Update derivation

docs/src/triangle.md

@@ -1,69 +1,17 @@
-# Surface Integral Over a Triangle
+# Integrating a Triangle
 
-**Note: This explanation reflects my original method but is no longer valid.
-An updated derivation is planned. **
-
-A linear transformation can be applied that maps any triangle onto a Barycentric
-coordinate system. A linear correction factor is then applied to correct for the
-domain transformation, where the area of the original triangle is $A$ and the area
-of the Barycentric triangle is $1/2$.
-```math
-\int_\triangle f(\bar{r}) \text{d}A
-    = \frac{A}{1/2} \int_0^1 \int_0^{1-v} f(u,v) \text{d}v \text{d}u
-```
-
-This Barycentric integral can be directly estimated using a nested application of
-the h-adaptive Gauss-Kronrod quadrature rules from QuadGK.jl (`quadgk_surface`).
-Alternatively, with some additional modification it can be transformed onto a
-fixed rectangular domain for integration via cubature rules or other nested
-quadrature rules.
-
-Let $g$ be a wrapper function for $f$ that is only non-zero at valid Barycentric
-coordinates.
+For a specified `Meshes.Triangle` surface with area $A$, let $u$ and $v$ be Barycentric coordinates that span the surface.
 ```math
-g(u,v,) =
-    \begin{cases}
-        f(u,v) & \text{if } 0 \le u+v \le 1 \\
-        0 & \text{otherwise}
-    \end{cases}
+\int_\triangle f(\bar{r}) \, \text{d}A
+    = \iint_\triangle f\left( \bar{r}(u,v) \right) \, \left( \text{d}u \wedge \text{d}v \right)
 ```
 
-Then
+Since the geometric transformation from the originally-arbitrary domain to a Barycentric domain is linear, the magnitude of the surface element $\text{d}u \wedge \text{d}v$ is constant throughout the integration domain. This constant will be equal to twice the magnitude of $A$.
 ```math
-\int_0^1 \int_0^{1-v} f(u,v) \text{d}v \text{d}u
-    = \int_0^1 \int_0^1 g(u,v) \text{d}v \text{d}u
+\int_\triangle f(\bar{r}) \, \text{d}A
+    = 2A \int_0^1 \int_0^{1-v} f\left( \bar{r}(u,v) \right) \, \text{d}u \, \text{d}v
 ```
 
-A domain transformation can be applied to map the Barycentric coordinate domains
-from $u,v \in [0,1]$ to $s,t \in [-1,1]$, enabling the application of Gauss-Legendre
-quadrature rules.
-```math
-s(u) = 2u - 1 \\
-u(s) = \frac{s+1}{2}
-\text{d}u = \frac{1}{2}\text{d}s
-```
-```math
-t(v) = 2v - 1 \\
-v(t) = \frac{t+1}{2}
-\text{d}v = \frac{1}{2}\text{d}t
-```
+This non-rectangular Barycentric domain prevents a direct application of most numerical integration methods. It can be directly integrated, albeit inefficiently, using nested Gauss-Kronrod quadrature rules. Alternatively, additional transformation could be applied to map this domain onto a rectangular domain.
 
-Leading to
-```math
-\int_0^1 \int_0^1 g(u,v) \text{d}v \text{d}u
-    = \frac{1}{4} \int_{-1}^1 \int_{-1}^1 g(\frac{s+1}{2},\frac{t+1}{2}) \text{d}t \text{d}s
-```
-
-Gauss-Legendre nodes ($x \in [-1,1]$) and weights ($w$) for a rule of order $N$
-can be efficiently calculated using the FastGaussQuadrature.jl package.
-```math
-\int_{-1}^1 \int_{-1}^1 g(\frac{s+1}{2},\frac{t+1}{2}) \text{d}t \text{d}s
-    \approx \sum_{i=1}^N \sum_{j=1}_N w_i w_j f(x_i,x_j)
-```
-
-This approximation can be rolled back up the stack of equations, leading to an
-expression that numerically approximates the original integral problem as
-```math
-\int_\triangle f(\bar{r}) \text{d}A
-    = \frac{A}{4(1/2)} \sum_{i=1}^N \sum_{j=1}_N w_i w_j f(\frac{x_i+1}{2}, \frac{x_j+1}{2})
-```
+**WORK IN PROGRESS:** continued derivation to detail this barycentric-rectangular domain transformation