feat: add barycentric helpers

Author: mrdav30

src/FixedMathSharp/Core/FixedMath.cs

@@ -379,6 +379,34 @@ public static Fixed64 BarycentricCoordinate(
             Fixed64 weightC
         ) => coordA + (coordB - coordA) * weightB + (coordC - coordA) * weightC;
 
+        /// <summary>
+        /// Returns the second-order scalar product sum for three barycentric vertices.
+        /// </summary>
+        /// <remarks>
+        /// Computes <c>a * a + b * b + c * c + a * b + a * c + b * c</c>.
+        /// </remarks>
+        [MethodImpl(MethodImplOptions.AggressiveInlining)]
+        public static Fixed64 SumSquaredBarycentricProducts(Fixed64 a, Fixed64 b, Fixed64 c) =>
+            (a * a) + (b * b) + (c * c) + (a * b) + (a * c) + (b * c);
+
+        /// <summary>
+        /// Returns the cross scalar product sum for two sets of three barycentric vertices.
+        /// </summary>
+        [MethodImpl(MethodImplOptions.AggressiveInlining)]
+        public static Fixed64 SumBarycentricProducts(
+            Fixed64 firstA,
+            Fixed64 firstB,
+            Fixed64 firstC,
+            Fixed64 secondA,
+            Fixed64 secondB,
+            Fixed64 secondC)
+        {
+            Fixed64 firstSum = firstA + firstB + firstC;
+            Fixed64 secondSum = secondA + secondB + secondC;
+            Fixed64 matchingProducts = (firstA * secondA) + (firstB * secondB) + (firstC * secondC);
+            return firstSum * secondSum + matchingProducts;
+        }
+
         /// <summary>
         /// Adds two fixed-point numbers by adding their raw Q32.32 payloads without saturation.
         /// </summary>

src/FixedMathSharp/Numerics/Matrices/Fixed3x3.cs

@@ -359,6 +359,28 @@ public static Fixed3x3 CreateScale(Vector3d scale) =>
     public static Fixed3x3 CreateScale(Fixed64 scaleFactor) =>
         CreateScale(new Vector3d(scaleFactor, scaleFactor, scaleFactor));
 
+    /// <summary>
+    /// Creates a symmetric matrix containing second-order barycentric product sums for three vectors.
+    /// </summary>
+    /// <remarks>
+    /// The diagonal contains squared component sums, and the off-diagonal elements contain cross-component sums.
+    /// </remarks>
+    [MethodImpl(MethodImplOptions.AggressiveInlining)]
+    public static Fixed3x3 CreateBarycentricProductSums(Vector3d a, Vector3d b, Vector3d c)
+    {
+        Fixed64 xx = FixedMath.SumSquaredBarycentricProducts(a.X, b.X, c.X);
+        Fixed64 yy = FixedMath.SumSquaredBarycentricProducts(a.Y, b.Y, c.Y);
+        Fixed64 zz = FixedMath.SumSquaredBarycentricProducts(a.Z, b.Z, c.Z);
+        Fixed64 xy = FixedMath.SumBarycentricProducts(a.X, b.X, c.X, a.Y, b.Y, c.Y);
+        Fixed64 xz = FixedMath.SumBarycentricProducts(a.X, b.X, c.X, a.Z, b.Z, c.Z);
+        Fixed64 yz = FixedMath.SumBarycentricProducts(a.Y, b.Y, c.Y, a.Z, b.Z, c.Z);
+
+        return new Fixed3x3(
+            xx, xy, xz,
+            xy, yy, yz,
+            xz, yz, zz);
+    }
+
     /// <summary>
     /// Normalizes the basis vectors of a 3x3 matrix to ensure they are orthogonal and unit length.
     /// </summary>

tests/FixedMathSharp.Tests/Core/FixedMath.Tests.cs

@@ -458,6 +458,28 @@ public void BarycentricCoordinate_WeightsSecondAndThirdValues()
         Assert.Equal(new Fixed64(25), FixedMath.BarycentricCoordinate(new Fixed64(10), new Fixed64(20), new Fixed64(30), Fixed64.Half, Fixed64.Half));
     }
 
+    [Fact]
+    public void SumSquaredBarycentricProducts_ReturnsSecondOrderSimplexProductSum()
+    {
+        var result = FixedMath.SumSquaredBarycentricProducts(new Fixed64(2), new Fixed64(3), new Fixed64(5));
+
+        Assert.Equal(new Fixed64(69), result);
+    }
+
+    [Fact]
+    public void SumBarycentricProducts_ReturnsCrossSimplexProductSum()
+    {
+        var result = FixedMath.SumBarycentricProducts(
+            new Fixed64(1),
+            new Fixed64(2),
+            new Fixed64(3),
+            new Fixed64(4),
+            new Fixed64(5),
+            new Fixed64(6));
+
+        Assert.Equal(new Fixed64(122), result);
+    }
+
     #endregion
 
     #region Test: MoveTowards Method

tests/FixedMathSharp.Tests/Numerics/Matrices/Fixed3x3.Tests.cs

@@ -73,6 +73,22 @@ public void CreateScale_WithUniformFixed64_SetsAllDiagonalComponents()
             matrix);
     }
 
+    [Fact]
+    public void CreateBarycentricProductSums_ReturnsSymmetricProductSumMatrix()
+    {
+        var result = Fixed3x3.CreateBarycentricProductSums(
+            new Vector3d(1, 2, 3),
+            new Vector3d(4, 5, 6),
+            new Vector3d(7, 8, 9));
+
+        Assert.Equal(
+            new Fixed3x3(
+                new Fixed64(105), new Fixed64(258), new Fixed64(306),
+                new Fixed64(258), new Fixed64(159), new Fixed64(378),
+                new Fixed64(306), new Fixed64(378), new Fixed64(225)),
+            result);
+    }
+
     [Fact]
     public void InvertDiagonal_WorksCorrectly()
     {