diff --git a/Project.toml b/Project.toml
index 80683af..e48cb40 100644
--- a/Project.toml
+++ b/Project.toml
@@ -1,7 +1,7 @@
 name = "Cartan"
 uuid = "e24af43f-9fd5-4672-9264-a75f3ae40eb2"
 authors = ["Michael Reed"]
-version = "0.3.5"
+version = "0.3.6"
 
 [deps]
 Requires = "ae029012-a4dd-5104-9daa-d747884805df"
diff --git a/src/Cartan.jl b/src/Cartan.jl
index 0bd43bd..52fd14c 100644
--- a/src/Cartan.jl
+++ b/src/Cartan.jl
@@ -27,7 +27,7 @@ module Cartan
 using SparseArrays, LinearAlgebra, Base.Threads, Grassmann, Requires
 import Grassmann: value, vector, valuetype, tangent, istangent, Derivation, radius, ⊕
 import Grassmann: realvalue, imagvalue, points, metrictensor, metricextensor
-import Grassmann: Values, Variables, FixedVector, list, volume
+import Grassmann: Values, Variables, FixedVector, list, volume, compound
 import Grassmann: Scalar, GradedVector, Bivector, Trivector
 import Base: @pure, OneTo, getindex
 import LinearAlgebra: cross
@@ -163,6 +163,11 @@ Grassmann.grade(::GradedField{G}) where G = G
 resize(t::TensorField) = TensorField(resize(domain(t)),codomain(t))
 resize(t::GlobalSection) = GlobalSection(resize(domain(t)),codomain(t))
 
+function resample(t::TensorField,i::NTuple)
+    rg = resample(base(t),i)
+    TensorField(rg,t.(points(rg)))
+end
+
 @pure Base.eltype(::Type{<:TensorField{B,F}}) where {B,F} = LocalTensor{B,F}
 Base.getindex(m::TensorField,i::Vararg{Int}) = LocalTensor(getindex(domain(m),i...), getindex(codomain(m),i...))
 Base.getindex(m::TensorField,i::Vararg{Union{Int,Colon}}) = TensorField(domain(m)(i...), getindex(codomain(m),i...))
@@ -176,6 +181,10 @@ function Base.setindex!(m::TensorField{B,F,N,<:IntervalRange} where {B,F,N},s::L
     setindex!(codomain(m),fiber(s),i...)
     return s
 end
+function Base.setindex!(m::TensorField{B,F,N,<:AlignedSpace} where {B,F,N},s::LocalTensor,i::Vararg{Int})
+    setindex!(codomain(m),fiber(s),i...)
+    return s
+end
 function Base.setindex!(m::TensorField,s::LocalTensor,i::Vararg{Int})
     setindex!(domain(m),base(s),i...)
     setindex!(codomain(m),fiber(s),i...)
@@ -221,6 +230,11 @@ function Base.similar(bc::Broadcast.Broadcasted{Broadcast.ArrayStyle{GlobalSecti
     # Use the domain field of t to create the output
     GlobalSection(domain(t), similar(Array{fibertype(ElType),N}, axes(bc)))
 end
+function Base.similar(bc::Broadcast.Broadcasted{Broadcast.ArrayStyle{<:AbstractFrameBundle{P,N}}}, ::Type{ElType}) where {N,P,ElType}
+    t = find_gf(bc)
+    # Use the data type to create the output
+    TensorField(t,similar(Array{ElType,N}, axes(bc)))
+end
 
 "`A = find_tf(As)` returns the first TensorField among the arguments."
 find_tf(bc::Base.Broadcast.Broadcasted) = find_tf(bc.args)
@@ -247,10 +261,19 @@ for fun ∈ (:Open,:Mirror,:Clamped,:Torus,:Ribbon,:Wing,:Mobius,:Klein,:Cone,:P
         @eval begin
             $top(m::TensorField{B,F,N,<:GridFrameBundle} where {B,F,N}) = TensorField($top(base(m)),fiber(m))
             $top(m::GridFrameBundle) = m($top(size(m)))
+            $top(p::PointArray) = TensorField(GridFrameBundle(p,$top(size(p))))
+            $(Symbol(fun,:Parameter))(p::PointArray) = $top(p)
         end
     end
 end
 
+spacing(x::AbstractVector) = sum(norm.(diff(fiber(x))))/(length(x)-1)
+spacing(x::AbstractArray{T,N}) where {T,N} = minimum(spacing.(Ref(x),list(1,N)))
+function spacing(x::AbstractArray,i)
+    n = norm.(diff(fiber(x),dims=i))
+    sum(n)/length(n)
+end
+
 linterp(x,x1,x2,f1,f2) = f1 + (f2-f1)*(x-x1)/(x2-x1)
 function bilinterp(x,y,x1,x2,y1,y2,f11,f21,f12,f22)
     f1 = linterp(x,x1,x2,f11,f21)
@@ -277,6 +300,13 @@ reposition_odd(p,x,t) = @inbounds (iseven(p) ? x[end]-x[1]+t : 2x[1]-t)
 reposition_even(p,x,t) = @inbounds (isodd(p) ? x[1]-x[end]+t : 2x[end]-t)
 @inline reposition(i1,i2,p1,p2,x,t) = i1 ? reposition_odd(p1,x,t) : i2 ? reposition_even(p2,x,t) : eltype(x)(t)
 
+function searchpoints(p,t)
+    i = searchsortedfirst(p,t)-1
+    i01 = iszero(i)
+    i01 && t==(@inbounds p[1]) ? (i+1,false) : (i,i01)
+end
+
+(m::TensorField)(s::Coordinate) = m(base(s))
 (m::TensorField)(s::LocalTensor) = LocalTensor(base(s), m(fiber(s)))
 (m::Grid{1})(t::Chain) = linterp(m,t)
 (m::Grid{1})(t::AbstractFloat) = linterp(m,t)
@@ -285,7 +315,7 @@ reposition_even(p,x,t) = @inbounds (isodd(p) ? x[1]-x[end]+t : 2x[end]-t)
 function linterp(m,t)
     p,f,t1 = points(m),fiber(m),(@inbounds t[1])
     isnan(t1) && (return zero(fibertype(m))/0)
-    i = searchsortedfirst(p,@inbounds t1)-1
+    i,i0 = searchpoints(p,t1)
     if !isopen(m)
         q = immersion(m)
         if iszero(i)
@@ -311,7 +341,7 @@ end
 end=#
 function parametric(t,m,d=diff(codomain(m))./diff(domain(m)))
     p = points(m)
-    i = searchsortedfirst(p,t)-1
+    i,i0 = searchpoints(p,t)
     codomain(m)[i]+(t-p[i])*d[i]
 end
 
@@ -319,7 +349,7 @@ end
 leaf(m::RectangleMap,i::Int,j::Int=2) = isone(j) ? m[i,:] : m[:,i]
 function leaf(m::RectangleMap,t::AbstractFloat,j::Int=2)
     Q,p = isone(j),points(m).v[j]
-    i = searchsortedfirst(p,t)-1
+    i,i0 = searchpoints(p,t)
     TensorField(points(m).v[Q ? 2 : 1],linterp(t,p[i],p[i+1],Q ? m.cod[i,:] : m.cod[:,i],Q ? m.cod[i+1,:] : m.cod[:,i+1]))
 end
 
@@ -334,14 +364,15 @@ end
 leaf(m::TensorField{B,F,2,<:FiberProductBundle} where {B,F},i::Int) = TensorField(base(base(m)),fiber(m)[:,i])
 function (m::TensorField{B,F,2,<:FiberProductBundle} where {B,F})(t::Real)
     k = 2; p = base(m).cod.v[1]
-    i = searchsortedfirst(p,t)-1
+    i,i0 = searchpoints(p,t)
     TensorField(base(base(m)),linterp(t,p[i],p[i+1],m.cod[:,i],m.cod[:,i+1]))
 end
 
 #(m::TensorField)(t::TensorField) = TensorField(base(t),m.(fiber(t)))
 #(m::GridFrameBundle)(t::TensorField) = GridFrameBundle(PointArray(points(t),m.(fiber(t))),immersion(m))
 (X::VectorField{B,F,N} where {B,F})(Y::VectorField{B,<:Chain{V,1,T,N} where {V,T},1} where B) where N = TensorField(base(Y),X.(fiber(Y)))
-(m::GridFrameBundle{N})(t::VectorField{B,<:Chain{V,1,T,N} where {V,T},1} where B) where N = GridFrameBundle(PointArray(points(t),m.(fiber(t))),immersion(m))
+(m::GridFrameBundle{N})(t::VectorField{B,<:Chain{V,1,T,N} where {V,T},1} where B) where N = TensorField(GridFrameBundle(PointArray(points(t),m.(fiber(t))),immersion(t)),fiber(t))
+#(m::GridFrameBundle{N})(t::VectorField{B,<:Chain{V,1,T,N} where {V,T},1} where B) where N = GridFrameBundle(PointArray(points(t),m.(fiber(t))),immersion(t))
 
 (m::SimplexFrameBundle)(t::Chain) = sinterp(m,t)
 (m::TensorField{B,F,N,<:SimplexFrameBundle} where {B,F,N})(t::Chain) = sinterp(m,t)
@@ -359,12 +390,12 @@ end
 function bilinterp(m,t::Chain{V,G,T,2} where {G,T}) where V
     x,y,f,t1,t2 = @inbounds (points(m).v[1],points(m).v[2],fiber(m),t[1],t[2])
     (isnan(t1) || isnan(t2)) && (return zero(fibertype(m))/0)
-    i,j = searchsortedfirst(x,t1)-1,searchsortedfirst(y,t2)-1
+    (i,i01),(j,j01) = searchpoints(x,t1),searchpoints(y,t2)
     if !isopen(m)
         q = immersion(m)
-        i01,i02,iq1,iq2,j01,j02,jq1,jq2 = @inbounds (
-            iszero(i),i==length(x),iszero(q.r[1]),iszero(q.r[2]),
-            iszero(j),j==length(y),iszero(q.r[3]),iszero(q.r[4]))
+        i02,iq1,iq2,j02,jq1,jq2 = @inbounds (
+            i==length(x),iszero(q.r[1]),iszero(q.r[2]),
+            j==length(y),iszero(q.r[3]),iszero(q.r[4]))
         if (i01 && iq1) || (i02 && iq2) || (j01 && jq1) || (j02 && jq2)
             return zero(fibertype(m))
         else
@@ -377,7 +408,7 @@ function bilinterp(m,t::Chain{V,G,T,2} where {G,T}) where V
                     (@inbounds reposition(j1,j2,q.p[3],q.p[4],y,t2))))
             end
         end
-    elseif iszero(i) || iszero(j) || i==length(x) || j==length(y)
+    elseif i01 || j01 || i==length(x) || j==length(y)
         # elseif condition creates 1 allocation, as opposed to 0 ???
         return zero(fibertype(m))
     end
@@ -395,16 +426,13 @@ end
 function trilinterp(m,t::Chain{V,G,T,3} where {G,T}) where V
     x,y,z,f,t1,t2,t3 = @inbounds (points(m).v[1],points(m).v[2],points(m).v[3],fiber(m),t[1],t[2],t[3])
     (isnan(t1) || isnan(t2) || isnan(t3)) && (return zero(fibertype(m))/0)
-    i,j,k = (
-        searchsortedfirst(x,t1)-1,
-        searchsortedfirst(y,t2)-1,
-        searchsortedfirst(z,t3)-1)
+    (i,i01),(j,j01),(k,k01) = (searchpoints(x,t1),searchpoints(y,t2),searchpoints(z,t3))
     if !isopen(m)
         q = immersion(m)
-        i01,i02,iq1,iq2,j01,j02,jq1,jq2,k01,k02,kq1,kq2 = @inbounds (
-            iszero(i),i==length(x),iszero(q.r[1]),iszero(q.r[2]),
-            iszero(j),j==length(y),iszero(q.r[3]),iszero(q.r[4]),
-            iszero(k),k==length(z),iszero(q.r[5]),iszero(q.r[6]))
+        i02,iq1,iq2,j02,jq1,jq2,k02,kq1,kq2 = @inbounds (
+            i==length(x),iszero(q.r[1]),iszero(q.r[2]),
+            j==length(y),iszero(q.r[3]),iszero(q.r[4]),
+            k==length(z),iszero(q.r[5]),iszero(q.r[6]))
         if (i01 && iq1) || (i02 && iq2) || (j01 && jq1) || (j02 && jq2) || (k01 && kq1) || (k02 && kq2)
             return zero(fibertype(m))
         else
@@ -419,7 +447,7 @@ function trilinterp(m,t::Chain{V,G,T,3} where {G,T}) where V
                     (@inbounds reposition(k1,k2,q.p[5],q.p[6],z,t3))))
             end
         end
-    elseif iszero(i) || iszero(j) || iszero(k) || i==length(x) || j==length(y) || k==length(z)
+    elseif i01 || j01 || k01 || i==length(x) || j==length(y) || k==length(z)
         # elseif condition creates 1 allocation, as opposed to 0 ???
         return zero(fibertype(m))
     end
@@ -442,18 +470,14 @@ end
 function (m)(t::Chain{V,G,T,4} where {G,T}) where V
     x,y,z,w,f,t1,t2,t3,t4 = @inbounds (points(m).v[1],points(m).v[2],points(m).v[3],points(m).v[4],fiber(m),t[1],t[2],t[3],t[4])
     (isnan(t1) || isnan(t2) || isnan(t3) ||isnan(t4)) && (return zero(fibertype(m))/0)
-    i,j,k,l = (
-        searchsortedfirst(x,t1)-1,
-        searchsortedfirst(y,t2)-1,
-        searchsortedfirst(z,t3)-1,
-        searchsortedfirst(w,t4)-1)
+    (i,i01),(j,j01),(k,k01),(l,l01) = (searchpoints(x,t1),searchpoints(y,t2),searchpoints(z,t3),searchpoints(w,t4))
     if !isopen(m)
         q = immersion(m)
-        i01,i02,iq1,iq2,j01,j02,jq1,jq2,k01,k02,kq1,kq2,l01,l02,lq1,lq2 = @inbounds (
-            iszero(i),i==length(x),iszero(q.r[1]),iszero(q.r[2]),
-            iszero(j),j==length(y),iszero(q.r[3]),iszero(q.r[4]),
-            iszero(k),k==length(z),iszero(q.r[5]),iszero(q.r[6]),
-            iszero(l),l==length(w),iszero(q.r[7]),iszero(q.r[8]))
+        i02,iq1,iq2,j02,jq1,jq2,k02,kq1,kq2,l02,lq1,lq2 = @inbounds (
+            i==length(x),iszero(q.r[1]),iszero(q.r[2]),
+            j==length(y),iszero(q.r[3]),iszero(q.r[4]),
+            k==length(z),iszero(q.r[5]),iszero(q.r[6]),
+            l==length(w),iszero(q.r[7]),iszero(q.r[8]))
         if (i01 && iq1) || (i02 && iq2) || (j01 && jq1) || (j02 && jq2) || (k01 && kq1) || (k02 && kq2) || (l01 && lq1) || (l02 && lq2)
             return zero(fibertype(m))
         else
@@ -470,7 +494,7 @@ function (m)(t::Chain{V,G,T,4} where {G,T}) where V
                     (@inbounds reposition(l1,l2,q.p[7],q.p[8],w,t4))))
             end
         end
-    elseif iszero(i) || iszero(j) || iszero(k) || iszero(l) || i==length(x) || j==length(y) || k==length(z) || l==length(w)
+    elseif i01 || j01 || k01 || l01 || i==length(x) || j==length(y) || k==length(z) || l==length(w)
         # elseif condition creates 1 allocation, as opposed to 0 ???
         return zero(fibertype(m))
     end
@@ -503,20 +527,15 @@ end
 function quintlinterp(m,t::Chain{V,G,T,5} where {G,T}) where V
     x,y,z,w,v,f,t1,t2,t3,t4,t5 = @inbounds (points(m).v[1],points(m).v[2],points(m).v[3],points(m).v[4],points(m).v[5],fiber(m),t[1],t[2],t[3],t[4],t[5])
     (isnan(t1) || isnan(t2) || isnan(t3) || isnan(t4) || isnan(t5)) && (return zero(fibertype(m))/0)
-    i,j,k,l,o = (
-        searchsortedfirst(x,t1)-1,
-        searchsortedfirst(y,t2)-1,
-        searchsortedfirst(z,t3)-1,
-        searchsortedfirst(w,t4)-1,
-        searchsortedfirst(v,t5)-1)
+    (i,i01),(j,j01),(k,k01),(l,l01),(o,o01) = (searchpoints(x,t1),searchpoints(y,t2),searchpoints(z,t3),searchpoints(w,t4),searchpoints(v,t5))
     if !isopen(m)
         q = immersion(m)
-        i01,i02,iq1,iq2,j01,j02,jq1,jq2,k01,k02,kq1,kq2,l01,l02,lq1,lq2,o01,o02,oq1,oq2 = @inbounds (
-            iszero(i),i==length(x),iszero(q.r[1]),iszero(q.r[2]),
-            iszero(j),j==length(y),iszero(q.r[3]),iszero(q.r[4]),
-            iszero(k),k==length(z),iszero(q.r[5]),iszero(q.r[6]),
-            iszero(l),l==length(w),iszero(q.r[7]),iszero(q.r[8]),
-            iszero(o),o==length(v),iszero(q.r[9]),iszero(q.r[10]))
+        i02,iq1,iq2,j02,jq1,jq2,k02,kq1,kq2,l02,lq1,lq2,o02,oq1,oq2 = @inbounds (
+            i==length(x),iszero(q.r[1]),iszero(q.r[2]),
+            j==length(y),iszero(q.r[3]),iszero(q.r[4]),
+            k==length(z),iszero(q.r[5]),iszero(q.r[6]),
+            l==length(w),iszero(q.r[7]),iszero(q.r[8]),
+            o==length(v),iszero(q.r[9]),iszero(q.r[10]))
         if (i01 && iq1) || (i02 && iq2) || (j01 && jq1) || (j02 && jq2) || (k01 && kq1) || (k02 && kq2) || (l01 && lq1) || (l02 && lq2) || (o01 && oq1) || (o02 && oq2)
             return zero(fibertype(m))
         else
@@ -535,7 +554,7 @@ function quintlinterp(m,t::Chain{V,G,T,5} where {G,T}) where V
                     (@inbounds reposition(o1,o2,q.p[9],q.p[10],v,t5))))
             end
         end
-    elseif iszero(i) || iszero(j) || iszero(k) || iszero(l) || iszero(o) || i==length(x) || j==length(y) || k==length(z) || l==length(w) || o==length(v)
+    elseif i01 || j01 || k01 || l01 || o01 || i==length(x) || j==length(y) || k==length(z) || l==length(w) || o==length(v)
         # elseif condition creates 1 allocation, as opposed to 0 ???
         return zero(fibertype(m))
     end
@@ -572,14 +591,27 @@ for op ∈ (:+,:-,:&,:∧,:∨)
         end
     end
 end
+(m::TensorNested)(t::TensorField) = TensorField(base(t), m.(fiber(t)))
+(m::TensorField{B,<:TensorNested} where B)(t::TensorField) = m⋅t
 @inline Base.:<<(a::GlobalFiber,b::GlobalFiber) = contraction(b,~a)
 @inline Base.:>>(a::GlobalFiber,b::GlobalFiber) = contraction(~a,b)
 @inline Base.:<(a::GlobalFiber,b::GlobalFiber) = contraction(b,a)
 Base.sign(a::TensorField) = TensorField(base(a), sign.(fiber(Real(a))))
 Base.inv(a::TensorField{B,<:Real} where B) = TensorField(base(a), inv.(fiber(a)))
 Base.inv(a::TensorField{B,<:Complex} where B) = TensorField(base(a), inv.(fiber(a)))
+Base.:*(a::TensorField{B,<:Real} where B,b::TensorField{B,<:Real} where B) = TensorField(base(a), fiber(a).*fiber(b))
+Base.:*(a::TensorField{B,<:Complex} where B,b::TensorField{B,<:Complex} where B) = TensorField(base(a), fiber(a).*fiber(b))
+Base.:*(a::TensorField{B,<:Real} where B,b::TensorField{B,<:Complex} where B) = TensorField(base(a), fiber(a).*fiber(b))
+Base.:*(a::TensorField{B,<:Complex} where B,b::TensorField{B,<:Real} where B) = TensorField(base(a), fiber(a).*fiber(b))
+Base.:*(a::TensorField{B,<:Real} where B,b::TensorField) = TensorField(base(a), fiber(a).*fiber(b))
+Base.:*(a::TensorField{B,<:Complex} where B,b::TensorField) = TensorField(base(a), fiber(a).*fiber(b))
+Base.:*(a::TensorField,b::TensorField{B,<:Real} where B) = TensorField(base(a), fiber(a).*fiber(b))
+Base.:*(a::TensorField,b::TensorField{B,<:Complex} where B) = TensorField(base(a), fiber(a).*fiber(b))
 Base.:/(a::TensorField,b::TensorField{B,<:Real} where B) = TensorField(base(a), fiber(a)./fiber(b))
 Base.:/(a::TensorField,b::TensorField{B,<:Complex} where B) = TensorField(base(a), fiber(a)./fiber(b))
+LinearAlgebra.:×(a::TensorField{R},b::TensorField{R}) where R = TensorField(base(a), .⋆(fiber(a).∧fiber(b),refmetric(base(a))))
+Grassmann.compound(t::TensorField,i::Val) = TensorField(base(t), compound.(fiber(t),i))
+Grassmann.compound(t::TensorField,i::Int) = TensorField(base(t), compound.(fiber(t),Val(i)))
 Grassmann.eigen(t::TensorField,i::Val) = TensorField(base(t), eigen.(fiber(t),i))
 Grassmann.eigen(t::TensorField,i::Int) = TensorField(base(t), eigen.(fiber(t),Val(i)))
 Grassmann.eigvals(t::TensorField,i::Val) = TensorField(base(t), eigvals.(fiber(t),i))
@@ -591,19 +623,19 @@ for type ∈ (:TensorField,)
     for (op,mop) ∈ ((:*,:wedgedot_metric),(:wedgedot,:wedgedot_metric),(:veedot,:veedot_metric),(:⋅,:contraction_metric),(:contraction,:contraction_metric),(:>,:contraction_metric),(:⊘,:⊘),(:>>>,:>>>),(:/,:/),(:^,:^))
         let bop = op ∈ (:*,:>,:>>>,:/,:^) ? :(Base.$op) : :(Grassmann.$op)
         @eval begin
-            $bop(a::$type{R},b::$type{R}) where R = $type(base(a),Grassmann.$mop.(fiber(a),fiber(b),ref(metricextensor(base(a)))))
+            $bop(a::$type{R},b::$type{R}) where R = $type(base(a),Grassmann.$mop.(fiber(a),fiber(b),refmetric(base(a))))
             $bop(a::Number,b::$type) = $type(base(b), Grassmann.$op.(a,fiber(b)))
-            $bop(a::$type,b::Number) = $type(base(a), Grassmann.$op.(fiber(a),b,$((op≠:^ ? () : (:(ref(metricextensor(base(a)))),))...)))
+            $bop(a::$type,b::Number) = $type(base(a), Grassmann.$op.(fiber(a),b,$((op≠:^ ? () : (:(refmetric(base(a))),))...)))
         end end
     end
 end
-for fun ∈ (:-,:!,:~,:real,:imag,:conj,:deg2rad)
+for fun ∈ (:-,:!,:~,:real,:imag,:conj,:deg2rad,:transpose)
     @eval Base.$fun(t::TensorField) = TensorField(domain(t), $fun.(codomain(t)))
 end
 for fun ∈ (:exp,:exp2,:exp10,:log,:log2,:log10,:sinh,:cosh,:abs,:sqrt,:cbrt,:cos,:sin,:tan,:cot,:sec,:csc,:asec,:acsc,:sech,:csch,:asech,:tanh,:coth,:asinh,:acosh,:atanh,:acoth,:asin,:acos,:atan,:acot,:sinc,:cosc,:cis,:abs2,:inv)
     @eval Base.$fun(t::TensorField) = TensorField(domain(t), $fun.(codomain(t),ref(metricextensor(t))))
 end
-for fun ∈ (:reverse,:involute,:clifford,:even,:odd,:scalar,:vector,:bivector,:volume,:value,:complementleft,:realvalue,:imagvalue,:outermorphism,:Outermorphism,:DiagonalOperator,:TensorOperator,:eigen,:eigvecs,:eigvals,:eigvalsreal,:eigvalscomplex,:eigvecsreal,:eigvecscomplex,:eigpolys)
+for fun ∈ (:reverse,:clifford,:even,:odd,:scalar,:vector,:bivector,:volume,:value,:complementleft,:realvalue,:imagvalue,:outermorphism,:Outermorphism,:DiagonalOperator,:TensorOperator,:eigen,:eigvecs,:eigvals,:eigvalsreal,:eigvalscomplex,:eigvecsreal,:eigvecscomplex,:eigpolys,:∧)
     @eval Grassmann.$fun(t::TensorField) = TensorField(domain(t), $fun.(codomain(t)))
 end
 for fun ∈ (:⋆,:angle,:radius,:complementlefthodge,:pseudoabs,:pseudoabs2,:pseudoexp,:pseudolog,:pseudoinv,:pseudosqrt,:pseudocbrt,:pseudocos,:pseudosin,:pseudotan,:pseudocosh,:pseudosinh,:pseudotanh,:metric,:unit)
@@ -778,11 +810,48 @@ function __init__()
                 Makie.lines!(getindex.(t,i),f;args...)
             end
         end
+        export scaledarrows, scaledarrows!
+        for fun ∈ (:arrows,:arrows!)
+            @eval begin
+                function $(Symbol(:scaled,fun))(M::VectorField,t::VectorField;args...)
+                    kwargs = if haskey(args,:gridsize)
+                        wargs = Dict(args)
+                        delete!(wargs,:gridsize)
+                        return $(Symbol(:scaled,fun))(resample(M,args[:gridsize]),resample(t,args[:gridsize]);(;wargs...)...)
+                    elseif haskey(args,:arcgridsize)
+                        wargs = Dict(args)
+                        delete!(wargs,:arcgridsize)
+                        aM = arcresample(M,args[:arcgridsize])
+                        return $(Symbol(:scaled,fun))(aM,TensorField(base(aM),t.(points(aM)));(;wargs...)...)
+                    else
+                        args
+                    end
+                    s = spacing(M)/(sum(fiber(norm(t)))/length(t))
+                    Makie.$fun(M,t;lengthscale=s/3,arrowsize=s/17,kwargs...)
+                end
+                function $(Symbol(:scaled,fun))(M::VectorField,t::TensorField{B,<:TensorOperator} where B;args...)
+                    kwargs = if haskey(args,:gridsize)
+                        wargs = Dict(args)
+                        delete!(wargs,:gridsize)
+                        return $(Symbol(:scaled,fun))(resample(M,args[:gridsize]),resample(t,args[:gridsize]);(;wargs...)...)
+                    elseif haskey(args,:arcgridsize)
+                        wargs = Dict(args)
+                        delete!(wargs,:arcgridsize)
+                        aM = arcresample(M,args[:arcgridsize])
+                        return $(Symbol(:scaled,fun))(aM,TensorField(base(aM),t.(points(aM)));(;wargs...)...)
+                    else
+                        args
+                    end
+                    s = spacing(M)/maximum(value(sum(map.(norm,fiber(value(t))))/length(t)))
+                    Makie.$fun(M,t;lengthscale=s/3,arrowsize=s/17,kwargs...)
+                end
+            end
+        end
         function Makie.arrows(M::VectorField,t::TensorField{B,<:TensorOperator,N,<:GridFrameBundle} where B;args...) where N
             Makie.arrows(TensorField(fiber(M),fiber(t));args...)
         end
         function Makie.arrows!(M::VectorField,t::TensorField{B,<:TensorOperator,N,<:GridFrameBundle} where B;args...) where N
-            Makie.arrows!(TensorField(fiber(M),fiber(t)))
+            Makie.arrows!(TensorField(fiber(M),fiber(t));args...)
         end
         function Makie.arrows(t::VectorField{<:Coordinate{<:Chain},<:TensorOperator,2,<:RealSpace{2}};args...)
             display(Makie.arrows(getindex.(t,1);args...))
@@ -964,6 +1033,7 @@ function __init__()
             end
         end
         Makie.mesh(M::TensorField,f::Function;args...) = Makie.mesh(M,f(M);args...)
+        Makie.mesh!(M::TensorField,f::Function;args...) = Makie.mesh!(M,f(M);args...)
         function Makie.mesh(M::TensorField{B,<:Chain,2,<:GridFrameBundle} where B;args...)
             Makie.mesh(GridFrameBundle(fiber(M));args...)
         end
diff --git a/src/diffgeo.jl b/src/diffgeo.jl
index 18ded25..4523147 100644
--- a/src/diffgeo.jl
+++ b/src/diffgeo.jl
@@ -706,10 +706,23 @@ isregular(f::IntervalMap) = prod(.!iszero.(fiber(speed(f))))
 Grassmann.metric(f::TensorField,g::TensorField) = maximum(fiber(abs(f-g)))
 LinearAlgebra.norm(f::IntervalMap,g::IntervalMap) = arclength(f-g)
 
-arctime(f) = inv(arclength(f))
-totalarclength(f::IntervalMap) = sum(abs.(diff(codomain(f)),refdiff(metricextensor(f))))
+function arcresample(f::IntervalMap,i::Int=length(f))
+    at = arctime(f)
+    ts = at.(LinRange(0,points(at)[end],i))
+    TensorField(ts, f.(ts))
+end
+function arcsample(f::IntervalMap,i::Int=length(f))
+    at = arctime(f)
+    ral = LinRange(0,points(at)[end],i)
+    ts = at.(ral)
+    TensorField(ral, f.(ts))
+end
+
+arctime(f) = (al = arclength(f); TensorField(fiber(al), points(al)))
+arcsteps(f::IntervalMap) = Real.(abs.(diff(fiber(f)),refdiff(metricextensor(f))))
+totalarclength(f::IntervalMap) = sum(arcsteps(f))
 function arclength(f::IntervalMap)
-    int = cumsum(abs.(diff(codomain(f)),refdiff(metricextensor(f))))
+    int = cumsum(arcsteps(f))
     pushfirst!(int,zero(eltype(int)))
     TensorField(domain(f), int)
 end # cumtrapz(speed(f))
@@ -810,8 +823,8 @@ CovariantDerivative(∇::Connection,X) = ∇(X)
 
 export arclength, arctime, totalarclength, trapz, cumtrapz, linecumtrapz, psum, pcumsum
 export centraldiff, tangent, tangent_fast, unittangent, speed, normal, unitnormal
-export curvenormal, unitcurvenormal, ribbon, tangentsurface, planecurve, link, linkmap
-export normalnorm, area, surfacearea, weingarten, gausssign, jacobian
+export arcresample, arcsample, ribbon, tangentsurface, planecurve, link, linkmap
+export normalnorm, area, surfacearea, weingarten, gausssign, jacobian, evolute, involute
 export normalnorm_slow, area_slow, surfacearea_slow, weingarten_slow, gausssign_slow
 export gaussintrinsic, gaussextrinsic, gaussintrinsicnorm, gaussextrinsicnorm
 export gaussintrinsic_slow, gausseintrinsicnorm_slow, curvatures, meancurvature
@@ -823,24 +836,29 @@ export sector_slow, sectordet_slow, sectorintegral_slow, sectorintegrate_slow
 # use graph for IntervalMap? or RealFunction!
 tangent(f::IntervalMap) = gradient(f)
 tangent(f::ScalarField) = tangent(graph(f))
-tangent(f::VectorField) = det(gradient(f))
+tangent(f::VectorField) = ∧(gradient(f))
 normal(f::ScalarField) = ⋆tangent(f)
 normal(f::VectorField) = ⋆tangent(f)
-unittangent(f::ScalarField,n=tangent(f)) = TensorField(domain(f), codomain(n)./norm.(codomain(n)))
-unittangent(f::VectorField,n=tangent(f)) = TensorField(domain(f), codomain(n)./norm.(codomain(n)))
+unittangent(f::ScalarField,n=tangent(f)) = unit(n)
+unittangent(f::VectorField,n=tangent(f)) = unit(n)
 unittangent(f::IntervalMap) = unitgradient(f)
 unitnormal(f) = ⋆unittangent(f)
 normalnorm(f) = Real(abs(normal(f)))
+jacobian(f::IntervalMap) = gradient(f)
+jacobian(f::ScalarField) = jacobian(graph(f))
 jacobian(f::VectorField) = TensorOperator(gradient(f))
 weingarten(f::VectorField) = jacobian(unitnormal(f))
+Base.adjoint(f::IntervalMap) = jacobian(f)
+Base.adjoint(f::ScalarField) = jacobian(f)
+Base.adjoint(f::VectorField) = jacobian(f)
 
 tangent_slow(f::IntervalMap) = gradient_slow(f)
 tangent_slow(f::ScalarField) = tangent_slow(graph(f))
-tangent_slow(f::VectorField) = det(gradient_slow(f))
+tangent_slow(f::VectorField) = ∧(gradient_slow(f))
 normal_slow(f::ScalarField) = ⋆tangent_slow(f)
 normal_slow(f::VectorField) = ⋆tangent_slow(f)
-unittangent_slow(f::ScalarField,n=tangent_slow(f)) = TensorField(domain(f), codomain(n)./norm.(codomain(n)))
-unittangent_slow(f::VectorField,n=tangent_slow(f)) = TensorField(domain(f), codomain(n)./norm.(codomain(n)))
+unittangent_slow(f::ScalarField,n=tangent_slow(f)) = unit(n)
+unittangent_slow(f::VectorField,n=tangent_slow(f)) = unit(n)
 unittangent_slow(f::IntervalMap) = unitgradient_slow(f)
 unitnormal_slow(f) = ⋆unittangent_slow(f)
 normalnorm_slow(f) = Real(abs(normal_slow(f)))
@@ -860,7 +878,7 @@ sector(f::RealFunction) = f
 sector(f::TensorField) = TensorOperator(sector(f,gradient(f)))
 sectordet(f::RealFunction) = f
 sectordet(f::PlaneCurve) = f∧gradient(f)
-sectordet(f::VectorField) = f∧det(gradient(f))
+sectordet(f::VectorField) = f∧(∧(gradient(f)))
 sectorintegral(f::RealFunction) = integral(f)
 sectorintegral(f::TensorField) = integral(sectordet(f))/mdims(fibertype(f))
 sectorintegrate(f::RealFunction) = integrate(f)
@@ -869,7 +887,7 @@ sector_slow(f::RealFunction) = f
 sector_slow(f::TensorField) = TensorOperator(sector(f,gradient_slow(f)))
 sectordet_slow(f::RealFunction) = f
 sectordet_slow(f::PlaneCurve) = f∧gradient_slow(f)
-sectordet_slow(f::VectorField) = f∧det(gradient_slow(f))
+sectordet_slow(f::VectorField) = f∧(∧(gradient_slow(f)))
 sectorintegral_slow(f::RealFunction) = integral(f)
 sectorintegral_slow(f::TensorField) = integral(sectordet_slow(f))/mdims(fibertype(f))
 sectorintegrate_slow(f::RealFunction) = integrate(f)
@@ -890,7 +908,7 @@ gaussintrinsicnorm(f::VectorField,N=normal(f)) = norm(gaussintrinsic(f,N))
 gaussextrinsic(f::VectorField) = sectordet(unitnormal(f))
 function gaussintrinsic(f::VectorField,N=normal(f))
     n,T = Real(abs(N)),pointtype(f)
-    Chain{Manifold(T),mdims(T)}(Real(sectordet(N/n))/n)
+    Chain{Manifold(T),mdims(T)}.(Real(sectordet(N/n))/n)
 end
 
 area_slow(f::VectorField) = integral(normalnorm_slow(f))
@@ -910,73 +928,197 @@ sectorarea(f::PlaneCurve) = sectorintegrate(f)
 sectorvolume(f::VectorField) = sectorintegrate(f)
 
 function speed(f::IntervalMap,d=centraldiffpoints(f),t=centraldifffiber(f,d))
-    TensorField(domain(f), abs.(t))
+    TensorField(base(f), Real.(abs.(t,refmetric(f))))
 end
-function curvenormal(f::IntervalMap,d=centraldiffpoints(f),t=centraldifffiber(f,d))
-    TensorField(domain(f), centraldiff(t,d))
+function normal(f::IntervalMap,d=centraldiffpoints(f),t=centraldifffiber(f,d))
+    s = abs.(t,refmetric(f))
+    TensorField(domain(f), centraldiff(t./s,d)./s)
 end
-function unitcurvenormal(f::IntervalMap,d=centraldiffpoints(f),t=centraldifffiber(f,d),n=centraldiff(t,d))
-    TensorField(domain(f), (n./abs.(n)))
+function unitnormal(f::IntervalMap,d=centraldiffpoints(f),t=centraldifffiber(f,d))
+    TensorField(domain(f), unit.(centraldiff(unit.(t,refmetric(f)),d),refmetric(f)))
 end
 
-export curvature, radius, osculatingplane, unitosculatingplane, binormal, unitbinormal, curvaturebivector, torsion, curvaturetrivector, frame, unitframe, frenet, wronskian, bishoppolar, bishop, curvaturetorsion
+export curvature, radius, osculatingplane, unitosculatingplane, binormal, unitbinormal, torsion, frame, unitframe, frenet, darboux, wronskian, bishoppolar, bishop, bishopframe, bishopunitframe
 
-function curvature(f::PlaneCurve,d=centraldiffpoints(f),t=centraldifffiber(f,d),a=abs.(t))
-    TensorField(domain(f), abs.(centraldiff(t./a,d))./a)
+function curvature(f::AbstractCurve,d=centraldiffpoints(f),t=centraldifffiber(f,d))
+    s = abs.(t,refmetric(f))
+    TensorField(domain(f), Real.(abs.(centraldiff(t./s,d),refmetric(f))./s))
 end
-
-function curvature(f::SpaceCurve,d=centraldiffpoints(f),t=centraldifffiber(f,d),a=abs.(t))
-    TensorField(domain(f), abs.(centraldiff(t./a,d))./a)
+function radius(f::AbstractCurve,d=centraldiffpoints(f),t=centraldifffiber(f,d))
+    s = abs.(t,refmetric(f))
+    TensorField(domain(f), Real.(s./abs.(centraldiff(t./s,d),refmetric(f))))
 end
-function radius(f::SpaceCurve,d=centraldiffpoints(f),t=centraldifffiber(f,d),a=abs.(t))
-    TensorField(domain(f), a./abs.(centraldiff(t./a,d)))
+function localevolute(f::AbstractCurve,d=centraldiffpoints(f),t=centraldifffiber(f,d))
+    s = abs.(t,refmetric(f))
+    n = centraldiff(t./s,d)
+    an2 = abs2.(n,refmetric(f))
+    TensorField(domain(f), (s./an2).*n)
 end
-function osculatingplane(f::SpaceCurve,d=centraldiffpoints(f),t=centraldifffiber(f,d),n=centraldiff(t,d))
-    TensorField(domain(f), t.∧n)
+evolute(f::AbstractCurve) = f+localevolute(f)
+Grassmann.involute(f::AbstractCurve) = f-unittangent(f)*arclength(f)
+function osculatingplane(f::AbstractCurve,d=centraldiffpoints(f),t=centraldifffiber(f,d))
+    TensorField(domain(f), TensorOperator.(Chain.(t,fiber(normal(f,t,d)))))
 end
-function unitosculatingplane(f::SpaceCurve,d=centraldiffpoints(f),t=centraldifffiber(f,d),n=centraldiff(t,d))
-    TensorField(domain(f), (t./abs.(t)).∧(n./abs.(n)))
+function unitosculatingplane(f::AbstractCurve,d=centraldiffpoints(f),t=centraldifffiber(f,d))
+    T = unit.(t,refmetric(f))
+    TensorField(domain(f),TensorOperator.(Chain.(T,unit.(centraldiff(T,d),refmetric(f)))))
 end
-function binormal(f::SpaceCurve,d=centraldiffpoints(f),t=centraldifffiber(f,d),n=centraldiff(t,d))
-    TensorField(domain(f), .⋆(t.∧n))
+function binormal(f::SpaceCurve,d=centraldiffpoints(f),t=centraldifffiber(f,d))
+    TensorField(domain(f), .⋆(t.∧fiber(normal(f,d,t)),refmetric(f)))
 end
 function unitbinormal(f::SpaceCurve,d=centraldiffpoints(f),t=centraldifffiber(f,d))
-    a = t./abs.(t)
-    n = centraldiff(t./abs.(t),d)
-    TensorField(domain(f), .⋆(a.∧(n./abs.(n))))
+    T = unit.(t)
+    TensorField(domain(f), .⋆(T.∧(unit.(centraldiff(T,d),refmetric(f))),refmetric(f)))
 end
-function curvaturebivector(f::SpaceCurve,d=centraldiffpoints(f),t=centraldifffiber(f,d),n=centraldiff(t,d))
-    a=abs.(t); ut=t./a
-    TensorField(domain(f), abs.(centraldiff(ut,d))./a.*(ut.∧(n./abs.(n))))
-end
-function torsion(f::SpaceCurve,d=centraldiffpoints(f),t=centraldifffiber(f,d),n=centraldiff(t,d),b=t.∧n)
-    TensorField(domain(f), (b.∧centraldiff(n,d))./abs.(.⋆b).^2)
-end
-function curvaturetrivector(f::SpaceCurve,d=centraldiffpoints(f),t=centraldifffiber(f,d),n=centraldiff(t,d),b=t.∧n)
-    a=abs.(t); ut=t./a
-    TensorField(domain(f), (abs.(centraldiff(ut,d)./a).^2).*(b.∧centraldiff(n,d))./abs.(.⋆b).^2)
+function torsion(f::SpaceCurve,d=centraldiffpoints(f),t=centraldifffiber(f,d))
+    n = centraldiff(t,d)
+    b = t.∧n
+    TensorField(domain(f), Real.((b.∧centraldiff(n,d))./abs2.(.⋆b,refmetric(f))))
 end
+#=function curvaturetrivector(f::SpaceCurve,d=centraldiffpoints(f),t=centraldifffiber(f,d),n=centraldiff(t,d),b=t.∧n)
+    a=abs.(t,refmetric(f)); ut=t./a
+    TensorField(domain(f), (abs2.(centraldiff(ut,d)./a)).*(b.∧centraldiff(n,d))./abs2.(.⋆b))
+end=#
 #torsion(f::TensorField,d=centraldiffpoints(f),t=centraldifffiber(f,d),n=centraldiff(t,d),a=abs.(t)) = TensorField(domain(f), abs.(centraldiff(.⋆((t./a).∧(n./abs.(n))),d))./a)
-function frame(f::PlaneCurve,d=centraldiffpoints(f),t=centraldifffiber(f,d),n=centraldiff(t,d))
-    TensorField(domain(f), TensorOperator.(Chain.(t,n)))
-end
-function frame(f::SpaceCurve,d=centraldiffpoints(f),t=centraldifffiber(f,d),n=centraldiff(t,d))
-    TensorField(domain(f), TensorOperator.(Chain.(t,n,.⋆(t.∧n))))
-end
-function unitframe(f::PlaneCurve,d=centraldiffpoints(f),t=centraldifffiber(f,d),n=centraldiff(t,d))
-    TensorField(domain(f), TensorOperator.(Chain.(t./abs.(t),n./abs.(n))))
-end
-function unitframe(f::SpaceCurve,d=centraldiffpoints(f),t=centraldifffiber(f,d),n=centraldiff(t,d))
-    b = .⋆(t.∧n)
-    TensorField(domain(f), TensorOperator.(Chain.(t./abs.(t),n./abs.(n),b./abs.(b))))
+frame(f::PlaneCurve,d=centraldiffpoints(f),e1=centraldifffiber(f,d)) = osculatingplane(f,d,e1)
+@generated function frame(f::AbstractCurve,d=centraldiffpoints(f),e1=centraldifffiber(f,d))
+    N = mdims(fibertype(f))
+    syms = Symbol.(:e,list(1,N-1))
+    Expr(:block,:(s = abs.(e1,refmetric(f))),
+        [:($(syms[i]) = centraldiff(unit.($(syms[i-1]),refmetric(f)),d)./s) for i ∈ list(2,N-1)]...,
+        :(TensorField(base(f),TensorOperator.(Chain.($(syms...),.⋆(.∧($(syms...)),refmetric(f)))))))
+end
+unitframe(f::PlaneCurve,d=centraldiffpoints(f),t=centraldifffiber(f,d)) = unitosculatingplane(f,d,t)
+@generated function unitframe(f::AbstractCurve,d=centraldiffpoints(f),t=centraldifffiber(f,d))
+    N = mdims(fibertype(f))
+    syms = Symbol.(:e,list(1,N-1))
+    Expr(:block,:(e1 = unit.(t,refmetric(f))),
+        [:($(syms[i]) = unit.(centraldiff($(syms[i-1]),d),refmetric(f))) for i ∈ list(2,N-1)]...,
+        :($(Symbol(:e,N)) = .⋆(.∧($(syms...)),refmetric(f))),
+        :(TensorField(base(f),TensorOperator.(Chain.($([:($(Symbol(:e,i))) for i ∈ list(1,N)]...))))))
+end
+function cartan(f::PlaneCurve,d=centraldiffpoints(f),t=centraldifffiber(f,d))
+    κ = curvature(f,d,t)
+    TensorField(domain(f), Chain.(Chain.(0,κ),Chain.(.-κ,0)))
+end
+function cartan(f::SpaceCurve,d=centraldiffpoints(f),t=centraldifffiber(f,d))
+    n = centraldiff(t,d)
+    s,b = abs.(t,refmetric(f)),t.∧n
+    κ = Real.(abs.(centraldiff(t./s,d),refmetric(f))./s)
+    τ = Real.((b.∧centraldiff(n,d))./abs2.(.⋆(b,refmetric(f)),refmetric(f)))
+    TensorField(base(f),TensorOperator.(Chain.(Chain.(0,κ,0),Chain.(.-κ,0,τ),Chain.(0,.-τ,0))))
+end
+@generated function cartan(f::AbstractCurve,d=centraldiffpoints(f),t=centraldifffiber(f,d))
+    V = Manifold(fibertype(f))
+    N = mdims(V)
+    bas = Grassmann.bladeindex.(N,UInt.(Λ(V).b[list(2,N)].∧Λ(V).b[list(3,N+1)]))
+    syms,curvs = Symbol.(:e,list(1,N)),Symbol.(:κ,list(1,N-1))
+    vals = [:($(curvs[i]) = Real.(Grassmann.contraction_metric.($(syms[i+1]),centraldiff($(syms[i]),d),refmetric(f))./s)) for i ∈ list(1,N-1)]
+    Expr(:block,:(s=abs.(t)),:(uf = fiber(unitframe(f,d,t))),
+        [:($(syms[i]) = getindex.(uf,$i)) for i ∈ list(1,N)]...,vals...,
+        :(TensorField(base(f),TensorOperator.(Chain.($([:(Chain.($([j ∈ (i-1,i+1) ? j==i-1 ? :(.-$(curvs[j])) : curvs[j-1] : 0 for j ∈ list(1,N)]...))) for i ∈ list(1,N)]...))))))
+end # cartan(unitframe(f))
+function frenet(f::PlaneCurve,d=centraldiffpoints(f),t=centraldifffiber(f,d))
+    s = abs.(t,refmetric(f))
+    T = t./s
+    N = unit.(centraldiff(T,d),refmetric(f))
+    TensorField(domain(f), TensorOperator.(centraldiff(Chain.(T,N),d)./s))
+end
+function frenet(f::SpaceCurve,d=centraldiffpoints(f),t=centraldifffiber(f,d))
+    s = abs.(t,refmetric(f))
+    T = t./s
+    N = unit.(centraldiff(T,d),refmetric(f))
+    TensorField(domain(f), TensorOperator.(centraldiff(Chain.(T,N,.⋆(T.∧N,refmetric(f))),d)./s))
+end
+frenet(f::AbstractCurve) = (F = unitframe(f); F⋅cartan(F))
+
+# curvature, torsion, etc... invariant vector
+curvatures(f::PlaneCurve,i::Int,args...) = curvatures(f,Val(i),args...)
+curvatures(f::SpaceCurve,i::Int,args...) = curvatures(f,Val(i),args...)
+curvatures(f::AbstractCurve,i::Int,args...) = curvatures(f,Val(i),args...)
+curvatures(f::PlaneCurve,d=centraldiffpoints(f),t=centraldifffiber(f,d)) = curvatures(f,Val(1),d,t)
+function curvatures(f::SpaceCurve,::Val{2},d=centraldiffpoints(f),t=centraldifffiber(f,d))
+    V = Manifold(fibertype(f))
+    torsion(f,d,t)*Λ(V).b[7]
+end
+function curvatures(f::SpaceCurve,d::Vector=centraldiffpoints(f),t=centraldifffiber(f,d))
+    V = Manifold(fibertype(f))
+    s = abs.(t,refmetric(f))
+    n = centraldiff(t./s,d)./s
+    b = t.∧n
+    TensorField(domain(f), Chain{V,2}.(value.(abs.(n,refmetric(f))),0,getindex.((b.∧centraldiff(n,d))./abs.(.⋆(b,refmetric(f)),refmetric(f)).^2,1)))
+end
+@generated function curvatures(f::AbstractCurve,d=centraldiffpoints(f),t=centraldifffiber(f,d))
+    V = Manifold(fibertype(f))
+    N = mdims(V)
+    bas = Grassmann.bladeindex.(N,UInt.(Λ(V).b[list(2,N)].∧Λ(V).b[list(3,N+1)]))
+    syms = Symbol.(:e,list(1,N))
+    vals = [:(Real.(Grassmann.contraction_metric.($(syms[i+1]),centraldiff($(syms[i]),d),refmetric(f))./s)) for i ∈ list(1,N-1)]
+    Expr(:block,:(s=abs.(t)),:(uf = fiber(unitframe(f,d,t))),
+        [:($(syms[i]) = getindex.(uf,$i)) for i ∈ list(1,N)]...,
+        :(TensorField(base(f),Chain{$V,2}.($([j ∈ bas ? vals[searchsortedfirst(bas,j)] : 0 for j ∈ list(1,Grassmann.binomial(N,2))]...)))))
+end
+@generated function curvatures(f::AbstractCurve,::Val{j},d=centraldiffpoints(f),e1=centraldifffiber(f,d)) where j
+    V = Manifold(fibertype(f))
+    N = mdims(V)
+    j==1 && (return :(curvature(f,d,e1)*$(Λ(V).b[N+2])))
+    syms = Symbol.(:e,list(1,N-1))
+    Expr(:block,:(s=abs.(e1)),
+        [:($(syms[i]) = centraldiff($(syms[i-1])./s,d)) for i ∈ list(2,j<N-1 ? j+1 : N-1)]...,
+        j==N-1 ? :($(Symbol(:e,N)) = .⋆(.∧($(syms...)),refmetric(f))) : nothing,
+        :(TensorField(base(f),Real.(Grassmann.contraction_metric.(unit.($(Symbol(:e,j+1)),refmetric(f)),centraldiff(unit.($(syms[j]),refmetric(f)),d),refmetric(f))./abs.(e1)).*$(Λ(V).b[j+1]∧Λ(V).b[j+2]))))
+end
+darboux(f::AbstractCurve) = compound(unitframe(f),Val(2))⋅curvatures(f)
+darboux(f::AbstractCurve,j) = compound(unitframe(f),Val(2))⋅curvatures(f,j)
+
+function bishopframe(f::SpaceCurve,θ0=0.0,d=centraldiffpoints(f),t=centraldifffiber(f,d))
+    s,b = Real.(abs.(t,refmetric(f)))
+    T = t./s
+    N = centraldiff(T,d)./s
+    B = .⋆(T.∧N,refmetric(f))
+    τs = fiber(torsion(f,d,t)).*s
+    θ = (diff(points(f))/2).*cumsum(τs[2:end]+τs[1:end-1]).+θ0
+    pushfirst!(θ,θ0)
+    cθ,sθ = cos.(θ),sin.(θ)
+    TensorField(domain(f), TensorOperator.(Chain.(T,cθ.*N.-sθ.*B,sθ.*N+cθ.*B)))
+end
+function bishopunitframe(f::SpaceCurve,θ0=0.0,d=centraldiffpoints(f),t=centraldifffiber(f,d))
+    s,b = Real.(abs.(t,refmetric(f)))
+    T = unit.(t)
+    N = unit.(centraldiff(t./s,d))
+    B = .⋆(T.∧N,refmetric(f))
+    τs = fiber(torsion(f,d,t)).*s
+    θ = (diff(points(f))/2).*cumsum(τs[2:end]+τs[1:end-1]).+θ0
+    pushfirst!(θ,θ0)
+    cθ,sθ = cos.(θ),sin.(θ)
+    TensorField(domain(f), TensorOperator.(Chain.(T,cθ.*N.-sθ.*B,sθ.*N+cθ.*B)))
+end
+function bishoppolar(f::SpaceCurve,θ0=0.0,d=centraldiffpoints(f),t=centraldifffiber(f,d))
+    n = centraldiff(t,d)
+    s,b = abs.(t,refmetric(f)),t.∧n
+    κ = Real.(abs.(centraldiff(t./s,d),refmetric(f))./s)
+    τs = Real.(((b.∧centraldiff(n,d))./abs2.(.⋆(b,refmetric(f)),refmetric(f))).*s)
+    θ = (diff(points(f))/2).*cumsum(τs[2:end]+τs[1:end-1]).+θ0
+    pushfirst!(θ,θ0)
+    TensorField(domain(f), Chain.(κ,θ))
 end
-function frenet(f::PlaneCurve,d=centraldiffpoints(f),t=centraldifffiber(f,d),n=centraldiff(t,d))
-    TensorField(domain(f), TensorOperator.(centraldiff(Chain.(t./abs.(t),n./abs.(n)),d)))
+function bishop(f::SpaceCurve,θ0=0.0,d=centraldiffpoints(f),t=centraldifffiber(f,d))
+    n = centraldiff(t,d)
+    s,b = abs.(t,refmetric(f)),t.∧n
+    κ = Real.(abs.(centraldiff(t./s,d),refmetric(f))./s)
+    τs = Real.(((b.∧centraldiff(n,d))./abs2.(.⋆(b,refmetric(f)),refmetric(f))).*s)
+    θ = (diff(points(f))/2).*cumsum(τs[2:end]+τs[1:end-1]).+θ0
+    pushfirst!(θ,θ0)
+    TensorField(domain(f), Chain.(κ.*cos.(θ),κ.*sin.(θ)))
 end
-function frenet(f::SpaceCurve,d=centraldiffpoints(f),t=centraldifffiber(f,d),n=centraldiff(t,d))
-    (ut,un)=(t./abs.(t),n./abs.(n))
-    TensorField(domain(f), TensorOperator.(centraldiff(Chain.(ut,un,.⋆(ut.∧un)),d)))
+#bishoppolar(f::TensorField) = TensorField(domain(f), Chain.(value.(codomain(curvature(f))),getindex.(codomain(cumtrapz(torsion(f))),1)))
+#bishop(f::TensorField,κ=value.(codomain(curvature(f))),θ=getindex.(codomain(cumtrapz(torsion(f))),1)) = TensorField(domain(f), Chain.(κ.*cos.(θ),κ.*sin.(θ)))
+
+function planecurve(κ::RealFunction,φ::Real=0.0)
+    int = iszero(φ) ? integral(κ) : integral(κ)+φ
+    integral(Chain.(cos(int),sin(int)))
 end
+
 function wronskian(f::SpaceCurve,d=centraldiffpoints(f),t=centraldifffiber(f,d),n=centraldiff(t,d))
     TensorField(domain(f), f.cod.∧t.∧n)
 end
@@ -984,7 +1126,7 @@ end
 function linkmap(f::SpaceCurve,g::SpaceCurve)
     TensorField(base(f)×base(g),[fiber(g)[j]-fiber(f)[i] for i ∈ OneTo(length(f)), j ∈ OneTo(length(g))])
 end
-link(tf::Chain,tg::Chain,f::Chain,g::Chain) = (gf = g-f; ∧(gf,tf,tg)/norm(gf)^3)
+link(tf::Chain,tg::Chain,f::Chain,g::Chain) = (gf = g-f; ngf = norm(gf); ∧(gf,tf,tg)/(ngf*ngf*ngf))
 function link(f::SpaceCurve,g::SpaceCurve)
     tf,tg = fiber(tangent(f)),fiber(tangent(g))
     TensorField(base(f)×base(g),[link(tf[i],tg[j],fiber(f)[i],fiber(g)[j]) for i ∈ OneTo(length(f)), j ∈ OneTo(length(g))])
@@ -992,53 +1134,27 @@ end
 linkintegral(f::SpaceCurve,g::SpaceCurve) = integral(link(f,g))/4π
 linknumber(f::SpaceCurve,g::SpaceCurve) = integrate(link(f,g))/4π
 
+#???
+function compare(f::TensorField)#???
+    d = centraldiffpoints(f)
+    t = centraldifffiber(f,d)
+    n = centraldiff(t,d)
+    s = abs.(t)
+    TensorField(domain(f), centraldiff(t./s,d).-n./s)
+end #????
+
 function frame(f::VectorField{<:Coordinate{<:Chain},<:Chain,2,<:RealSpace{2}})#,d=centraldiff(points(f)),t=centraldiff(codomain(f),d),n=centraldiff(t,d))
     Ψ = gradient(f)
     Ψu,Ψv = getindex.(Ψ,1),getindex.(Ψ,2)
     ξ3 = ⋆(Ψu∧Ψv)
-    TensorField(domain(f), TensorOperator.(Chain.(fiber(Ψu),fiber(ξ3),fiber(⋆(ξ3∧Ψu)))))
+    TensorField(domain(f), TensorOperator.(Chain.(fiber(Ψu),fiber(⋆(ξ3∧Ψu)),fiber(ξ3))))
 end
 function unitframe(f::VectorField{<:Coordinate{<:Chain},<:Chain,2,<:RealSpace{2}})#,d=centraldiff(points(f)),t=centraldiff(codomain(f),d),n=centraldiff(t,d))
     Ψ = gradient(f)
     Ψu,Ψv = getindex.(Ψ,1),getindex.(Ψ,2)
     ξ3 = ⋆(Ψu∧Ψv)
     ξ2 = ⋆(ξ3∧Ψu)
-    TensorField(domain(f), TensorOperator.(Chain.(fiber(Ψu/abs(Ψu)),fiber(ξ3/abs(ξ3)),fiber(ξ2/abs(ξ2)))))
-end
-
-#???
-function compare(f::TensorField)#???
-    d = centraldiffpoints(f)
-    t = centraldifffiber(f,d)
-    n = centraldiff(t,d)
-    TensorField(domain(f), centraldiff(t./abs.(t)).-n./abs.(t))
-end #????
-
-function curvaturetorsion(f::SpaceCurve,d=centraldiffpoints(f),t=centraldifffiber(f,d),n=centraldiff(t,d),b=t.∧n)
-    a = abs.(t)
-    TensorField(domain(f), Chain.(value.(abs.(centraldiff(t./a,d))./a),getindex.((b.∧centraldiff(n,d))./abs.(.⋆b).^2,1)))
-end
-
-function bishoppolar(f::SpaceCurve,κ=value.(codomain(curvature(f))))
-    d = diff(points(f))
-    τs = getindex.(codomain(torsion(f)).*codomain(speed(f)),1)
-    θ = (d/2).*cumsum(τs[2:end]+τs[1:end-1])
-    pushfirst!(θ,zero(eltype(θ)))
-    TensorField(domain(f), Chain.(κ,θ))
-end
-function bishop(f::SpaceCurve,κ=value.(codomain(curvature(f))))
-    d = diff(points(f))
-    τs = getindex.(codomain(torsion(f)).*codomain(speed(f)),1)
-    θ = (d/2).*cumsum(τs[2:end]+τs[1:end-1])
-    pushfirst!(θ,zero(eltype(θ)))
-    TensorField(domain(f), Chain.(κ.*cos.(θ),κ.*sin.(θ)))
-end
-#bishoppolar(f::TensorField) = TensorField(domain(f), Chain.(value.(codomain(curvature(f))),getindex.(codomain(cumtrapz(torsion(f))),1)))
-#bishop(f::TensorField,κ=value.(codomain(curvature(f))),θ=getindex.(codomain(cumtrapz(torsion(f))),1)) = TensorField(domain(f), Chain.(κ.*cos.(θ),κ.*sin.(θ)))
-
-function planecurve(κ::RealFunction,φ::Real=0.0)
-    int = iszero(φ) ? integral(κ) : integral(κ)+φ
-    integral(Chain.(cos(int),sin(int)))
+    TensorField(domain(f), TensorOperator.(Chain.(fiber(Ψu/abs(Ψu)),fiber(ξ2/abs(ξ2)),fiber(ξ3/abs(ξ3)))))
 end
 
 export surfacemetric, surfacemetricdiag, surfaceframe, shape
@@ -1083,7 +1199,7 @@ end
 
 secondform(dom,f::Function) = secondform(TensorField(dom,f))
 function secondform(t,g=gradient(t),V=Submanifold(2))
-    n = ⋆unittangent(t,det(g))
+    n = ⋆unittangent(t,∧(g))
     dfdx,dfdy = getindex.(g,1),getindex.(g,2)
     ddfdx,ddfdy2 = gradient(dfdx),gradient(dfdy,Val(2))
     ddfdx2,ddfdxdy = getindex.(ddfdx,1),getindex.(ddfdx,2)
@@ -1092,7 +1208,7 @@ end
 
 firstsecondform(dom,f::Function) = firstsecondform(TensorField(dom,f))
 function firstsecondform(t,g=gradient(t),V=Submanifold(2))
-    n = ⋆unittangent(t,det(g))
+    n = ⋆unittangent(t,∧(g))
     dfdx,dfdy = getindex.(g,1),getindex.(g,2)
     ddfdx,ddfdy2 = gradient(dfdx),gradient(dfdy,Val(2))
     ddfdx2,ddfdxdy = getindex.(ddfdx,1),getindex.(ddfdx,2)
diff --git a/src/topology.jl b/src/topology.jl
index afbe5a6..a92af0c 100644
--- a/src/topology.jl
+++ b/src/topology.jl
@@ -12,7 +12,7 @@
 #   https://github.com/chakravala
 #   https://crucialflow.com
 
-export FiberProduct, FiberProductBundle, HomotopyBundle
+export FiberProduct, FiberProductBundle, HomotopyBundle, resample
 export LocalSection, GlobalFiber, LocalFiber, localfiber, globalfiber
 export base, fiber, domain, codomain, ↦, →, ←, ↤, basetype, fibertype, graph
 export ProductSpace, RealRegion, NumberLine, Rectangle, Hyperrectangle, ⧺, ⊕
@@ -46,6 +46,14 @@ Base.iterate(t::RealRegion,state) = (s=state+1; s≤length(t) ? (getindex(t,s),s
 
 resize_lastdim!(m::ProductSpace,i) = (resize!(m.v[end],i); m)
 
+resample(m::OneTo,i::Int) = LinRange(1,m.stop,i)
+resample(m::UnitRange,i::Int) = LinRange(m.start,m.stop,i)
+resample(m::StepRange,i::Int) = LinRange(m.start,m.stop,i)
+resample(m::LinRange,i::Int) = LinRange(m.start,m.stop,i)
+resample(m::StepRangeLen,i::Int) = StepRangeLen(m.ref,(m.step*(m.len-1))/(i-1),i)
+resample(m::AbstractRange,i::NTuple{1,Int}) = resample(m,i...)
+resample(m::ProductSpace,i::NTuple) = ProductSpace(resample.(m.v,i))
+
 @generated Base.size(m::RealRegion{V}) where V = :(($([:(size(@inbounds m.v[$i])...) for i ∈ 1:mdims(V)]...),))
 @generated Base.getindex(m::RealRegion{V,T,N},i::Vararg{Int}) where {V,T,N} = :(Chain{V,1,T}(Values{N,T}($([:((@inbounds m.v[$j])[@inbounds i[$j]]) for j ∈ 1:N]...))))
 Base.getindex(m::NumberLine{V,T},i::Int) where {V,T} = Chain{V,1,T}(Values(((@inbounds m.v[1])[i],)))
@@ -182,6 +190,12 @@ resize(m::ProductTopology{1},i) = ProductTopology(@inbounds resize(m.v[1],i))
     Expr(:call,:ProductTopology,Expr(:call,:Values,[j≠N ? :(@inbounds m.v[$j]) : :(@inbounds resize(m.v[$j],i)) for j ∈ list(1,N)]...))
 end
 
+resample(m,i::Int...) = resample(m,i)
+resample(m::ProductTopology{1},i::NTuple{1}) = ProductTopology(@inbounds resize(m.v[1],i[1]))
+@generated function resample(m::ProductTopology{N},i::NTuple{N}) where N
+    Expr(:call,:ProductTopology,Expr(:call,:Values,[:(@inbounds resize(m.v[$j],i[$j])) for j ∈ list(1,N)]...))
+end
+
 @generated Base.size(m::ProductTopology{N}) where N = :(($([:(size(@inbounds m.v[$i])...) for i ∈ 1:N]...),))
 @generated Base.getindex(m::ProductTopology{N},i::Vararg{Int}) where N = :(Values{N,Int}($([:((@inbounds m.v[$j])[@inbounds i[$j]]) for j ∈ 1:N]...)))
 Base.getindex(m::ProductTopology{1},i::Int) = Values(((@inbounds m.v[1])[i],))
@@ -202,9 +216,28 @@ function Base._ind2sub(A::ProductTopology, ind)
     Base._ind2sub(axes(A), ind)
 end
 
-exclude(m::ProductTopology) = m
-exclude(m::ProductTopology{N},n::Int) where N = ProductTopology(m.v[vcat([i≠n ? Values((i,)) : Values{0,Int}() for i ∈ list(1,N)]...)])
-exclude(m::ProductTopology{N},n::Int...) where N = ProductTopology(m.v[vcat([i∉n ? Values((i,)) : Values{0,Int}() for i ∈ list(1,N)]...)])
+#exclude(m::ProductTopology{N},n::Int) where N = ProductTopology(m.v[vcat([i≠n ? Values((i,)) : Values{0,Int}() for i ∈ list(1,N)]...)])
+#exclude(m::ProductTopology{N},n::Int...) where N = ProductTopology(m.v[vcat([i∉n ? Values((i,)) : Values{0,Int}() for i ∈ list(1,N)]...)])
+@inline exclude(m::ProductTopology) = m
+@inline exclude(m::ProductTopology{2},::Val{1}) = ProductTopology(m.v[2])
+@inline exclude(m::ProductTopology{2},::Val{2}) = ProductTopology(m.v[1])
+@generated function exclude(m::ProductTopology{N},::Val{n}) where {N,n}
+    N==2 && (return :(ProductTopology(m.v[$n])))
+    vals = vcat([i≠n ? Values((i,)) : Values{0,Int}() for i ∈ list(1,N)]...)
+    :(ProductTopology(m.v[$vals]))
+end
+@generated function exclude(m::ProductTopology{N},::Val{n1},::Val{n2}) where {N,n1,n2}
+    vals = vcat([i∉(n1,n2) ? Values((i,)) : Values{0,Int}() for i ∈ list(1,N)]...)
+    :(ProductTopology(m.v[$vals]))
+end
+@generated function exclude(m::ProductTopology{N},::Val{n1},::Val{n2},::Val{n3}) where {N,n1,n2,n3}
+    vals = vcat([i∉(n1,n2,n3) ? Values((i,)) : Values{0,Int}() for i ∈ list(1,N)]...)
+    :(ProductTopology(m.v[$vals]))
+end
+@generated function exclude(m::ProductTopology{N},::Val{n1},::Val{n2},::Val{n3},::Val{n4}) where {N,n1,n2,n3,n4}
+    vals = vcat([i∉(n1,n2,n3,n4) ? Values((i,)) : Values{0,Int}() for i ∈ list(1,N)]...)
+    :(ProductTopology(m.v[$vals]))
+end
 
 cross(a::ProductTopology,b::ProductTopology) = ProductTopology(Values(a.v...,b.v...))
 cross(a::ProductTopology,b::AbstractVector{Int}) = ProductTopology(Values(a.v...,b))
@@ -217,9 +250,17 @@ struct QuotientTopology{N,L,M,O,LA<:ImmersedTopology{L,L}} <: ImmersedTopology{N
     q::Values{O,LA}
     r::Values{M,Int}
     s::Values{N,Int}
+    #t::Values{O,Int}
     #QuotientTopology(p::Values{O,Int},q::Values{O,LA},r::Values{M,Int},n::Values{N,Int}) where {O,L,LA<:ImmersedTopology{L,L},M,N} = QuotientTopology{N,L,M,O,LA}(p,q,r,n)
 end
 
+#QuotientTopology(p::Values{O,Int},q::Values{O,LA},r::Values{M,Int},n::Values{N,Int}) where {O,L,LA<:ImmersedTopology{L,L},M,N} = QuotientTopology{N,L,M,O,LA}(p,q,r,n,invert_q(Val(O),r))
+
+invert_q(s::Int,i::Int) = iszero(s) ? () : (i,)
+invert_q(::Val{M},s::Values{M,Int}) where M = s
+invert_q(::Val{0},s::Values{M,Int}) where M = Values{0,Int}()
+@generated invert_q(::Val{O},s::Values{M,Int}) where {O,M} = Expr(:call,:(Values{O,Int}),[:(invert_q((@inbounds s[$i]),$i)...) for i ∈ list(1,M)]...)
+
 const OpenTopology{N,L,M,LA} = QuotientTopology{N,L,M,0,LA}
 const CompactTopology{N,L,M,LA} = QuotientTopology{N,L,M,M,LA}
 QuotientTopology(n::ProductTopology) = OpenTopology(n.v)
@@ -257,48 +298,57 @@ SphereTopology(n::Values{5,Int}) = QuotientTopology(Values(1,2,3,4,5,6,7,8,10,9)
 GeographicTopology(n::Values{2,Int}) = QuotientTopology(Values(2,1,3,4),Values(ProductTopology(n[2]),ProductTopology(n[2]),ProductTopology(CrossRange(n[1])),ProductTopology(CrossRange(n[1]))),Values(1,2,3,4),n)
 
 OpenParameter(n::ProductTopology) = OpenParameter(n.v)
-OpenParameter(n::Values{1,Int}) = TensorField(GridFrameBundle(PointArray(0,LinRange(0,1,n[1])),OpenTopology(n)))
-OpenParameter(n::Values{2,Int}) = TensorField(GridFrameBundle(PointArray(LinRange(0,1,n[1])⊕LinRange(0,1,n[2])),OpenTopology(n)))
-OpenParameter(n::Values{3,Int}) = TensorField(GridFrameBundle(PointArray((LinRange(0,1,n[1])⊕LinRange(0,1,n[2]))⊕LinRange(0,1,n[3])),OpenTopology(n)))
-OpenParameter(n::Values{4,Int}) = TensorField(GridFrameBundle(PointArray(((LinRange(0,1,n[1])⊕LinRange(0,1,n[2]))⊕LinRange(0,1,n[3]))⊕LinRange(0,1,n[4])),OpenTopology(n)))
-OpenParameter(n::Values{5,Int}) = TensorField(GridFrameBundle(PointArray((((LinRange(0,1,n[1])⊕LinRange(0,1,n[2]))⊕LinRange(0,1,n[3]))⊕LinRange(0,1,n[4]))⊕LinRange(0,1,n[5])),OpenTopology(n)))
-RibbonParameter(n::Values{2,Int}) = TensorField(GridFrameBundle(PointArray(LinRange(0,2π,n[1])⊕LinRange(-1,1,n[2])),RibbonTopology(n)))
-MobiusParameter(n::Values{2,Int}) = TensorField(GridFrameBundle(PointArray(LinRange(0,2π,n[1])⊕LinRange(-1,1,n[2])),MobiusTopology(n)))
-WingParameter(n::Values{2,Int}) = TensorField(GridFrameBundle(PointArray(LinRange(0,1,n[1])⊕LinRange(-1,1,n[2])),WingTopology(n)))
-MirrorParameter(n::Values{1,Int}) = TensorField(GridFrameBundle(PointArray(0,LinRange(0,2π,n[1])),MirrorTopology(n)))
-MirrorParameter(n::Values{2,Int}) = TensorField(GridFrameBundle(PointArray(LinRange(0,2π,n[1])⊕LinRange(0,1,n[2])),MirrorTopology(n)))
-MirrorParameter(n::Values{3,Int}) = TensorField(GridFrameBundle(PointArray((LinRange(0,2π,n[1])⊕LinRange(0,1,n[2]))⊕LinRange(0,1,n[3])),MirrorTopology(n)))
-MirrorParameter(n::Values{4,Int}) = TensorField(GridFrameBundle(PointArray(((LinRange(0,2π,n[1])⊕LinRange(0,1,n[2]))⊕LinRange(0,1,n[3]))⊕LinRange(0,1,n[4])),MirrorTopology(n)))
-MirrorParameter(n::Values{5,Int}) = TensorField(GridFrameBundle(PointArray((((LinRange(0,2π,n[1])⊕LinRange(0,1,n[2]))⊕LinRange(0,1,n[3]))⊕LinRange(0,1,n[4]))⊕LinRange(0,1,n[5])),MirrorTopology(n)))
-ClampedParameter(n::Values{1,Int}) = TensorField(GridFrameBundle(PointArray(0,LinRange(0,2π,n[1])),ClampedTopology(n)))
-ClampedParameter(n::Values{2,Int}) = TensorField(GridFrameBundle(PointArray(LinRange(0,2π,n[1])⊕LinRange(0,2π,n[2])),ClampedTopology(n)))
-ClampedParameter(n::Values{3,Int}) = TensorField(GridFrameBundle(PointArray((LinRange(0,2π,n[1])⊕LinRange(0,2π,n[2]))⊕LinRange(0,2π,n[3])),ClampedTopology(n)))
-ClampedParameter(n::Values{4,Int}) = TensorField(GridFrameBundle(PointArray(((LinRange(0,2π,n[1])⊕LinRange(0,2π,n[2]))⊕LinRange(0,2π,n[3]))⊕LinRange(0,2π,n[4])),ClampedTopology(n)))
-ClampedParameter(n::Values{5,Int}) = TensorField(GridFrameBundle(PointArray((((LinRange(0,2π,n[1])⊕LinRange(0,2π,n[2]))⊕LinRange(0,2π,n[3]))⊕LinRange(0,2π,n[4]))⊕LinRange(0,2π,n[5])),ClampedTopology(n)))
-TorusParameter(n::Values{1,Int}) = TensorField(GridFrameBundle(PointArray(0,LinRange(0,2π,n[1])),TorusTopology(n)))
-TorusParameter(n::Values{2,Int}) = TensorField(GridFrameBundle(PointArray(LinRange(0,2π,n[1])⊕LinRange(0,2π,n[2])),TorusTopology(n)))
-TorusParameter(n::Values{3,Int}) = TensorField(GridFrameBundle(PointArray((LinRange(0,2π,n[1])⊕LinRange(0,2π,n[2]))⊕LinRange(0,2π,n[3])),TorusTopology(n)))
-TorusParameter(n::Values{4,Int}) = TensorField(GridFrameBundle(PointArray(((LinRange(0,2π,n[1])⊕LinRange(0,2π,n[2]))⊕LinRange(0,2π,n[3]))⊕LinRange(0,2π,n[4])),TorusTopology(n)))
-TorusParameter(n::Values{5,Int}) = TensorField(GridFrameBundle(PointArray((((LinRange(0,2π,n[1])⊕LinRange(0,2π,n[2]))⊕LinRange(0,2π,n[3]))⊕LinRange(0,2π,n[4]))⊕LinRange(0,2π,n[5])),TorusTopology(n)))
-HopfParameter(n::Values{2,Int}) = TensorField(GridFrameBundle(PointArray(LinRange(0,2π,n[2])⊕LinRange(0,4π,n[3])),HopfTopology(n)))
-HopfParameter(n::Values{3,Int}) = TensorField(GridFrameBundle(PointArray((LinRange(7π/16/n[1],7π/16,n[1])⊕LinRange(0,2π,n[2]))⊕LinRange(0,4π,n[3])),HopfTopology(n)))
-KleinParameter(n::Values{2,Int}) = TensorField(GridFrameBundle(PointArray(LinRange(0,2π,n[1])⊕LinRange(0,2π,n[2])),KleinTopology(n)))
-ConeParameter(n::Values{2,Int}) = TensorField(GridFrameBundle(PointArray(LinRange(0,1,n[1])⊕LinRange(0,2π,n[2])),ConeTopology(n)))
-PolarParameter(n::Values{2,Int}) = TensorField(GridFrameBundle(PointArray(LinRange(0,1,n[1])⊕LinRange(0,2π,n[2])),PolarTopology(n)))
+OpenParameter(n::Values{1,Int}) = OpenTopology(PointArray(0,LinRange(0,1,n[1])))
+OpenParameter(n::Values{2,Int}) = OpenTopology(LinRange(0,1,n[1])⊕LinRange(0,1,n[2]))
+OpenParameter(n::Values{3,Int}) = OpenTopology(LinRange(0,1,n[1])⊕LinRange(0,1,n[2])⊕LinRange(0,1,n[3]))
+OpenParameter(n::Values{4,Int}) = OpenTopology(LinRange(0,1,n[1])⊕LinRange(0,1,n[2])⊕LinRange(0,1,n[3])⊕LinRange(0,1,n[4]))
+OpenParameter(n::Values{5,Int}) = OpenTopology(LinRange(0,1,n[1])⊕LinRange(0,1,n[2])⊕LinRange(0,1,n[3])⊕LinRange(0,1,n[4])⊕LinRange(0,1,n[5]))
+RibbonParameter(n::Values{2,Int}) = RibbonTopology(LinRange(0,2π,n[1])⊕LinRange(-1,1,n[2]))
+MobiusParameter(n::Values{2,Int}) = MobiusTopology(LinRange(0,2π,n[1])⊕LinRange(-1,1,n[2]))
+WingParameter(n::Values{2,Int}) = WingParameter(LinRange(0,1,n[1])⊕LinRange(-1,1,n[2]))
+MirrorParameter(n::Values{1,Int}) = MirrorTopology(PointArray(0,LinRange(0,2π,n[1])))
+MirrorParameter(n::Values{2,Int}) = MirrorTopology(LinRange(0,2π,n[1])⊕LinRange(0,1,n[2]))
+MirrorParameter(n::Values{3,Int}) = MirrorTopology(LinRange(0,2π,n[1])⊕LinRange(0,1,n[2])⊕LinRange(0,1,n[3]))
+MirrorParameter(n::Values{4,Int}) = MirrorTopology(LinRange(0,2π,n[1])⊕LinRange(0,1,n[2])⊕LinRange(0,1,n[3])⊕LinRange(0,1,n[4]))
+MirrorParameter(n::Values{5,Int}) = MirrorTopology(LinRange(0,2π,n[1])⊕LinRange(0,1,n[2])⊕LinRange(0,1,n[3])⊕LinRange(0,1,n[4])⊕LinRange(0,1,n[5]))
+ClampedParameter(n::Values{1,Int}) = ClampedTopology(PointArray(0,LinRange(0,2π,n[1])))
+ClampedParameter(n::Values{2,Int}) = ClampedTopology(LinRange(0,2π,n[1])⊕LinRange(0,2π,n[2]))
+ClampedParameter(n::Values{3,Int}) = ClampedTopology(LinRange(0,2π,n[1])⊕LinRange(0,2π,n[2])⊕LinRange(0,2π,n[3]))
+ClampedParameter(n::Values{4,Int}) = ClampedTopology(LinRange(0,2π,n[1])⊕LinRange(0,2π,n[2])⊕LinRange(0,2π,n[3])⊕LinRange(0,2π,n[4]))
+ClampedParameter(n::Values{5,Int}) = ClampedTopology(LinRange(0,2π,n[1])⊕LinRange(0,2π,n[2])⊕LinRange(0,2π,n[3])⊕LinRange(0,2π,n[4])⊕LinRange(0,2π,n[5]))
+TorusParameter(n::Values{1,Int}) = TorusTopology(PointArray(0,LinRange(0,2π,n[1])))
+TorusParameter(n::Values{2,Int}) = TorusTopology(LinRange(0,2π,n[1])⊕LinRange(0,2π,n[2]))
+TorusParameter(n::Values{3,Int}) = TorusTopology(LinRange(0,2π,n[1])⊕LinRange(0,2π,n[2])⊕LinRange(0,2π,n[3]))
+TorusParameter(n::Values{4,Int}) = TorusTopology(LinRange(0,2π,n[1])⊕LinRange(0,2π,n[2])⊕LinRange(0,2π,n[3])⊕LinRange(0,2π,n[4]))
+TorusParameter(n::Values{5,Int}) = TorusTopology(LinRange(0,2π,n[1])⊕LinRange(0,2π,n[2])⊕LinRange(0,2π,n[3])⊕LinRange(0,2π,n[4])⊕LinRange(0,2π,n[5]))
+HopfParameter(n::Values{2,Int}) = HopfTopology(LinRange(0,2π,n[2])⊕LinRange(0,4π,n[3]))
+HopfParameter(n::Values{3,Int}) = HopfTopology(LinRange(7π/16/n[1],7π/16,n[1])⊕LinRange(0,2π,n[2])⊕LinRange(0,4π,n[3]))
+KleinParameter(n::Values{2,Int}) = KleinTopology(LinRange(0,2π,n[1])⊕LinRange(0,2π,n[2]))
+ConeParameter(n::Values{2,Int}) = ConeTopology(LinRange(0,1,n[1])⊕LinRange(0,2π,n[2]))
+PolarParameter(n::Values{2,Int}) = PolarTopology(LinRange(0,1,n[1])⊕LinRange(0,2π,n[2]))
 SphereParameter(n::Values{1,Int}) = TorusParameter(n)
-SphereParameter(n::Values{2,Int}) = TensorField(GridFrameBundle(PointArray(LinRange(0,π,n[1])⊕LinRange(0,2π,n[2])),SphereTopology(n)))
-SphereParameter(n::Values{3,Int}) = TensorField(GridFrameBundle(PointArray((LinRange(0,π,n[1])⊕LinRange(0,π,n[2]))⊕LinRange(0,2π,n[3])),SphereTopology(n)))
-SphereParameter(n::Values{4,Int}) = TensorField(GridFrameBundle(PointArray(((LinRange(0,π,n[1])⊕LinRange(0,π,n[2]))⊕LinRange(0,π,n[3]))⊕LinRange(0,2π,n[4])),SphereTopology(n)))
-SphereParameter(n::Values{5,Int}) = TensorField(GridFrameBundle(PointArray((((LinRange(0,π,n[1])⊕LinRange(0,π,n[2]))⊕LinRange(0,π,n[3]))⊕LinRange(0,π,n[4]))⊕LinRange(0,2π,n[5])),SphereTopology(n)))
-GeographicParameter(n::Values{2,Int}) = TensorField(GridFrameBundle(PointArray(LinRange(-π,π,n[1])⊕LinRange(-π/2,π/2,n[2])),GeographicTopology(n)))
-
+SphereParameter(n::Values{2,Int}) = SphereTopology(LinRange(0,π,n[1])⊕LinRange(0,2π,n[2]))
+SphereParameter(n::Values{3,Int}) = SphereTopology(LinRange(0,π,n[1])⊕LinRange(0,π,n[2])⊕LinRange(0,2π,n[3]))
+SphereParameter(n::Values{4,Int}) = SphereTopology(LinRange(0,π,n[1])⊕LinRange(0,π,n[2])⊕LinRange(0,π,n[3])⊕LinRange(0,2π,n[4]))
+SphereParameter(n::Values{5,Int}) = SphereTopology(LinRange(0,π,n[1])⊕LinRange(0,π,n[2])⊕LinRange(0,π,n[3])⊕LinRange(0,π,n[4])⊕LinRange(0,2π,n[5]))
+GeographicParameter(n::Values{2,Int}) = GeographicTopology(LinRange(-π,π,n[1])⊕LinRange(-π/2,π/2,n[2]))
+
+for fun ∈ (:Open,:Ribbon,:Mobius,:Wing,:Mirror,:Clamped,:Torus,:Hopf,:Klein,:Cone,:Polar,:Sphere,:Geographic)
+    for typ ∈ (Symbol(fun,:Topology),Symbol(fun,:Parameter))
+        @eval begin
+            export $typ
+            $typ(p::ProductSpace) = $typ(PointArray(p))
+            $typ(p::Values{N,<:AbstractVector} where N) = $typ(ProductSpace(p))
+            $typ(p::T...) where T<:AbstractVector = $typ(ProductSpace(Values(p)))
+            $typ(n::NTuple) = $typ(Values(n))
+        end
+    end
+end
 for mod ∈ (:Topology,:Parameter)
     for fun ∈ (:Hopf,)
         for typ ∈ (Symbol(fun,mod),)
             @eval begin
-                export $typ
                 $typ(n::Int...) = $typ(Values(n...))
-                $typ(n::NTuple) = $typ(Values(n))
                 $typ() = $typ(Values(7,60,61))
             end
         end
@@ -306,19 +356,15 @@ for mod ∈ (:Topology,:Parameter)
     for fun ∈ (:Open,:Mirror,:Clamped,:Torus)
         for typ ∈ (Symbol(fun,mod),)
             @eval begin
-                export $typ
                 $typ() = $typ(60,60)
                 $typ(n::Int...) = $typ(Values(n...))
-                $typ(n::NTuple) = $typ(Values(n))
             end
         end
     end
-    for (fun,(n,m)) ∈ ((:Ribbon,(60,20)),(:Wing,(60,20)),(:Mobius,(60,20)),(:Klein,(60,60)),(:Cone,(30,:(2n+1))),(:Polar,(30,:(2n))),(:Sphere,(30,:(2n+1))),(:Geographic,(:(2n+1),30)))
+    for (fun,n,m) ∈ ((:Ribbon,60,20),(:Wing,60,20),(:Mobius,60,20),(:Klein,60,60),(:Cone,30,:(2n+1)),(:Polar,30,:(2n)),(:Sphere,30,:(2n+1)),(:Geographic,61,:(n÷2)))
         for typ ∈ (Symbol(fun,mod),)
             @eval begin
-                export $typ
                 $typ(n=$n,m=$m) = $typ(Values(n,m))
-                $typ(n::NTuple{2,Int}) = $typ(Values(n))
             end
         end
     end
@@ -407,13 +453,28 @@ end
          :(m.r),
          Expr(:call,:Values,[j≠N ? :(@inbounds m.s[$j]) : :i for j ∈ list(1,N)]...))
 end
-@generated function resize(m::CompactTopology{N,L,M,O},i) where {N,L,M,O}
+@generated function resize(m::QuotientTopology{N,L,O,O},i) where {N,L,O}
     Expr(:call,:QuotientTopology,:(m.p),
          Expr(:call,:Values,[j∉(O-1,O) ? :(@inbounds resize(m.q[$j],i)) : :(@inbounds m.q[$j]) for j ∈ list(1,O)]...),
          :(m.r),
          Expr(:call,:Values,[j≠N ? :(@inbounds m.s[$j]) : :i for j ∈ list(1,N)]...))
 end
 
+resample(m::QuotientTopology{N,L,M,0},i::NTuple{N}) where {N,L,M} = QuotientTopology(m.p,m.q,m.r,Values(i))
+@generated function resample(m::QuotientTopology{N,L,O,O},i::NTuple{N}) where {N,L,O}
+    Expr(:block,:(perms = $(reverse(Values{L}.(Values{N}(Grassmann.combo(N,L)))))),
+        Expr(:call,:QuotientTopology,:(m.p),
+            Expr(:call,:Values,Expr(:tuple,[:(@inbounds resample(m.q[$j],i[perms[$((j+1)÷2)]])) for j ∈ list(1,O)]...)),
+            :(m.r),Expr(:call,:Values,:i)))
+end
+@generated function resample(m::QuotientTopology{N,L,M,O},i::NTuple{N}) where {N,L,M,O}
+    Expr(:block,:(t = invert_q($(Val(O)),m.r)),
+        :(perms = $(reverse(Values{L}.(Values{N}(Grassmann.combo(N,L)))))),
+        Expr(:call,:QuotientTopology,:(m.p),
+            Expr(:call,:Values,Expr(:tuple,[:(@inbounds resample(m.q[$j],i[perms[t[$((j+1)÷2)]]])) for j ∈ list(1,O)]...)),
+            :(m.r),Expr(:call,:Values,:i)))
+end
+
 @pure Base.eltype(::Type{<:QuotientTopology{N}}) where N = Values{N,Int}
 Base.size(m::QuotientTopology) = m.s.v
 Base.iterate(t::QuotientTopology) = (getindex(t,1),1)
@@ -644,21 +705,11 @@ function subtopology(m::QuotientTopology,i::NTuple{N,Int},::Colon,j::NTuple{M,In
     N1,M1,s = N+1,N+M+2,size(m)
     r,vals = m.r[Values(2N1-1,2N1,2M1-1,2M1)],Values(i...,j...,k...)
     (p1,p2,p3,p4) = (
-        findface(m,r,1,vals,M1-1),findface(m,r,2,vals,M1-1),
-        findface(m,r,3,vals,N1),findface(m,r,4,vals,N1))
+        findface(m,r,1,vals,Val(M1-1)),findface(m,r,2,vals,Val(M1-1)),
+        findface(m,r,3,vals,Val(N1)),findface(m,r,4,vals,Val(N1)))
     p1z,p2z,p3z,p4z = iszero.((p1,p2,p3,p4))
     pz = !(p1z)+!(p2z)+!(p3z)+!(p4z)
-    vpz = if iszero(pz)
-        Values{0}
-    elseif isone(pz)
-        Values{1}
-    elseif pz==2
-        Values{2}
-    elseif pz==3
-        Values{3}
-    else
-        Values{4}
-    end
+    vpz = iszero(pz) ? Values{0} : isone(pz) ? Values{1} : pz==2 ? Values{2} : pz==3 ? Values{3} : Values{4}
     a = iszero(pz) ? Values{0,Int}() : vpz((p1z ? () : (p1,))...,(p2z ? () : (p2,))...,(p3z ? () : (p3,))...,(p4z ? () : (p4,))...)
     b = if iszero(pz)
         Values{0,Array{Values{1,Int},1}}()
@@ -669,21 +720,23 @@ function subtopology(m::QuotientTopology,i::NTuple{N,Int},::Colon,j::NTuple{M,In
         (p4z ? () : (ProductTopology(m.q[r[4]].v[N1]),))...)
     end
     c = Values{4,Int}(p1z ? 0 : 1,p2z ? 0 : !(p1z)+!(p2z),p3z ? 0 : !(p1z)+!(p2z)+!(p3z),p4z ? 0 : pz)
+    #e = iszero(pz) ? Values{0,Int} : vpz((p1z ? () : (1,))...,(p2z ? () : (!(p1z)+!(p2z),))...,(p3z ? () : (!(p1z)+!(p2z)+!(p3z),))...,(p4z ? () : (pz,))...)
     QuotientTopology(a,b,c,Values(s[N1],s[M1]))
 end
 function subtopology(m::QuotientTopology,i::NTuple{N,Int},::Colon,j::NTuple{M,Int},::Colon,k::NTuple{L,Int},::Colon,l::NTuple{O,Int} where O) where {N,M,L}
     N1,M1,L1,s = N+1,N+M+2,N+M+L+3,size(m)
     r,vals = m.r[Values(2N1-1,2N1,2M1-1,2M1,2L1-1,2L1)],Values(i...,j...,k...,l...)
     (p1,p2,p3,p4,p5,p6) = (
-        findface(m,r,1,vals,M1-1,L1-1),findface(m,r,2,vals,M1-1,L1-1),
-        findface(m,r,3,vals,N1,L1-1),findface(m,r,4,vals,N1,L1-1),
-        findface(m,r,5,vals,N1,M1),findface(m,r,6,vals,N1,M1))
+        findface(m,r,1,vals,Val(M1-1),Val(L1-1)),findface(m,r,2,vals,Val(M1-1),Val(L1-1)),
+        findface(m,r,3,vals,Val(N1),Val(L1-1)),findface(m,r,4,vals,Val(N1),Val(L1-1)),
+        findface(m,r,5,vals,Val(N1),Val(M1)),findface(m,r,6,vals,Val(N1),Val(M1)))
     p1z,p2z,p3z,p4z,p5z,p6z = iszero.((p1,p2,p3,p4,p5,p6))
     pz = !(p1z)+!(p2z)+!(p3z)+!(p4z)+!(p5z)+!(p6z)
-    a = Values{pz,Int}((p1z ? () : (p1,))...,(p2z ? () : (p2,))...,(p3z ? () : (p3,))...,(p4z ? () : (p4,))...,(p5z ? () : (p5,))...,(p6z ? () : (p6,))...)
+    vpz = iszero(pz) ? Values{0} : isone(pz) ? Values{1} : pz==2 ? Values{2} : pz==3 ? Values{3} : pz==4 ? Values{4} : pz==5 ? Values{5} : Values{6}
+    a = iszero(pz) ? Values{0,Int}() : vpz((p1z ? () : (p1,))...,(p2z ? () : (p2,))...,(p3z ? () : (p3,))...,(p4z ? () : (p4,))...,(p5z ? () : (p5,))...,(p6z ? () : (p6,))...)
     b = if iszero(pz)
-        Values{pz,Array{Values{2,Int},2}}()
-    else; Values{pz}(
+        Values{0,Array{Values{2,Int},2}}()
+    else; vpz(
         (p1z ? () : (ProductTopology(m.q[r[1]].v[Values(M1-1,L1-1)]),))...,
         (p2z ? () : (ProductTopology(m.q[r[2]].v[Values(M1-1,L1-1)]),))...,
         (p3z ? () : (ProductTopology(m.q[r[3]].v[Values(N1,L1-1)]),))...,
@@ -692,22 +745,24 @@ function subtopology(m::QuotientTopology,i::NTuple{N,Int},::Colon,j::NTuple{M,In
         (p6z ? () : (ProductTopology(m.q[r[6]].v[Values(N1,M1)]),))...)
     end
     c = Values(p1z ? 0 : 1,p2z ? 0 : !(p1z)+!(p2z),p3z ? 0 : !(p1z)+!(p2z)+!(p3z),p4z ? 0 : !(p1z)+!(p2z)+!(p3z)+!(p4z),p5z ? 0 : !(p1z)+!(p2z)+!(p3z)+!(p4z)+!(p5z),p6z ? 0 : pz)
+    #e = iszero(pz) ? Values{0,Int} : vpz((p1z ? () : (1,))...,(p2z ? () : (!(p1z)+!(p2z),))...,(p3z ? () : (!(p1z)+!(p2z)+!(p3z),))...,(p4z ? () : (!(p1z)+!(p2z)+!(p3z)+!(p4z),))...,(p5z ? () : (!(p1z)+!(p2z)+!(p3z)+!(p4z)+!(p5z),))...,(p6z ? () : (pz,))...)
     QuotientTopology(a,b,c,Values(s[N1],s[M1],s[L1]))
 end
 function subtopology(m::QuotientTopology,i::NTuple{N,Int},::Colon,j::NTuple{M,Int},::Colon,k::NTuple{L,Int},::Colon,l::NTuple{O,Int},::Colon,h::NTuple{H,Int} where H) where {N,M,L,O}
     N1,M1,L1,O1,s = N+1,N+M+2,N+M+L+3,N+M+L+O+4,size(m)
     r,vals = m.r[Values(2N1-1,2N1,2M1-1,2M1,2L1-1,2L1,2O1-1,2O1)],Values(i...,j...,k...,l...,h...)
     (p1,p2,p3,p4,p5,p6,p7,p8) = (
-        findface(m,r,1,vals,M1-1,L1-1,O1-1),findface(m,r,2,vals,M1-1,L1-1,O1-1),
-        findface(m,r,3,vals,N1,L1-1,O1-1),findface(m,r,4,vals,N1,L1-1,O1-1),
-        findface(m,r,5,vals,N1,M1,O1-1),findface(m,r,6,vals,N1,M1,O1-1),
-        findface(m,r,7,vals,N1,M1,L1),findface(m,r,8,vals,N1,M1,L1))
+        findface(m,r,1,vals,Val(M1-1),Val(L1-1),Val(O1-1)),findface(m,r,2,vals,Val(M1-1),Val(L1-1),Val(O1-1)),
+        findface(m,r,3,vals,Val(N1),Val(L1-1),Val(O1-1)),findface(m,r,4,vals,Val(N1),Val(L1-1),Val(O1-1)),
+        findface(m,r,5,vals,Val(N1),Val(M1),Val(O1-1)),findface(m,r,6,vals,Val(N1),Val(M1),Val(O1-1)),
+        findface(m,r,7,vals,Val(N1),Val(M1),Val(L1)),findface(m,r,8,vals,Val(N1),Val(M1),Val(L1)))
     p1z,p2z,p3z,p4z,p5z,p6z,p7z,p8z = iszero.((p1,p2,p3,p4,p5,p6,p7,p8))
     pz = !(p1z)+!(p2z)+!(p3z)+!(p4z)+!(p5z)+!(p6z)+!(p7z)+!(p8z)
-    a = Values{pz,Int}((p1z ? () : (p1,))...,(p2z ? () : (p2,))...,(p3z ? () : (p3,))...,(p4z ? () : (p4,))...,(p5z ? () : (p5,))...,(p6z ? () : (p6,))...,(p7z ? () : (p7,))...,(p8z ? () : (p8,))...)
+    vpz = iszero(pz) ? Values{0} : isone(pz) ? Values{1} : pz==2 ? Values{2} : pz==3 ? Values{3} : pz==4 ? Values{4} : pz==5 ? Values{5} : pz==6 ? Values{6} : pz==7 ? Values{7} : Values{8}
+    a = iszero(pz) ? Values{0,Int}() : vpz((p1z ? () : (p1,))...,(p2z ? () : (p2,))...,(p3z ? () : (p3,))...,(p4z ? () : (p4,))...,(p5z ? () : (p5,))...,(p6z ? () : (p6,))...,(p7z ? () : (p7,))...,(p8z ? () : (p8,))...)
     b = if iszero(pz)
-        Values{pz,Array{Values{3,Int},3}}()
-    else; Values{pz}(
+        Values{0,Array{Values{3,Int},3}}()
+    else; vpz(
         (p1z ? () : (ProductTopology(m.q[r[1]].v[Values(M1-1,L1-1,O1-1)]),))...,
         (p2z ? () : (ProductTopology(m.q[r[2]].v[Values(M1-1,L1-1,O1-1)]),))...,
         (p3z ? () : (ProductTopology(m.q[r[3]].v[Values(N1,L1-1,O1-1)]),))...,
@@ -718,6 +773,7 @@ function subtopology(m::QuotientTopology,i::NTuple{N,Int},::Colon,j::NTuple{M,In
         (p8z ? () : (ProductTopology(m.q[r[8]].v[Values(N1,M1,L1)]),))...)
     end
     c = Values(p1z ? 0 : 1,p2z ? 0 : !(p1z)+!(p2z),p3z ? 0 : !(p1z)+!(p2z)+!(p3z),p4z ? 0 : !(p1z)+!(p2z)+!(p3z)+!(p4z),p5z ? 0 : !(p1z)+!(p2z)+!(p3z)+!(p4z)+!(p5z),p6z ? 0 : !(p1z)+!(p2z)+!(p3z)+!(p4z)+!(p5z)+!(p6z),p7z ? 0 : !(p1z)+!(p2z)+!(p3z)+!(p4z)+!(p5z)+!(p6z)+!(p7z),p8z ? 0 : pz)
+    #e = iszero(pz) ? Values{0,Int} : vpz((p1z ? () : (1,))...,(p2z ? () : (!(p1z)+!(p2z),))...,(p3z ? () : (!(p1z)+!(p2z)+!(p3z),))...,(p4z ? () : (!(p1z)+!(p2z)+!(p3z)+!(p4z),))...,(p5z ? () : (!(p1z)+!(p2z)+!(p3z)+!(p4z)+!(p5z),))...,(p6z ? () : (!(p1z)+!(p2z)+!(p3z)+!(p4z)+!(p5z)+!(p6z),))...,(p7z ? () : (!(p1z)+!(p2z)+!(p3z)+!(p4z)+!(p5z)+!(p6z)+!(p7z),))...,(p8z ? () : (pz,))...)
     QuotientTopology(a,b,c,Values(s[N1],s[M1],s[L1],s[O1]))
 end
 
@@ -836,6 +892,7 @@ Base.show(io::IO, ::MIME"text/plain", t::Global) = show(io,t)
 ref(itr::InducedMetric) = Ref(itr)
 ref(itr::Global) = Ref(itr.v)
 ref(itr) = itr
+refmetric(x) = ref(metricextensor(x))
 
 # LocalFiber
 
@@ -943,6 +1000,7 @@ localfiber(x) = x
 localfiber(x::LocalTensor) = fiber(x)
 localfiber(x::LocalSection) = fiber(x)
 
+(m::TensorNested)(x::LocalTensor) = LocalTensor(base(x),m(fiber(x)))
 @inline Base.:<<(a::LocalFiber,b::LocalFiber) = contraction(b,~a)
 @inline Base.:>>(a::LocalFiber,b::LocalFiber) = contraction(~a,b)
 @inline Base.:<(a::LocalFiber,b::LocalFiber) = contraction(b,a)
@@ -951,6 +1009,9 @@ Base.inv(a::LocalTensor{B,<:Real} where B) = LocalTensor(base(a), inv(fiber(a)))
 Base.inv(a::LocalTensor{B,<:Complex} where B) = LocalTensor(base(a), inv(fiber(a)))
 Base.:/(a::LocalTensor,b::LocalTensor{B,<:Real} where B) = LocalTensor(base(a), fiber(a)/fiber(b))
 Base.:/(a::LocalTensor,b::LocalTensor{B,<:Complex} where B) = LocalTensor(base(a), fiber(a)/fiber(b))
+LinearAlgebra.:×(a::LocalTensor{R},b::LocalTensor{R}) where R = TensorField(base(a), ⋆(fiber(a)∧fiber(b),metricextensor(base(a))))
+Grassmann.compound(t::LocalTensor,i::Val) = LocalTensor(base(t), compound(fiber(t),i))
+Grassmann.compound(t::LocalTensor,i::Int) = LocalTensor(base(t), compound(fiber(t),i))
 Grassmann.eigen(t::LocalTensor,i::Val) = LocalTensor(base(t), eigen(fiber(t),i))
 Grassmann.eigen(t::LocalTensor,i::Int) = LocalTensor(base(t), eigen(fiber(t),i))
 Grassmann.eigvals(t::LocalTensor,i::Val) = LocalTensor(base(t), eigvals(fiber(t),i))
@@ -962,13 +1023,13 @@ for type ∈ (:Coordinate,:LocalSection,:LocalTensor)
     for tensor ∈ (:Single,:Couple,:PseudoCouple,:Chain,:Spinor,:AntiSpinor,:Multivector,:DiagonalOperator,:TensorOperator,:Outermorphism)
         @eval (T::Type{<:$tensor})(s::$type) = $type(base(s), T(fiber(s)))
     end
-    for fun ∈ (:-,:!,:~,:real,:imag,:conj,:deg2rad)
+    for fun ∈ (:-,:!,:~,:real,:imag,:conj,:deg2rad,:transpose)
         @eval Base.$fun(s::$type) = $type(base(s), $fun(fiber(s)))
     end
     for fun ∈ (:inv,:exp,:exp2,:exp10,:log,:log2,:log10,:sinh,:cosh,:abs,:sqrt,:cbrt,:cos,:sin,:tan,:cot,:sec,:csc,:asec,:acsc,:sech,:csch,:asech,:tanh,:coth,:asinh,:acosh,:atanh,:acoth,:asin,:acos,:atan,:acot,:sinc,:cosc,:cis,:abs2)
         @eval Base.$fun(s::$type) = $type(base(s), $fun(fiber(s),metricextensor(base(s))))
     end
-    for fun ∈ (:reverse,:involute,:clifford,:even,:odd,:scalar,:vector,:bivector,:volume,:value,:curl,:∂,:d,:complementleft,:realvalue,:imagvalue,:outermorphism,:Outermorphism,:DiagonalOperator,:TensorOperator,:eigen,:eigvecs,:eigvals,:eigvalsreal,:eigvalscomplex,:eigvecsreal,:eigvecscomplex,:eigpolys)
+    for fun ∈ (:reverse,:involute,:clifford,:even,:odd,:scalar,:vector,:bivector,:volume,:value,:curl,:∂,:d,:complementleft,:realvalue,:imagvalue,:outermorphism,:Outermorphism,:DiagonalOperator,:TensorOperator,:eigen,:eigvecs,:eigvals,:eigvalsreal,:eigvalscomplex,:eigvecsreal,:eigvecscomplex,:eigpolys,:∧)
         @eval Grassmann.$fun(s::$type) = $type(base(s), $fun(fiber(s)))
     end
     for fun ∈ (:⋆,:angle,:radius,:complementlefthodge,:pseudoabs,:pseudoabs2,:pseudoexp,:pseudolog,:pseudoinv,:pseudosqrt,:pseudocbrt,:pseudocos,:pseudosin,:pseudotan,:pseudocosh,:pseudosinh,:pseudotanh,:metric,:unit)
@@ -1059,6 +1120,7 @@ basetype(::PointArray{B}) where B = B
 basetype(::Type{<:PointArray{B}}) where B = B
 fibertype(::PointArray{B,F} where B) where F = F
 fibertype(::Type{<:PointArray{B,F} where B}) where F = F
+isinduced(::Array) = false
 isinduced(::Global) = false
 isinduced(::Global{N,<:InducedMetric} where N) = true
 isinduced(p::PointArray) = isinduced(metricextensor(p))
@@ -1284,6 +1346,7 @@ GridFrameBundle(dom::AbstractArray,fun::Function) = GridFrameBundle(dom, fun.(do
 
 isopen(t::GridFrameBundle) = isopen(immersion(t))
 iscompact(t::GridFrameBundle) = iscompact(immersion(t))
+isinduced(t::GridFrameBundle) = isinduced(base(t))
 bundle(m::GridFrameBundle) = m.id
 basetype(m::GridFrameBundle) = basetype(base(m))
 basetype(::Type{<:GridFrameBundle{N,C} where N}) where C = basetype(C)
@@ -1295,6 +1358,17 @@ fibertype(::Type{<:GridFrameBundle{N,C} where N}) where C = fibertype(C)
 cross(a::GridFrameBundle,b::GridFrameBundle) = a⊕b
 cross(a::GridFrameBundle,b::AbstractVector{<:Real}) = a⊕b
 
+function resample(m::GridFrameBundle,i::NTuple)
+    rp,rq = resample(points(m),i),resample(immersion(m),i)
+    gid = iszero(m.id) ? 0 : (global grid_id+=1)
+    pid = iszero(base(m).id) ? 0 : (global point_id+=1)
+    if isinduced(m)
+        GridFrameBundle(gid,PointArray(pid,rp),rq)
+    else
+        GridFrameBundle(gid,PointArray(pid,rp,m.(rp)),rq)
+    end
+end
+
 resize_lastdim!(m::GridFrameBundle,i) = (resize_lastdim!(base(m),i); m)
 resize(m::GridFrameBundle) = GridFrameBundle(m.id,base(m),resize(immersion(m),size(base(m))[end]))
 Base.resize!(m::GridFrameBundle,i) = (resize!(base(m),i); m)
@@ -1327,11 +1401,11 @@ function Base.similar(bc::Broadcast.Broadcasted{Broadcast.ArrayStyle{GridFrameBu
     # Use the data type to create the output
     GridFrameBundle(similar(Array{pointtype(ElType),N}, ax), similar(Array{metrictype(ElType),N}, ax))
 end
-function Base.similar(bc::Broadcast.Broadcasted{Broadcast.ArrayStyle{GridFrameBundle{N,C,PA}}}, ::Type{ElType}) where {N,C,PA,ElType}
+#=function Base.similar(bc::Broadcast.Broadcasted{Broadcast.ArrayStyle{GridFrameBundle{N,C,PA}}}, ::Type{ElType}) where {N,C,PA,ElType}
     t = find_gf(bc)
     # Use the data type to create the output
     GridFrameBundle(similar(Array{ElType,N}, axes(bc)), metricextensor(t))
-end
+end=#
 
 "`A = find_gf(As)` returns the first GridFrameBundle among the arguments."
 find_gf(bc::Base.Broadcast.Broadcasted) = find_gf(bc.args)
@@ -1339,7 +1413,7 @@ find_gf(bc::Base.Broadcast.Extruded) = find_gf(bc.x)
 find_gf(args::Tuple) = find_gf(find_gf(args[1]), Base.tail(args))
 find_gf(x) = x
 find_gf(::Tuple{}) = nothing
-find_gf(a::GridFrameBundle, rest) = a
+find_gf(a::AbstractFrameBundle, rest) = a
 find_gf(::Any, rest) = find_gf(rest)
 
 # SimplexFrameBundle
@@ -1365,6 +1439,8 @@ fibertype(::SimplexFrameBundle{P}) where P = metrictype(P)
 fibertype(::Type{<:SimplexFrameBundle{P}}) where P = metrictype(P)
 Base.size(m::SimplexFrameBundle) = size(vertices(m))
 
+isinduced(t::SimplexFrameBundle) = isinduced(base(t))
+
 Base.broadcast(f,t::SimplexFrameBundle) = SimplexFrameBundle(f.(PointCloud(t)),ImmersedTopology(t))
 
 Base.firstindex(m::SimplexFrameBundle) = 1