Improve docs (#61)

* Incorporate triangle doc

* Update derivation

* Add Support Matrix and API pages

* Remove external references

docs/make.jl

@@ -1,6 +1,18 @@
 using Documenter
 using MeshIntegrals
 
-makedocs(sitename="MeshIntegrals.jl Documentation")
+makedocs(
+    sitename="MeshIntegrals.jl",
+    pages = [
+        "Home" => [
+            "About" => "index.md",
+            "Support Matrix" => "supportmatrix.md"
+        ],
+        "Derivations" => [
+            "Integrating a Triangle" => "triangle.md"
+        ],
+        "Public API" => "api.md"
+    ]
+)
 
 deploydocs(repo = "github.com/mikeingold/MeshIntegrals.jl.git")

docs/src/api.md

@@ -0,0 +1,23 @@
+# Public API
+
+## Integrals
+
+```@docs
+MeshIntegrals.integral
+```
+
+### Aliases
+
+```@docs
+MeshIntegrals.lineintegral
+MeshIntegrals.surfaceintegral
+MeshIntegrals.volumeintegral
+```
+
+## Integration Algorithms
+
+```@docs
+MeshIntegrals.GaussKronrod
+MeshIntegrals.GaussLegendre
+MeshIntegrals.HAdaptiveCubature
+```

docs/src/supportmatrix.md

@@ -0,0 +1,38 @@
+# Support Matrix
+
+| Symbol | Meaning |
+|--------|---------|
+| :white_check_mark: | Implemented, passes tests |
+| :x: | Not yet supported |
+| :stop_sign: | Not supported |
+
+### Integral
+| Geometry | Gauss-Legendre | Gauss-Kronrod | H-Adaptive Cubature |
+|----------|----------------|---------------|---------------------|
+| `Ball` in `𝔼{2}` | :white_check_mark: | :white_check_mark: | :white_check_mark: |
+| `Ball` in `𝔼{3}` | :white_check_mark: | :stop_sign: | :white_check_mark: |
+| `BezierCurve` | :white_check_mark: | :white_check_mark: | :white_check_mark: |
+| `Box` in `𝔼{1}` | :white_check_mark: | :white_check_mark: | :white_check_mark: |
+| `Box` in `𝔼{2}` | :white_check_mark: | :white_check_mark: | :white_check_mark: |
+| `Box` in `𝔼{3}` | :white_check_mark: | :stop_sign: | :white_check_mark: |
+| `Circle` | :white_check_mark: | :white_check_mark: | :white_check_mark: |
+| `Cone` | :white_check_mark: | :white_check_mark: | :white_check_mark: |
+| `ConeSurface` | :white_check_mark: | :white_check_mark: | :white_check_mark: |
+| `Cylinder` | :white_check_mark: | :stop_sign: | :white_check_mark: |
+| `CylinderSurface` | :white_check_mark: | :white_check_mark: | :white_check_mark: |
+| `Disk` | :white_check_mark: | :white_check_mark: | :white_check_mark: |
+| `Frustum` | :stop_sign: | :stop_sign: | :stop_sign: |
+| `FrustumSurface` | :stop_sign: | :stop_sign: | :stop_sign: |
+| `Line` | :white_check_mark: | :white_check_mark: | :white_check_mark: |
+| `ParaboloidSurface` | :white_check_mark: | :white_check_mark: | :white_check_mark: |
+| `Plane` | :white_check_mark: | :white_check_mark: | :white_check_mark: |
+| `Ray` | :white_check_mark: | :white_check_mark: | :white_check_mark: |
+| `Ring` | :white_check_mark: | :white_check_mark: | :white_check_mark: |
+| `Rope` | :white_check_mark: | :white_check_mark: | :white_check_mark: |
+| `Segment` | :white_check_mark: | :white_check_mark: | :white_check_mark: |
+| `SimpleMesh` | :x: | :x: | :x: |
+| `Sphere` in `𝔼{2}` | :white_check_mark: | :white_check_mark: | :white_check_mark: |
+| `Sphere` in `𝔼{3}` | :white_check_mark: | :white_check_mark: | :white_check_mark: |
+| `Tetrahedron` in `𝔼{3}` | :x: | :white_check_mark: | :x: |
+| `Triangle` | :white_check_mark: | :white_check_mark: | :white_check_mark: |
+| `Torus` | :white_check_mark: | :white_check_mark: | :white_check_mark: |

docs/src/triangle.md

@@ -0,0 +1,17 @@
+# Integrating a Triangle
+
+For a specified `Meshes.Triangle` surface with area $A$, let $u$ and $v$ be Barycentric coordinates that span the surface.
+```math
+\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)
+```
+
+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_\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
+```
+
+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.
+
+**WORK IN PROGRESS:** continued derivation to detail this barycentric-rectangular domain transformation

src/integral.jl

@@ -8,7 +8,7 @@ abstract type IntegrationAlgorithm end
     GaussKronrod(kwargs...)
 
 Numerically integrate using the h-adaptive Gauss-Kronrod quadrature rule implemented
-by QuadGK.jl. All standard [`QuadGK.quadgk`](@ref) keyword arguments are supported.
+by QuadGK.jl. All standard `QuadGK.quadgk` keyword arguments are supported.
 """
 struct GaussKronrod <: IntegrationAlgorithm
     kwargs
@@ -35,7 +35,7 @@ end
     GaussKronrod(kwargs...)
 
 Numerically integrate areas and surfaces using the h-adaptive cubature rule
-implemented by HCubature.jl. All standard [`HCubature.hcubature`](@ref) keyword
+implemented by HCubature.jl. All standard `HCubature.hcubature` keyword
 arguments are supported.
 """
 struct HAdaptiveCubature <: IntegrationAlgorithm