feat(physics): added area & inertia calculations for the shape and circles

Engine.Core/Primitives/Circle.cs

@@ -34,6 +34,16 @@ public readonly struct Circle(Vector2D center, float radius) : IEquatable<Circle
     /// </summary>
     public readonly float Diameter => 2f * Radius;
 
+    /// <summary>
+    /// Gets the area of the <see cref="Circle"/>.
+    /// </summary>
+    public readonly float Area => Math.Pi * RadiusSquared;
+
+    /// <summary>
+    /// Gets the geometric interia of the <see cref="Circle"/>.
+    /// </summary>
+    public readonly float GeometricInertia => .5f * RadiusSquared;
+
     /// <summary>
     /// A predefined unit <see cref="Circle"/> with a center at the origin and a radius of 1.
     /// </summary>

Engine.Core/Primitives/Shape2D.cs

@@ -22,6 +22,31 @@ public class Shape2D(List<Vector2D> vertices) : IEnumerable<Vector2D>
 
     private readonly List<Vector2D> _vertices = vertices;
 
+    /// <summary>
+    /// Gets the area of the <see cref="Shape2D"/>.
+    /// </summary>
+    public float Area
+    {
+        get
+        {
+            float area = 0f;
+
+            for (int i = 0; i < _vertices.Count; i++)
+            {
+                Vector2D a = _vertices[i];
+                Vector2D b = _vertices[(i + 1) % _vertices.Count];
+                area += a.Cross(b);
+            }
+
+            return area.Abs() * .5f;
+        }
+    }
+
+    /// <summary>
+    /// Gets the geometric interia of the <see cref="Shape2D"/>.
+    /// </summary>
+    public float GeometricInertia => GetGeometricInertia(this, Vector2D.Zero);
+
     /// <summary>
     /// Gets the vertices of the <see cref="Shape2D"/>.
     /// </summary>
@@ -112,6 +137,40 @@ public class Shape2D(List<Vector2D> vertices) : IEnumerable<Vector2D>
         return new Triangle(p1, p2, p3);
     }
 
+    /// <summary>
+    /// Gets the geometric interia of the <see cref="Shape2D"/>.
+    /// </summary>
+    /// <param name="shape">The shape to get the geometrical interia of.</param>
+    /// <param name="centerOfMass">The point in space to calculate the geometrical interia from.</param>
+    /// <returns>The geometrical interia of the <see cref="Shape2D"/>.</returns>
+    public static float GetGeometricInertia(Shape2D shape, Vector2D centerOfMass)
+    {
+        float geometricInertia = 0f;
+
+        for (int i = 0; i < shape._vertices.Count; i++)
+        {
+            Vector2D p1 = centerOfMass.FromTo(shape._vertices[i]);
+            Vector2D p2 = centerOfMass.FromTo(shape._vertices[(i + 1) % shape._vertices.Count]);
+
+            float cross = p1.Cross(p2);
+            float dot = p1.Dot(p1) + p1.Dot(p2) + p2.Dot(p2);
+
+            geometricInertia += cross * dot;
+        }
+
+        return geometricInertia.Abs() / 12f;
+    }
+
+    /// <summary>
+    /// Gets the  interia of the <see cref="Shape2D"/>.
+    /// </summary>
+    /// <param name="shape">The shape to get the interia of.</param>
+    /// <param name="centerOfMass">The point in space to calculate the geometrical interia from.</param>
+    /// <param name="mass">Mass of the shape.</param>
+    /// <returns>The interia of the <see cref="Shape2D"/>.</returns>
+    public static float GetInertia(Shape2D shape, Vector2D centerOfMass, float mass)
+        => GetGeometricInertia(shape, centerOfMass) * mass;
+
     /// <summary>
     /// Triangulates the given convex <see cref="Shape2D"/>.
     /// </summary>
@@ -297,6 +356,12 @@ public static class Shape2DExtensions
     /// <inheritdoc cref="Shape2D.GetSuperTriangle(Shape2D)" />
     public static Triangle ToSuperTriangle(this Shape2D shape) => Shape2D.GetSuperTriangle(shape);
 
+    /// <inheritdoc cref="Shape2D.GetGeometricInertia(Shape2D, Vector2D)" />
+    public static float GetGeometricInertia(this Shape2D shape, Vector2D centerOfMass) => Shape2D.GetGeometricInertia(shape, centerOfMass);
+
+    /// <inheritdoc cref="Shape2D.GetInertia(Shape2D, Vector2D, float)" />
+    public static float GetInertia(this Shape2D shape, Vector2D centerOfMass, float mass) => Shape2D.GetInertia(shape, centerOfMass, mass);
+
     /// <inheritdoc cref="Shape2D.TriangulateConvex(Shape2D, IList{Triangle})" />
     public static void ToTrianglesConvex(this Shape2D shape, IList<Triangle> triangles) => Shape2D.TriangulateConvex(shape, triangles);