
+
+        var hr = PInvoke.GetDpiForMonitor(
+            new HMONITOR(hMonitor),
+            MONITOR_DPI_TYPE.MDT_EFFECTIVE_DPI,
+            out var dpiX,
+            out _);
+
+        return hr.Succeeded && dpiX > 0 ? (int)dpiX : DefaultDpi;
+    }
+
+    private static SizeInt32 ScaleSize(SizeInt32 size, int fromDpi, int toDpi)
+    {
+        if (fromDpi <= 0 || toDpi <= 0 || fromDpi == toDpi)
+        {
+            return size;
+        }
+
+        var scale = (double)toDpi / fromDpi;
+        return new SizeInt32(
+            (int)Math.Round(size.Width * scale),
+            (int)Math.Round(size.Height * scale));
+    }
+
+    private static RectInt32 ScaleRect(RectInt32 rect, int fromDpi, int toDpi)
+    {
+        var scale = (double)toDpi / fromDpi;
+        return new RectInt32(
+            (int)Math.Round(rect.X * scale),
+            (int)Math.Round(rect.Y * scale),
+            (int)Math.Round(rect.Width * scale),
+            (int)Math.Round(rect.Height * scale));
+    }
+
+    private static SizeInt32 ClampSize(int width, int height, RectInt32 workArea) =>
+        new(Math.Min(width, workArea.Width), Math.Min(height, workArea.Height));
+
+    private static RectInt32 CenterRectInWorkArea(SizeInt32 size, RectInt32 workArea) =>
+        new(
+            workArea.X + ((workArea.Width - size.Width) / 2),
+            workArea.Y + ((workArea.Height - size.Height) / 2),
+            size.Width,
+            size.Height);
+
+    private static bool IsOffscreen(RectInt32 rect, RectInt32 workArea) =>
+        rect.X + MinimumVisibleSize > workArea.X + workArea.Width ||
+        rect.X + rect.Width - MinimumVisibleSize < workArea.X ||
+        rect.Y + MinimumVisibleSize > workArea.Y + workArea.Height ||
+        rect.Y + rect.Height - MinimumVisibleSize < workArea.Y;
+}