Set default tolerances to match CoordRefSystems.jl values (#1231)

* Set default tolerances to match CoordRefSystems.jl values

* Update src/tolerances.jl

Co-authored-by: github-actions[bot] <41898282+github-actions[bot]@users.noreply.github.com>

* Update test/tolerances.jl

Co-authored-by: github-actions[bot] <41898282+github-actions[bot]@users.noreply.github.com>

* Adjust tests

* Adjust tests

* Explain tolerance formula in docs

---------

Co-authored-by: github-actions[bot] <41898282+github-actions[bot]@users.noreply.github.com>

Author: juliohm

docs/src/tolerances.md

@@ -1,9 +1,10 @@
 # Tolerances
 
 The absolute tolerance used for floating point comparisons is hard-coded in
-the project to `1e-10` for `Float64` and to `1f-5` for `Float32`. You can use
-[ScopedValues.jl](https://github.com/vchuravy/ScopedValues.jl) to customize
-these tolerance values in specific computations:
+the project to `eps(T) ^ (3 // 4)` where `T` is either `Float64` or `Float32`.
+This formula preserves roughly 3/4 of the significant digits of the type `T`.
+You can use [ScopedValues.jl](https://github.com/vchuravy/ScopedValues.jl) to
+customize these tolerance values in specific computations:
 
 ```julia
 using Meshes

src/tolerances.jl

@@ -2,8 +2,8 @@
 # Licensed under the MIT License. See LICENSE in the project root.
 # ------------------------------------------------------------------
 
-const ATOL64 = ScopedValue(1.0e-10)
-const ATOL32 = ScopedValue(1.0f-5)
+const ATOL64 = ScopedValue(eps(Float64)^(3 // 4))
+const ATOL32 = ScopedValue(eps(Float32)^(3 // 4))
 
 """
     atol(T)

test/intersections.jl

@@ -1011,9 +1011,7 @@ end
   # Special case: the direction vector is not long enough to cross triangle
   r = Ray(cart(0.2, 0.2, 1.0), vector(0.0, 0.0, -0.00001))
   @test intersection(r, t) |> type == Crossing
-  if T === Float64
-    @test r ∩ t ≈ cart(0.2, 0.2, 0.0)
-  end
+  @test evaluate(Euclidean(), r ∩ t, cart(0.2, 0.2, 0.0)) < 1000atol(ℳ)
   # Special case: reverse direction vector should not hit the triangle
   r = Ray(cart(0.2, 0.2, 1.0), vector(0.0, 0.0, 1.0))
   @test intersection(r, t) |> type == NotIntersecting

test/polytopes.jl

@@ -247,8 +247,8 @@ end
   @test nvertices(ur1) < nvertices(r1)
   @test nvertices(ur2) < nvertices(r2)
   if T === Float32
-    @test nvertices(ur1) == 10
-    @test nvertices(ur2) == 10
+    @test nvertices(ur1) == 12
+    @test nvertices(ur2) == 12
   else
     @test nvertices(ur1) == 17
     @test nvertices(ur2) == 17
@@ -652,7 +652,7 @@ end
   upoly = unique(poly)
   @test nvertices(upoly) < nvertices(poly)
   if T === Float32
-    @test nvertices(upoly) == 10
+    @test nvertices(upoly) == 12
   else
     @test nvertices(upoly) == 17
   end

test/tolerances.jl

@@ -2,21 +2,17 @@
   ℒ = ℳ
   𝒜 = typeof(zero(ℳ)^2)
   𝒱 = typeof(zero(ℳ)^3)
-  if T === Float32
-    @test atol(T) == 1.0f-5
-    @test atol(ℒ) == 1.0f-5 * u"m"
-    @test atol(𝒜) == 1.0f-5^2 * u"m^2"
-    @test atol(𝒱) == 1.0f-5^3 * u"m^3"
-  else
-    @test atol(T) == 1.0e-10
-    @test atol(ℒ) == 1.0e-10 * u"m"
-    @test atol(𝒜) == 1.0e-10^2 * u"m^2"
-    @test atol(𝒱) == 1.0e-10^3 * u"m^3"
-  end
+
+  @test atol(T) == eps(T)^(3 // 4)
+  @test atol(ℒ) == atol(T) * u"m"
+  @test atol(𝒜) == atol(T)^2 * u"m^2"
+  @test atol(𝒱) == atol(T)^3 * u"m^3"
+
   @test atol(zero(T)) == atol(T)
   @test atol(zero(ℒ)) == atol(ℒ)
   @test atol(zero(𝒜)) == atol(𝒜)
   @test atol(zero(𝒱)) == atol(𝒱)
+
   @inferred atol(T)
   @inferred atol(ℒ)
   @inferred atol(𝒜)

test/transforms.jl

@@ -2233,7 +2233,7 @@ end
   @test nvertices(rpoly) == nvertices(poly)
   @test length(first(rings(rpoly))) > length(first(rings(poly)))
   opoly = TB.revert(repair, rpoly, cache)
-  @test opoly == poly
+  @test opoly ≈ poly
 end
 
 @testitem "Repair(11)" setup = [Setup] begin