Update documentation (#222)

* update docs for B-spline manifold

* remove citation

* update documentation for refinement

* fix typo

README.md

@@ -172,19 +172,3 @@ save_svg("sine-curve_no-points.svg", M, unitlength=50, xlims=(-2,2), ylims=(-8,8
 This is useful when you edit graphs (or curves) with your favorite vector graphics editor.
 
 ![](docs/src/img/inkscape.png)
-
-## Citation
-If you use BasicBSpline.jl in your work, please consider citing it by
-
-```bibtex
-@software{yuto_horikawa_2022_6465801,
-  author       = {Yuto Horikawa},
-  title        = {hyrodium/BasicBSpline.jl: v0.5.6},
-  month        = apr,
-  year         = 2022,
-  publisher    = {Zenodo},
-  version      = {v0.5.6},
-  doi          = {10.5281/zenodo.6465801},
-  url          = {https://doi.org/10.5281/zenodo.6465801}
-}
-```

docs/src/math-bsplinemanifold.md

@@ -27,7 +27,7 @@ Higher dimensional tensor products ``\mathcal{P}[p^1,k^1]\otimes\cdots\otimes\ma
 B-spline manifold is a parametric representation of a shape.
 
 !!! tip "Def.  B-spline manifold"
-    For given ``d``-dimensional B-spline basis functions ``B_{(i^1,p^1,k^1)} \otimes \cdots \otimes B_{(i^d,p^d,k^d)}`` and given points ``\bm{a}_{i^1,\dots,i^d} \in \mathbb{R}^{\hat{d}}``, B-spline manifold is defined by following equality:
+    For given ``d``-dimensional B-spline basis functions ``B_{(i^1,p^1,k^1)} \otimes \cdots \otimes B_{(i^d,p^d,k^d)}`` and given points ``\bm{a}_{i^1,\dots,i^d} \in V``, B-spline manifold is defined by following equality:
     ```math
     \bm{p}(t^1,\dots,t^d;\bm{a}_{i^1,\dots,i^d})
     =\sum_{i^1,\dots,i^d}(B_{(i^1,p^1,k^1)} \otimes \cdots \otimes B_{(i^d,p^d,k^d)})(t^1,\dots,t^d) \bm{a}_{i^1,\dots,i^d}
@@ -88,33 +88,27 @@ savefig("2dim-manifold.html") # hide
 
 ## Affine commutativity
 !!! info "Thm.  Affine commutativity"
-    If ``T`` is a affine transform ``\mathbb{R}^d\to\mathbb{R}^d``, then the following equality holds.
+    Let ``T`` be a affine transform ``V \to W``, then the following equality holds.
     ```math
     T(\bm{p}(t; \bm{a}))
     =\bm{p}(t; T(\bm{a}))
     ```
 
-## Currying
+## Fixing arguments (currying)
+Just like fixing first index such as `A[3,:]::Array{1}` for matrix `A::Array{2}`, fixing first argument `M(4.3,:)` will create `BSplineManifold{1}` for B-spline surface `M::BSplineManifold{2}`.
 
 ```@repl math
-p = 2; # degree of polynomial
-k = KnotVector(1:8); # knot vector
-P = BSplineSpace{p}(k); # B-spline space
-rand_a = [SVector(rand(), rand(), rand()) for i in 1:dim(P), j in 1:dim(P)];
-a = [SVector(2*i-6.5, 2*j-6.5, 0) for i in 1:dim(P), j in 1:dim(P)] + rand_a; # random generated control points
-M = BSplineManifold(a,(P,P)); # Define B-spline manifold
+p = 2;
+k = KnotVector(1:8);
+P = BSplineSpace{p}(k);
+a = [SVector(5i+5j+rand(), 5i-5j+rand(), rand()) for i in 1:dim(P), j in 1:dim(P)];
+M = BSplineManifold(a,(P,P));
 M isa BSplineManifold{2}
 M(:,:) isa BSplineManifold{2}
-M(4.3,:) isa BSplineManifold{1}
+M(4.3,:) isa BSplineManifold{1}  # Fix first argument
 ```
 
 ```@example math
-p = 2 # degree of polynomial
-k = KnotVector(1:8) # knot vector
-P = BSplineSpace{p}(k) # B-spline space
-rand_a = [SVector(rand(), rand(), rand()) for i in 1:dim(P), j in 1:dim(P)]
-a = [SVector(2*i-6.5, 2*j-6.5, 0) for i in 1:dim(P), j in 1:dim(P)] + rand_a # random generated control points
-M = BSplineManifold(a,(P,P)) # Define B-spline manifold
 plot(M)
 plot!(M(4.3,:), linewidth = 5, color=:white)
 savefig("2dim-manifold-currying.html") # hide

docs/src/math-refinement.md

@@ -7,11 +7,16 @@ using StaticArrays
 using Plots; plotly()
 ```
 
+## Documentation
+
 ```@docs
 refinement
 ```
 
-```@repl math
+## Example
+### Define original manifold
+
+```@example math
 p = 2 # degree of polynomial
 k = KnotVector(1:8) # knot vector
 P = BSplineSpace{p}(k) # B-spline space
@@ -20,7 +25,7 @@ a = [SVector(2*i-6.5, 2*j-6.5) for i in 1:dim(P), j in 1:dim(P)] + rand_a # rand
 M = BSplineManifold(a,(P,P)) # Define B-spline manifold
 ```
 
-## h-refinemnet
+### h-refinemnet
 Insert additional knots to knot vector.
 
 ```@repl math
@@ -33,7 +38,7 @@ save_png("2dim_h-refinement.png", M_h) # save image
 Note that this shape and the last shape are identical.
 
 
-## p-refinemnet
+### p-refinemnet
 Increase the polynomial degree of B-spline manifold.
 
 ```@repl math

src/_Fitting.jl

@@ -365,7 +365,7 @@ Similarly, for the two-dimensional case, minimize the following integral.
 ```math
 \int_{I^1 \times I^2} \left\|f(t^1, t^2)-\sum_{i,j} B_{(i,p^1,k^1)}(t^1)B_{(j,p^2,k^2)}(t^2) \bm{a}_{ij}\right\|^2 dt^1dt^2
 ```
-Currently, this function currently supports up to three dimensions.
+Currently, this function supports up to three dimensions.
 
 # Examples
 ```jldoctest