From dce95a209de964642786963250536bb8d2349161 Mon Sep 17 00:00:00 2001
From: Michael Heuer <michael.heuer@rwth-aachen.de>
Date: Thu, 8 Feb 2024 22:46:54 +0100
Subject: [PATCH 1/2] docs: added equations and documentation

---
 .gitignore                                    |   4 +
 README.md                                     |   2 +-
 doc-images/equations/_preamble.tex            |  17 +
 doc-images/equations/basis-function-zero.svg  | 262 ++++++++++
 doc-images/equations/basis-function-zero.tex  |  11 +
 doc-images/equations/basis-function.svg       | 309 ++++++++++++
 doc-images/equations/basis-function.tex       |  10 +
 doc-images/equations/basis-prefactor.svg      | 282 +++++++++++
 doc-images/equations/basis-prefactor.tex      |  11 +
 doc-images/equations/control-points.svg       | 388 ++++++++++++++
 doc-images/equations/control-points.tex       |  13 +
 doc-images/equations/curve-deriv.tex          |  13 -
 doc-images/equations/curve-deriv.tex.svg      | 268 ----------
 doc-images/equations/curve.svg                | 314 ++++++++++++
 doc-images/equations/curve.tex                |  19 +-
 doc-images/equations/curve.tex.svg            | 152 ------
 .../equations/generate-equations-images.sh    |  10 +
 doc-images/equations/knots.svg                | 474 ++++++++++++++++++
 doc-images/equations/knots.tex                |  16 +
 src/curve/basis/mod.rs                        | 191 +++++++
 src/curve/knots/mod.rs                        | 182 +------
 src/curve/mod.rs                              |  21 +
 src/curve/parameters/mod.rs                   |   2 +-
 src/curve/points/mod.rs                       |  17 +-
 src/lib.rs                                    |  21 +-
 src/manipulation/merge.rs                     |  28 +-
 src/manipulation/mod.rs                       |   2 +
 27 files changed, 2395 insertions(+), 644 deletions(-)
 create mode 100644 doc-images/equations/_preamble.tex
 create mode 100644 doc-images/equations/basis-function-zero.svg
 create mode 100644 doc-images/equations/basis-function-zero.tex
 create mode 100644 doc-images/equations/basis-function.svg
 create mode 100644 doc-images/equations/basis-function.tex
 create mode 100644 doc-images/equations/basis-prefactor.svg
 create mode 100644 doc-images/equations/basis-prefactor.tex
 create mode 100644 doc-images/equations/control-points.svg
 create mode 100644 doc-images/equations/control-points.tex
 delete mode 100644 doc-images/equations/curve-deriv.tex
 delete mode 100644 doc-images/equations/curve-deriv.tex.svg
 create mode 100644 doc-images/equations/curve.svg
 delete mode 100644 doc-images/equations/curve.tex.svg
 create mode 100755 doc-images/equations/generate-equations-images.sh
 create mode 100644 doc-images/equations/knots.svg
 create mode 100644 doc-images/equations/knots.tex
 create mode 100644 src/curve/basis/mod.rs

diff --git a/.gitignore b/.gitignore
index b80da91..041d818 100644
--- a/.gitignore
+++ b/.gitignore
@@ -5,6 +5,10 @@ target/
 # files
 .DS_Store
 **/*.rs.bk
+*.aux
+*.log
+*.gz
+*.pdf
 
 #/target
 /Cargo.lock
diff --git a/README.md b/README.md
index 2bc3579..beece4e 100644
--- a/README.md
+++ b/README.md
@@ -1,4 +1,4 @@
-# Bsplines
+# `bsplines` Rust Library
 
 [![Crates.io](https://img.shields.io/crates/v/bsplines)](https://crates.io/crates/bsplines)
 [![Docs.rs](https://docs.rs/bsplines/badge.svg)](https://docs.rs/bsplines)
diff --git a/doc-images/equations/_preamble.tex b/doc-images/equations/_preamble.tex
new file mode 100644
index 0000000..dbfb0b6
--- /dev/null
+++ b/doc-images/equations/_preamble.tex
@@ -0,0 +1,17 @@
+\documentclass[tikz]{standalone}
+\usepackage{amssymb}
+\usepackage{amsmath}
+\usepackage{oubraces}
+\usepackage[customcolors]{hf-tikz}
+\definecolor{docsrscolor}{HTML}{f5f5f5}
+
+\newcommand{\myeqs}[1]{
+    \Huge
+    \begin{tikzpicture}
+        \node[
+            fill=docsrscolor,
+            rounded corners=8mm,
+            inner sep=8mm
+        ]{$#1$};
+    \end{tikzpicture}
+}
diff --git a/doc-images/equations/basis-function-zero.svg b/doc-images/equations/basis-function-zero.svg
new file mode 100644
index 0000000..7a28935
--- /dev/null
+++ b/doc-images/equations/basis-function-zero.svg
@@ -0,0 +1,262 @@
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diff --git a/doc-images/equations/basis-function-zero.tex b/doc-images/equations/basis-function-zero.tex
new file mode 100644
index 0000000..bde9539
--- /dev/null
+++ b/doc-images/equations/basis-function-zero.tex
@@ -0,0 +1,11 @@
+\input{_preamble}
+\begin{document}
+    \myeqs{
+        \mathcal{N}_{i,0}^{\boldsymbol{U}^{(k)}}(u)
+        =
+        \begin{cases}
+            1, & u\in\left[u_i^{(k)},u_{i+1}^{(k)}\right)\,\lor\, (i=n-k \land u=1)\\
+            0, & \text{else}
+        \end{cases}
+    }
+\end{document}
diff --git a/doc-images/equations/basis-function.svg b/doc-images/equations/basis-function.svg
new file mode 100644
index 0000000..04fdac5
--- /dev/null
+++ b/doc-images/equations/basis-function.svg
@@ -0,0 +1,309 @@
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diff --git a/doc-images/equations/basis-function.tex b/doc-images/equations/basis-function.tex
new file mode 100644
index 0000000..d27943a
--- /dev/null
+++ b/doc-images/equations/basis-function.tex
@@ -0,0 +1,10 @@
+\input{_preamble}
+\begin{document}
+    \myeqs{
+        \myeqs{
+            \mathcal{N}_{i,p-k}^{\boldsymbol{U}^{(k)}}(u)
+            = \mu_{i,p-k-1}^{(k)}(u)\, \mathcal{N}_{i,p-k-1}^{\boldsymbol{U}^{(k)}}(u)
+            + \left[1-\mu_{i+1,p-k-1}^{(k)}(u)\right]\, \mathcal{N}_{i+1,p-k-1}^{\boldsymbol{U}^{(k)}}(u)
+        }
+    }
+\end{document}
diff --git a/doc-images/equations/basis-prefactor.svg b/doc-images/equations/basis-prefactor.svg
new file mode 100644
index 0000000..cd2caf1
--- /dev/null
+++ b/doc-images/equations/basis-prefactor.svg
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diff --git a/doc-images/equations/basis-prefactor.tex b/doc-images/equations/basis-prefactor.tex
new file mode 100644
index 0000000..0b93230
--- /dev/null
+++ b/doc-images/equations/basis-prefactor.tex
@@ -0,0 +1,11 @@
+\input{_preamble}
+\begin{document}
+    \myeqs{
+        \mu_{g,h}^{(k)}(u)
+        =
+        \begin{cases}
+            0 & \text{if}\quad u_{g+h+1}^{(k)} = u_{g}^{(k)}\\[2mm]
+            \frac{u-u_g^{(k)}}{u_{g+h+1}^{(k)}-u_g^{(k)}} & \text{else}
+        \end{cases}
+    }
+\end{document}
diff --git a/doc-images/equations/control-points.svg b/doc-images/equations/control-points.svg
new file mode 100644
index 0000000..5843c4c
--- /dev/null
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diff --git a/doc-images/equations/control-points.tex b/doc-images/equations/control-points.tex
new file mode 100644
index 0000000..c95df8d
--- /dev/null
+++ b/doc-images/equations/control-points.tex
@@ -0,0 +1,13 @@
+\input{_preamble}
+\begin{document}
+    \myeqs{
+        \boldsymbol{P}_i^{(k)}=
+        \begin{cases}
+            \boldsymbol{P}_{i}^{(0)} & k=0 \\[2mm]
+            \left.\begin{cases}
+                      \boldsymbol{0} & u_{i+p+1}^{{(0)}} = u_{i+k}^{{(0)}}\\[2mm]
+                      \frac{p-k+1}{u_{i+p+1}^{{(0)}} - u_{i+k}^{{(0)}}} \left( \boldsymbol{P}_{i+1}^{(k-1)} - \boldsymbol{P}_{i}^{(k-1)} \right) & \text{else}%u_{i+p+1} \neq u_{i+k}\\[2mm]
+            \end{cases}\right|& k>0 \\
+        \end{cases}
+    }
+\end{document}
diff --git a/doc-images/equations/curve-deriv.tex b/doc-images/equations/curve-deriv.tex
deleted file mode 100644
index fc5e42f..0000000
--- a/doc-images/equations/curve-deriv.tex
+++ /dev/null
@@ -1,13 +0,0 @@
-\documentclass[10pt]{article}
-\usepackage[usenames]{color}
-\usepackage{amssymb}
-\usepackage{amsmath}
-\usepackage{nicefrac}
-\definecolor{mygreen}{rgb}{0.454,0.824,0.208}
-\definecolor{myred}{rgb}{0.8,0.173,0.137}
-
-\usepackage[utf8]{inputenc}
-\begin{equation}\nonumber
-\mathcal{C}^{(k)}(u) = \frac{\partial^k\mathcal{C}^{(0)}(u)}{\partial u^k} = \sum_{i=0}^{n-k} \mathcal{N}_{i,p-k}^{\boldsymbol{U^{(k)}}} (u)\, \boldsymbol{P}^{(k)}_i,\quad u \in [u_{p-k},u_{n+1-k}]
-\end{equation}
-\end{document}
diff --git a/doc-images/equations/curve-deriv.tex.svg b/doc-images/equations/curve-deriv.tex.svg
deleted file mode 100644
index 332b428..0000000
--- a/doc-images/equations/curve-deriv.tex.svg
+++ /dev/null
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diff --git a/doc-images/equations/curve.svg b/doc-images/equations/curve.svg
new file mode 100644
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diff --git a/doc-images/equations/curve.tex b/doc-images/equations/curve.tex
index 8153ff8..393502a 100644
--- a/doc-images/equations/curve.tex
+++ b/doc-images/equations/curve.tex
@@ -1,13 +1,8 @@
-\documentclass[10pt]{article}
-\usepackage[usenames]{color}
-\usepackage{amssymb}
-\usepackage{amsmath}
-\usepackage{nicefrac}
-\definecolor{mygreen}{rgb}{0.454,0.824,0.208}
-\definecolor{myred}{rgb}{0.8,0.173,0.137}
-
-\usepackage[utf8]{inputenc}
-\begin{equation}\nonumber
-\mathcal{C}(u) = \sum_{i=0}^{n} \mathcal{N}_{i,p}^{\boldsymbol{U}} (u)\, \boldsymbol{P}_i,\quad u \in [u_p,u_{n+1}]
-\end{equation}
+\input{_preamble}
+\begin{document}
+    \myeqs{
+        \mathcal{C}^{(k)}(u) = \frac{\partial^k\mathcal{C}^{(0)}(u)}{\partial u^k} = \sum_{i=0}^{n-k} \mathcal{N}_{i,p-k}^{\boldsymbol{U^{(k)}}} (u)\, \boldsymbol{P}^{(k)}_i,\quad
+        u \in [u_{p-k},u_{n+1-k}],\quad
+        n > p
+    }
 \end{document}
diff --git a/doc-images/equations/curve.tex.svg b/doc-images/equations/curve.tex.svg
deleted file mode 100644
index 613d504..0000000
--- a/doc-images/equations/curve.tex.svg
+++ /dev/null
@@ -1,152 +0,0 @@
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diff --git a/doc-images/equations/generate-equations-images.sh b/doc-images/equations/generate-equations-images.sh
new file mode 100755
index 0000000..e31f89c
--- /dev/null
+++ b/doc-images/equations/generate-equations-images.sh
@@ -0,0 +1,10 @@
+for file in *.tex; do
+    if [ "$file" != "_preamble.tex" ]; then
+        echo "Processing file: $file"
+        pdflatex -shell-escape -synctex=1 "$file" | grep '^!.*' -A200
+        pdf2svg "${file%.tex}.pdf" "${file%.tex}.svg"
+        rm -f "${file%.tex}.aux" "${file%.tex}.log" "${file%.tex}.pdf" "${file%.tex}.pdf" "${file%.tex}.synctex.gz"
+    else
+        echo "Excluded file: $file"
+    fi
+done
diff --git a/doc-images/equations/knots.svg b/doc-images/equations/knots.svg
new file mode 100644
index 0000000..57c7812
--- /dev/null
+++ b/doc-images/equations/knots.svg
@@ -0,0 +1,474 @@
+<?xml version="1.0" encoding="UTF-8"?>
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+<use xlink:href="#glyph-5-0" x="965.44" y="81.153"/>
+</g>
+<g fill="rgb(0%, 0%, 0%)" fill-opacity="1">
+<use xlink:href="#glyph-1-0" x="979.254" y="67.467"/>
+</g>
+<g fill="rgb(0%, 0%, 0%)" fill-opacity="1">
+<use xlink:href="#glyph-2-0" x="986.571" y="67.467"/>
+</g>
+<g fill="rgb(0%, 0%, 0%)" fill-opacity="1">
+<use xlink:href="#glyph-1-1" x="997.787" y="67.467"/>
+</g>
+<g fill="rgb(0%, 0%, 0%)" fill-opacity="1">
+<use xlink:href="#glyph-2-1" x="979.254" y="90.127"/>
+</g>
+<g fill="rgb(0%, 0%, 0%)" fill-opacity="1">
+<use xlink:href="#glyph-1-9" x="986.156" y="90.127"/>
+<use xlink:href="#glyph-1-10" x="1001.028868" y="90.127"/>
+</g>
+<g fill="rgb(0%, 0%, 0%)" fill-opacity="1">
+<use xlink:href="#glyph-5-1" x="1011.004" y="81.153"/>
+</g>
+</svg>
diff --git a/doc-images/equations/knots.tex b/doc-images/equations/knots.tex
new file mode 100644
index 0000000..11fa279
--- /dev/null
+++ b/doc-images/equations/knots.tex
@@ -0,0 +1,16 @@
+\input{_preamble}
+\begin{document}
+    \myeqs{
+        \boldsymbol{U}^{(k)}
+        =\left\{u_i^{(k)}\right\}
+        \equiv\overunderbraces{
+            &&\br{5}{\text{domain}\vphantom{p}}
+        }{
+            \left\{\vphantom{u^{(k)}}\right.
+            &u_{0}^{(k)},\dots,&u_{p-k}^{(k)}&,
+            &u_{p-k+1}^{(k)}, \dots,u_{n-k}^{(k)}
+            &,&u_{n+1-k}^{(k)}&,\dots,u_{n+p+1-2k}^{(k)}& \left.\vphantom{u^{(k)}}\right\}}
+        {&\br{2}{p-k+1}&&\br{1}{n-p}&&\br{2}{1+p-k}},\quad
+        u_{i}^{(k)}\leq u_{i+1}^{(k)},
+    }
+\end{document}
diff --git a/src/curve/basis/mod.rs b/src/curve/basis/mod.rs
new file mode 100644
index 0000000..1b8be03
--- /dev/null
+++ b/src/curve/basis/mod.rs
@@ -0,0 +1,191 @@
+#![cfg_attr(feature = "doc-images",
+cfg_attr(all(),
+doc = ::embed_doc_image::embed_image!("eq-basis-function", "doc-images/equations/basis-function.svg"),
+doc = ::embed_doc_image::embed_image!("eq-basis-prefactor", "doc-images/equations/basis-prefactor.svg"),
+doc = ::embed_doc_image::embed_image!("eq-basis-function-zero", "doc-images/equations/basis-function-zero.svg")))]
+//! Evaluates the basis spline functions using the Cox-de Boor-Mansfield recurrence relation
+//!
+//! ![The Cox-de Boor-Mansfield recurrence relation][eq-basis-function]
+//!
+//! with the basis functions of degree `p = 0`
+//!
+//! ![Basis function of degree zero][eq-basis-function-zero]
+//!
+//! where the conditional `⋁ (i = n - k ⋀ u = U_{n+1-k)` closes the last interval
+//! and the pre-factors
+//!
+//! ![Pre-factors][eq-basis-prefactor]
+
+use crate::types::VecD;
+
+/// Evaluates the `i`-th basis spline function of degree `p`
+///
+/// ## Arguments
+///
+/// - `i` the index with `i ∈ {0, 1, ..., n}`
+/// - `p` the spline degree
+/// - `k` the derivative order
+/// - `U` the knot vector
+pub fn basis(Uk: &VecD, i: usize, p: usize, k: usize, n: usize, u: f64) -> f64 {
+    if p == 0 {
+        if (Uk[i] <= u && u < Uk[i + 1]) || (i == n - k && u == Uk[n + 1 - k]) {
+            return 1.0;
+        }
+        return 0.0;
+    }
+
+    let summand1 = if Uk[i + p] == Uk[i] {
+        0.0
+    } else {
+        let g = i;
+        let h = p - 1;
+        (u - Uk[g]) / (Uk[g + h + 1] - Uk[g]) * basis(Uk, i, h, k, n, u)
+    };
+
+    let summand2 = if Uk[i + 1 + p] == Uk[i + 1] {
+        0.0
+    } else {
+        let g = i + 1;
+        let h = p - 1;
+
+        // The following equation is numerically more stable than
+        // `(1.0 - ((u - Uk[g]) / (Uk[g + h + 1] - Uk[g]))) * self.evaluate(k, g, h, u)`
+        (Uk[g + p] - u) / (Uk[g + h + 1] - Uk[g]) * basis(Uk, g, h, k, n, u)
+    };
+
+    summand1 + summand2
+}
+
+#[cfg(test)]
+mod tests {
+    use approx::assert_relative_eq;
+    use nalgebra::dvector;
+
+    use crate::curve::knots::Knots;
+
+    const SEGMENTS: usize = 4;
+
+    #[test]
+    fn basis_func_degree3() {
+        let k = 0;
+        let p = 3;
+        let knots = Knots::new(p, dvector![0., 0., 0., 0., 1. / 3., 2. / 3., 1., 1., 1., 1.]);
+
+        // Basis function i = 0
+        let mut i = 0;
+        assert_eq!(knots.evaluate(k, i, p, 0.0), 1.0);
+        assert_eq!(knots.evaluate(k, i, p, 1. / 6.), 1. / 8.);
+        assert_eq!(knots.evaluate(k, i, p, 1. / 3.), 0.0);
+        assert_eq!(knots.evaluate(k, i, p, 1. / 2.), 0.0);
+        assert_eq!(knots.evaluate(k, i, p, 2. / 3.), 0.0);
+        assert_eq!(knots.evaluate(k, i, p, 5. / 6.), 0.0);
+        assert_eq!(knots.evaluate(k, i, p, 1.), 0.0);
+
+        i = 1;
+        assert_eq!(knots.evaluate(k, i, p, 0.), 0.0);
+        assert_eq!(knots.evaluate(k, i, p, 1. / 6.), 19. / 32.);
+        assert_eq!(knots.evaluate(k, i, p, 1. / 3.), 1. / 4.);
+        assert_relative_eq!(knots.evaluate(k, i, p, 1. / 2.), 1. / 32., epsilon = f64::EPSILON.sqrt());
+        assert_eq!(knots.evaluate(k, i, p, 2. / 3.), 0.0);
+        assert_eq!(knots.evaluate(k, i, p, 5. / 6.), 0.0);
+        assert_eq!(knots.evaluate(k, i, p, 1.), 0.0);
+
+        i = 2;
+        assert_eq!(knots.evaluate(k, i, p, 0.), 0.0);
+        assert_eq!(knots.evaluate(k, i, p, 1. / 6.), 25. / 96.);
+        assert_eq!(knots.evaluate(k, i, p, 1. / 3.), 7. / 12.);
+        assert_relative_eq!(knots.evaluate(k, i, p, 1. / 2.), 15. / 32., epsilon = f64::EPSILON.sqrt());
+        assert_relative_eq!(knots.evaluate(k, i, p, 2. / 3.), 1. / 6., epsilon = f64::EPSILON.sqrt());
+        assert_relative_eq!(knots.evaluate(k, i, p, 5. / 6.), 1. / 48., epsilon = f64::EPSILON.sqrt());
+        assert_eq!(knots.evaluate(k, i, p, 1.0), 0.0);
+
+        i = 3;
+        assert_eq!(knots.evaluate(k, i, p, 0.), 0.0);
+        assert_eq!(knots.evaluate(k, i, p, 1. / 6.), 1. / 48.);
+        assert_eq!(knots.evaluate(k, i, p, 1. / 3.), 1. / 6.);
+        assert_relative_eq!(knots.evaluate(k, i, p, 1. / 2.), 15. / 32., epsilon = f64::EPSILON.sqrt());
+        assert_relative_eq!(knots.evaluate(k, i, p, 2. / 3.), 7. / 12., epsilon = f64::EPSILON.sqrt());
+        assert_relative_eq!(knots.evaluate(k, i, p, 5. / 6.), 25. / 96., epsilon = f64::EPSILON.sqrt());
+        assert_eq!(knots.evaluate(k, i, p, 1.0), 0.0);
+
+        i = 4;
+        assert_eq!(knots.evaluate(k, i, p, 0.), 0.0);
+        assert_eq!(knots.evaluate(k, i, p, 1. / 6.), 0.0);
+        assert_eq!(knots.evaluate(k, i, p, 1. / 3.), 0.0);
+        assert_relative_eq!(knots.evaluate(k, i, p, 1. / 2.), 1. / 32., epsilon = f64::EPSILON.sqrt());
+        assert_relative_eq!(knots.evaluate(k, i, p, 2. / 3.), 1. / 4., epsilon = f64::EPSILON.sqrt());
+        assert_relative_eq!(knots.evaluate(k, i, p, 5. / 6.), 19. / 32., epsilon = f64::EPSILON.sqrt());
+        assert_eq!(knots.evaluate(k, i, p, 1.0), 0.0);
+
+        i = 5;
+        assert_eq!(knots.evaluate(k, i, p, 0.0), 0.0);
+        assert_eq!(knots.evaluate(k, i, p, 1. / 6.), 0.);
+        assert_eq!(knots.evaluate(k, i, p, 1. / 3.), 0.0);
+        assert_eq!(knots.evaluate(k, i, p, 1. / 2.), 0.0);
+        assert_eq!(knots.evaluate(k, i, p, 2. / 3.), 0.0);
+        assert_relative_eq!(knots.evaluate(k, i, p, 5. / 6.), 1. / 8., epsilon = f64::EPSILON.sqrt());
+        assert_eq!(knots.evaluate(k, i, p, 1.), 1.0);
+    }
+
+    #[test]
+    fn basis_func_degree4() {
+        let k = 1;
+        let p = 4;
+        let knots = Knots::new(p, dvector![0., 0., 0., 0., 0., 1. / 3., 2. / 3., 1., 1., 1., 1., 1.]);
+
+        // Basis function i = 0
+        let mut i = 0;
+        assert_eq!(knots.evaluate(k, i, p, 0.0), 1.0);
+        assert_eq!(knots.evaluate(k, i, p, 1. / 6.), 1. / 8.);
+        assert_eq!(knots.evaluate(k, i, p, 1. / 3.), 0.0);
+        assert_eq!(knots.evaluate(k, i, p, 1. / 2.), 0.0);
+        assert_eq!(knots.evaluate(k, i, p, 2. / 3.), 0.0);
+        assert_eq!(knots.evaluate(k, i, p, 5. / 6.), 0.0);
+        assert_eq!(knots.evaluate(k, i, p, 1.), 0.0);
+
+        i = 1;
+        assert_eq!(knots.evaluate(k, i, p, 0.), 0.0);
+        assert_eq!(knots.evaluate(k, i, p, 1. / 6.), 19. / 32.);
+        assert_eq!(knots.evaluate(k, i, p, 1. / 3.), 1. / 4.);
+        assert_relative_eq!(knots.evaluate(k, i, p, 1. / 2.), 1. / 32., epsilon = f64::EPSILON.sqrt());
+        assert_eq!(knots.evaluate(k, i, p, 2. / 3.), 0.0);
+        assert_eq!(knots.evaluate(k, i, p, 5. / 6.), 0.0);
+        assert_eq!(knots.evaluate(k, i, p, 1.), 0.0);
+
+        i = 2;
+        assert_eq!(knots.evaluate(k, i, p, 0.), 0.0);
+        assert_eq!(knots.evaluate(k, i, p, 1. / 6.), 25. / 96.);
+        assert_eq!(knots.evaluate(k, i, p, 1. / 3.), 7. / 12.);
+        assert_relative_eq!(knots.evaluate(k, i, p, 1. / 2.), 15. / 32., epsilon = f64::EPSILON.sqrt());
+        assert_relative_eq!(knots.evaluate(k, i, p, 2. / 3.), 1. / 6., epsilon = f64::EPSILON.sqrt());
+        assert_relative_eq!(knots.evaluate(k, i, p, 5. / 6.), 1. / 48., epsilon = f64::EPSILON.sqrt());
+        assert_eq!(knots.evaluate(k, i, p, 1.0), 0.0);
+
+        i = 3;
+        assert_eq!(knots.evaluate(k, i, p, 0.), 0.0);
+        assert_eq!(knots.evaluate(k, i, p, 1. / 6.), 1. / 48.);
+        assert_eq!(knots.evaluate(k, i, p, 1. / 3.), 1. / 6.);
+        assert_relative_eq!(knots.evaluate(k, i, p, 1. / 2.), 15. / 32., epsilon = f64::EPSILON.sqrt());
+        assert_relative_eq!(knots.evaluate(k, i, p, 2. / 3.), 7. / 12., epsilon = f64::EPSILON.sqrt());
+        assert_relative_eq!(knots.evaluate(k, i, p, 5. / 6.), 25. / 96., epsilon = f64::EPSILON.sqrt());
+        assert_eq!(knots.evaluate(k, i, p, 1.0), 0.0);
+
+        i = 4;
+        assert_eq!(knots.evaluate(k, i, p, 0.), 0.0);
+        assert_eq!(knots.evaluate(k, i, p, 1. / 6.), 0.0);
+        assert_eq!(knots.evaluate(k, i, p, 1. / 3.), 0.0);
+        assert_relative_eq!(knots.evaluate(k, i, p, 1. / 2.), 1. / 32., epsilon = f64::EPSILON.sqrt());
+        assert_relative_eq!(knots.evaluate(k, i, p, 2. / 3.), 1. / 4., epsilon = f64::EPSILON.sqrt());
+        assert_relative_eq!(knots.evaluate(k, i, p, 5. / 6.), 19. / 32., epsilon = f64::EPSILON.sqrt());
+        assert_eq!(knots.evaluate(k, i, p, 1.0), 0.0);
+
+        i = 5;
+        assert_eq!(knots.evaluate(k, i, p, 0.0), 0.0);
+        assert_eq!(knots.evaluate(k, i, p, 1. / 6.), 0.);
+        assert_eq!(knots.evaluate(k, i, p, 1. / 3.), 0.0);
+        assert_eq!(knots.evaluate(k, i, p, 1. / 2.), 0.0);
+        assert_eq!(knots.evaluate(k, i, p, 2. / 3.), 0.0);
+        assert_relative_eq!(knots.evaluate(k, i, p, 5. / 6.), 1. / 8., epsilon = f64::EPSILON.sqrt());
+        assert_eq!(knots.evaluate(k, i, p, 1.), 1.0);
+    }
+}
diff --git a/src/curve/knots/mod.rs b/src/curve/knots/mod.rs
index 3c153a8..03abdbd 100644
--- a/src/curve/knots/mod.rs
+++ b/src/curve/knots/mod.rs
@@ -1,13 +1,24 @@
-//! Knot vectors can be generated with different methods:
+#![cfg_attr(feature = "doc-images",
+cfg_attr(all(),
+doc = ::embed_doc_image::embed_image!("eq-knots", "doc-images/equations/knots.svg")))]
+//! Implements the knot vector defining the [spline basis functions][basis].
 //!
-//! - Uniform knots
-//! - Knot-averaging
-//! - De-Boor's method
+//! The knot vector parametrizing the `k`-th degree curve is composed of `n+p+2 - 2k` scalar values
+//! in ascending order, called 'knots'.
+//!
+//! ![The knot vector][eq-knots]
+//!
+//! The head and tail contains of `p-k+1` knots of value `0` and `1`, respectively.
+//! This leaves `n-p` internal knots in the center.
+//! The interval from index `i = p-k,..., n+1-k` is called 'domain'.
+//!
+//! Different [knot vector generation methods][methods] are available.
 
 use std::ops::MulAssign;
 
 use thiserror::Error;
 
+use crate::curve::basis;
 use crate::{
     curve::{parameters, parameters::Parameters, CurveError},
     types::{KnotVectorDerivatives, VecD, VecDView, VecHelpers},
@@ -64,7 +75,7 @@ impl Knots {
         let mut Uk: Vec<VecD> = Vec::with_capacity(degree + 1);
         Uk.push(knots);
 
-        // TODO Add multipliity check
+        // TODO Add multiplicity check
 
         let mut knots = Knots { Uk, p: degree, k_max: 0 };
         knots.derive();
@@ -168,7 +179,6 @@ impl Knots {
         for k in 1..=p {
             // obtain the `k`-th derivative knot vector from the previous `k-1`-th derivative knot vector segment
             // by dropping the first and last segment
-
             let segment_of_previous_order_knot_vector = self.Uk[k - 1].segment(1, self.len(k - 1) - 2).clone_owned();
 
             self.Uk.push(segment_of_previous_order_knot_vector);
@@ -261,43 +271,10 @@ impl Knots {
         let n = self.segments();
         let pk = p - k;
 
-        evaluate(i, pk, n, Uk, u, k)
+        basis::basis(Uk, i, pk, k, n, u)
     }
 }
 
-pub(crate) fn evaluate(i: usize, pk: usize, n: usize, Uk: &VecD, u: f64, k: usize) -> f64 {
-    // Err(ParameterOutOfBounds)
-
-    if pk == 0 {
-        // the conditional (i == n - k) &&  (u == Uk(n+1)) closes the last interval [u_n,u_n+1]
-        if (Uk[i] <= u && u < Uk[i + 1]) || (i == n - k && u == Uk[n + 1]) {
-            return 1.0;
-        }
-        return 0.0;
-    }
-
-    let summand1 = if Uk[i + pk] == Uk[i] {
-        0.0
-    } else {
-        let g = i;
-        let h = pk - 1;
-        (u - Uk[g]) / (Uk[g + h + 1] - Uk[g]) * evaluate(i, h, n, Uk, u, k) /* self.evaluate(k, i, h, u) */
-    };
-
-    let summand2 = if Uk[i + 1 + pk] == Uk[i + 1] {
-        0.0
-    } else {
-        let g = i + 1;
-        let h = pk - 1;
-
-        // The following equation is numerically more stable than
-        //(1.0 - ((u - Uk[g]) / (Uk[g + h + 1] - Uk[g]))) * self.evaluate(k, g, h, u)
-        (Uk[g + pk] - u) / (Uk[g + h + 1] - Uk[g]) * evaluate(g, h, n, Uk, u, k)
-    };
-
-    summand1 + summand2
-}
-
 fn is_valid(knots: Knots) -> bool {
     knots.segments() == knots.len(0) - (knots.degree() + 2)
 }
@@ -399,7 +376,6 @@ fn rescale(knots: &mut VecD, old_lim: (f64, f64), new_lim: (f64, f64)) {
 
 #[cfg(test)]
 mod tests {
-    use approx::assert_relative_eq;
     use nalgebra::dvector;
     use rstest::rstest;
 
@@ -461,130 +437,6 @@ mod tests {
         assert_eq!(knots_example(3).domain(), dvector![0.0, 0.5, 1.0]);
     }
 
-    #[test]
-    fn basis_func_degree3() {
-        let k = 0;
-        let p = 3;
-        let knots = Knots::new(p, dvector![0., 0., 0., 0., 1. / 3., 2. / 3., 1., 1., 1., 1.]);
-
-        // Basis function i = 0
-        let mut i = 0;
-        assert_eq!(knots.evaluate(k, i, p, 0.0), 1.0);
-        assert_eq!(knots.evaluate(k, i, p, 1. / 6.), 1. / 8.);
-        assert_eq!(knots.evaluate(k, i, p, 1. / 3.), 0.0);
-        assert_eq!(knots.evaluate(k, i, p, 1. / 2.), 0.0);
-        assert_eq!(knots.evaluate(k, i, p, 2. / 3.), 0.0);
-        assert_eq!(knots.evaluate(k, i, p, 5. / 6.), 0.0);
-        assert_eq!(knots.evaluate(k, i, p, 1.), 0.0);
-
-        i = 1;
-        assert_eq!(knots.evaluate(k, i, p, 0.), 0.0);
-        assert_eq!(knots.evaluate(k, i, p, 1. / 6.), 19. / 32.);
-        assert_eq!(knots.evaluate(k, i, p, 1. / 3.), 1. / 4.);
-        assert_relative_eq!(knots.evaluate(k, i, p, 1. / 2.), 1. / 32., epsilon = f64::EPSILON.sqrt());
-        assert_eq!(knots.evaluate(k, i, p, 2. / 3.), 0.0);
-        assert_eq!(knots.evaluate(k, i, p, 5. / 6.), 0.0);
-        assert_eq!(knots.evaluate(k, i, p, 1.), 0.0);
-
-        i = 2;
-        assert_eq!(knots.evaluate(k, i, p, 0.), 0.0);
-        assert_eq!(knots.evaluate(k, i, p, 1. / 6.), 25. / 96.);
-        assert_eq!(knots.evaluate(k, i, p, 1. / 3.), 7. / 12.);
-        assert_relative_eq!(knots.evaluate(k, i, p, 1. / 2.), 15. / 32., epsilon = f64::EPSILON.sqrt());
-        assert_relative_eq!(knots.evaluate(k, i, p, 2. / 3.), 1. / 6., epsilon = f64::EPSILON.sqrt());
-        assert_relative_eq!(knots.evaluate(k, i, p, 5. / 6.), 1. / 48., epsilon = f64::EPSILON.sqrt());
-        assert_eq!(knots.evaluate(k, i, p, 1.0), 0.0);
-
-        i = 3;
-        assert_eq!(knots.evaluate(k, i, p, 0.), 0.0);
-        assert_eq!(knots.evaluate(k, i, p, 1. / 6.), 1. / 48.);
-        assert_eq!(knots.evaluate(k, i, p, 1. / 3.), 1. / 6.);
-        assert_relative_eq!(knots.evaluate(k, i, p, 1. / 2.), 15. / 32., epsilon = f64::EPSILON.sqrt());
-        assert_relative_eq!(knots.evaluate(k, i, p, 2. / 3.), 7. / 12., epsilon = f64::EPSILON.sqrt());
-        assert_relative_eq!(knots.evaluate(k, i, p, 5. / 6.), 25. / 96., epsilon = f64::EPSILON.sqrt());
-        assert_eq!(knots.evaluate(k, i, p, 1.0), 0.0);
-
-        i = 4;
-        assert_eq!(knots.evaluate(k, i, p, 0.), 0.0);
-        assert_eq!(knots.evaluate(k, i, p, 1. / 6.), 0.0);
-        assert_eq!(knots.evaluate(k, i, p, 1. / 3.), 0.0);
-        assert_relative_eq!(knots.evaluate(k, i, p, 1. / 2.), 1. / 32., epsilon = f64::EPSILON.sqrt());
-        assert_relative_eq!(knots.evaluate(k, i, p, 2. / 3.), 1. / 4., epsilon = f64::EPSILON.sqrt());
-        assert_relative_eq!(knots.evaluate(k, i, p, 5. / 6.), 19. / 32., epsilon = f64::EPSILON.sqrt());
-        assert_eq!(knots.evaluate(k, i, p, 1.0), 0.0);
-
-        i = 5;
-        assert_eq!(knots.evaluate(k, i, p, 0.0), 0.0);
-        assert_eq!(knots.evaluate(k, i, p, 1. / 6.), 0.);
-        assert_eq!(knots.evaluate(k, i, p, 1. / 3.), 0.0);
-        assert_eq!(knots.evaluate(k, i, p, 1. / 2.), 0.0);
-        assert_eq!(knots.evaluate(k, i, p, 2. / 3.), 0.0);
-        assert_relative_eq!(knots.evaluate(k, i, p, 5. / 6.), 1. / 8., epsilon = f64::EPSILON.sqrt());
-        assert_eq!(knots.evaluate(k, i, p, 1.), 1.0);
-    }
-
-    #[test]
-    fn basis_func_degree4() {
-        let k = 1;
-        let p = 4;
-        let knots = Knots::new(p, dvector![0., 0., 0., 0., 0., 1. / 3., 2. / 3., 1., 1., 1., 1., 1.]);
-
-        // Basis function i = 0
-        let mut i = 0;
-        assert_eq!(knots.evaluate(k, i, p, 0.0), 1.0);
-        assert_eq!(knots.evaluate(k, i, p, 1. / 6.), 1. / 8.);
-        assert_eq!(knots.evaluate(k, i, p, 1. / 3.), 0.0);
-        assert_eq!(knots.evaluate(k, i, p, 1. / 2.), 0.0);
-        assert_eq!(knots.evaluate(k, i, p, 2. / 3.), 0.0);
-        assert_eq!(knots.evaluate(k, i, p, 5. / 6.), 0.0);
-        assert_eq!(knots.evaluate(k, i, p, 1.), 0.0);
-
-        i = 1;
-        assert_eq!(knots.evaluate(k, i, p, 0.), 0.0);
-        assert_eq!(knots.evaluate(k, i, p, 1. / 6.), 19. / 32.);
-        assert_eq!(knots.evaluate(k, i, p, 1. / 3.), 1. / 4.);
-        assert_relative_eq!(knots.evaluate(k, i, p, 1. / 2.), 1. / 32., epsilon = f64::EPSILON.sqrt());
-        assert_eq!(knots.evaluate(k, i, p, 2. / 3.), 0.0);
-        assert_eq!(knots.evaluate(k, i, p, 5. / 6.), 0.0);
-        assert_eq!(knots.evaluate(k, i, p, 1.), 0.0);
-
-        i = 2;
-        assert_eq!(knots.evaluate(k, i, p, 0.), 0.0);
-        assert_eq!(knots.evaluate(k, i, p, 1. / 6.), 25. / 96.);
-        assert_eq!(knots.evaluate(k, i, p, 1. / 3.), 7. / 12.);
-        assert_relative_eq!(knots.evaluate(k, i, p, 1. / 2.), 15. / 32., epsilon = f64::EPSILON.sqrt());
-        assert_relative_eq!(knots.evaluate(k, i, p, 2. / 3.), 1. / 6., epsilon = f64::EPSILON.sqrt());
-        assert_relative_eq!(knots.evaluate(k, i, p, 5. / 6.), 1. / 48., epsilon = f64::EPSILON.sqrt());
-        assert_eq!(knots.evaluate(k, i, p, 1.0), 0.0);
-
-        i = 3;
-        assert_eq!(knots.evaluate(k, i, p, 0.), 0.0);
-        assert_eq!(knots.evaluate(k, i, p, 1. / 6.), 1. / 48.);
-        assert_eq!(knots.evaluate(k, i, p, 1. / 3.), 1. / 6.);
-        assert_relative_eq!(knots.evaluate(k, i, p, 1. / 2.), 15. / 32., epsilon = f64::EPSILON.sqrt());
-        assert_relative_eq!(knots.evaluate(k, i, p, 2. / 3.), 7. / 12., epsilon = f64::EPSILON.sqrt());
-        assert_relative_eq!(knots.evaluate(k, i, p, 5. / 6.), 25. / 96., epsilon = f64::EPSILON.sqrt());
-        assert_eq!(knots.evaluate(k, i, p, 1.0), 0.0);
-
-        i = 4;
-        assert_eq!(knots.evaluate(k, i, p, 0.), 0.0);
-        assert_eq!(knots.evaluate(k, i, p, 1. / 6.), 0.0);
-        assert_eq!(knots.evaluate(k, i, p, 1. / 3.), 0.0);
-        assert_relative_eq!(knots.evaluate(k, i, p, 1. / 2.), 1. / 32., epsilon = f64::EPSILON.sqrt());
-        assert_relative_eq!(knots.evaluate(k, i, p, 2. / 3.), 1. / 4., epsilon = f64::EPSILON.sqrt());
-        assert_relative_eq!(knots.evaluate(k, i, p, 5. / 6.), 19. / 32., epsilon = f64::EPSILON.sqrt());
-        assert_eq!(knots.evaluate(k, i, p, 1.0), 0.0);
-
-        i = 5;
-        assert_eq!(knots.evaluate(k, i, p, 0.0), 0.0);
-        assert_eq!(knots.evaluate(k, i, p, 1. / 6.), 0.);
-        assert_eq!(knots.evaluate(k, i, p, 1. / 3.), 0.0);
-        assert_eq!(knots.evaluate(k, i, p, 1. / 2.), 0.0);
-        assert_eq!(knots.evaluate(k, i, p, 2. / 3.), 0.0);
-        assert_relative_eq!(knots.evaluate(k, i, p, 5. / 6.), 1. / 8., epsilon = f64::EPSILON.sqrt());
-        assert_eq!(knots.evaluate(k, i, p, 1.), 1.0);
-    }
-
     #[test]
     fn multiplicity() {
         let knots = Knots::new(2, dvector![0., 0., 0., 0.25, 0.5, 0.5, 0.75, 1., 1., 1.]);
diff --git a/src/curve/mod.rs b/src/curve/mod.rs
index badd1b4..8f4bd9b 100644
--- a/src/curve/mod.rs
+++ b/src/curve/mod.rs
@@ -1,3 +1,23 @@
+//#![warn(missing_docs)]
+//#![warn(missing_doc_code_examples)]
+#![cfg_attr(feature = "doc-images",
+cfg_attr(all(),
+doc = ::embed_doc_image::embed_image!("eq-curve", "doc-images/equations/curve.svg")))]
+//! Implements the B-spline curve.
+//!
+//! A B-spline curve can be defined by
+//!
+//! ![B-spline curve][eq-curve]
+//!
+//! with the
+//! - parameter `u ∈ [0,1]` defining a point on the curve,
+//! - derivative order `k`,
+//! - number of polynomial spline segments `n`,
+//! - spline degree `p`,
+//! - `k`-th derivative [knot vector][knots] `U`,
+//! - `n+1-k` [spline basis function][basis] `N` of degree `p` defined by the [knot vector][knots] `U`, and
+//! - `n+1-k`, `N`-dimensional [control points][points] `P`.
+
 use embed_doc_image::embed_doc_image;
 use thiserror::Error;
 
@@ -14,6 +34,7 @@ use crate::{
     types::VecD,
 };
 
+pub mod basis;
 pub mod generation;
 pub mod knots;
 pub mod parameters;
diff --git a/src/curve/parameters/mod.rs b/src/curve/parameters/mod.rs
index 8bae19d..fb795c1 100644
--- a/src/curve/parameters/mod.rs
+++ b/src/curve/parameters/mod.rs
@@ -1,4 +1,4 @@
-//! Curve parameters can be generated with different methods:
+//! Implements different parameter generation methods.
 //!
 //! - Equally spaced parameters
 //! - Centripetal method
diff --git a/src/curve/points/mod.rs b/src/curve/points/mod.rs
index 1995f67..dfcacb5 100644
--- a/src/curve/points/mod.rs
+++ b/src/curve/points/mod.rs
@@ -1,3 +1,16 @@
+#![cfg_attr(feature = "doc-images",
+cfg_attr(all(),
+doc = ::embed_doc_image::embed_image!("eq-control-points", "doc-images/equations/control-points.svg")))]
+//! Implements the control points constituting the control polygon of the curve.
+//!
+//! The control points of the `k`-th derivative B-spline curve can be derived
+//! from the zeroth order control points
+//!
+//! ![The control points][eq-control-points]
+//!
+//! Control points are generated and manipulated as part of different curve generation
+//! and curve manipulation methods.
+
 use std::ops::MulAssign;
 
 use crate::{
@@ -142,8 +155,8 @@ impl ControlPoints {
             return VecD::zeros(self.dimension());
         }
 
-        (p - k_max + 1) as f64 / (U0[i + p + 1] - U0[i + k_max]) *
-            (self.derive_single_point(i + 1, k_max - 1, knots) - self.derive_single_point(i, k_max - 1, knots))
+        (p - k_max + 1) as f64 / (U0[i + p + 1] - U0[i + k_max])
+            * (self.derive_single_point(i + 1, k_max - 1, knots) - self.derive_single_point(i, k_max - 1, knots))
     }
 
     pub fn reverse(&mut self) -> &mut Self {
diff --git a/src/lib.rs b/src/lib.rs
index be21eec..eca474d 100644
--- a/src/lib.rs
+++ b/src/lib.rs
@@ -2,9 +2,11 @@
 //#![warn(missing_doc_code_examples)]
 #![cfg_attr(feature = "doc-images",
 cfg_attr(all(),
-doc = ::embed_doc_image::embed_image!("eq-curve", "doc-images/equations/curve-deriv.tex.svg"),
+doc = ::embed_doc_image::embed_image!("eq-curve", "doc-images/equations/curve.svg"),
+doc = ::embed_doc_image::embed_image!("eq-knots", "doc-images/equations/knots.svg"),
+doc = ::embed_doc_image::embed_image!("eq-control-points", "doc-images/equations/control-points.svg"),
 doc = ::embed_doc_image::embed_image!("img-curve", "doc-images/plots/manipulation/insert-before.svg")))]
-//! `bsplines` is a Rust library for vectorized, N-dimensional B-spline curves and their derivatives based on
+//! **bsplines** is a library for vectorized, N-dimensional B-spline curves and their derivatives based on
 //! [nalgebra].
 //!
 //! ## Features
@@ -21,7 +23,7 @@ doc = ::embed_doc_image::embed_image!("img-curve", "doc-images/plots/manipulatio
 //!     - [splitting][manipulation::split]
 //!     - [merging][manipulation::split]
 //!
-//! ## Mathematical Definition
+//! ## What are B-Splines?
 //!
 //! B-splines are parametric functions composed of piecewise polynomials with a polynomial degree `p > 0`.
 //! These piecewise polynomials are joined so that the parametric function is `p-1` times continuously
@@ -33,19 +35,6 @@ doc = ::embed_doc_image::embed_image!("img-curve", "doc-images/plots/manipulatio
 //! complex-shaped and high-dimensional data, while maintaining a low polynomial degree. Because of the polynomial
 //! nature, all possible derivatives are accessible.
 //!
-//! A B-spline curve `C` can be defined by
-//!
-//! ![The mathematical definition of a B-spline curve.][eq-curve]
-//!
-//! with
-//! the derivative order `k`,
-//! the parameter `u ∈ [0,1]` defining a point on the curve,
-//! the number of polynomial spline segments `n`,
-//! the spline degree `p`,
-//! the `k`-th derivative knot vector `U^(k)`,
-//! the `n+1` spline basis function `N_{i,p}` of degree `p` defined by the knot vector `U^(k)`, and
-//! the `n+1` control points `P_i` that can be of arbitrary dimension.
-//!
 //! ![A 2D B-Spline curve.][img-curve]
 //!
 //! Still, evaluations or spatial manipulations can be executed fast because only local polynomial segments must be
diff --git a/src/manipulation/merge.rs b/src/manipulation/merge.rs
index 3a6979d..575038b 100644
--- a/src/manipulation/merge.rs
+++ b/src/manipulation/merge.rs
@@ -21,10 +21,11 @@ use std::ops::{AddAssign, DivAssign, SubAssign};
 use nalgebra::SVD;
 use thiserror::Error;
 
+use crate::curve::basis::basis;
 use crate::{
     curve,
     curve::{
-        knots::{evaluate, is_clamped, is_normed, reversed, Knots},
+        knots::{is_clamped, is_normed, reversed, Knots},
         points::{ControlPoints, Points},
         Curve, CurveError,
     },
@@ -185,7 +186,7 @@ fn calculateKV(a: &Curve) -> MatD {
             let mut sum = 0.;
 
             for a in m - p..=m - k {
-                sum += prefactor(p, a, i, k, S, &Vk[0]) * evaluate(a, p - k, m - k, &Vk[k], Vk[0][m + 1], 0);
+                sum += prefactor(p, a, i, k, S, &Vk[0]) * basis(&Vk[k], a, p - k, 0, m - k, Vk[0][m + 1]);
             }
             KV[(k, i - (m + 1 - p))] = sum;
         }
@@ -206,7 +207,7 @@ fn calculateKW(b: &Curve) -> MatD {
             let mut sum = 0.;
 
             for b in 0..=p - k {
-                sum += prefactor(p, b, j, k, T, &Wk[0]) * evaluate(b, p - k, o - k, &Wk[k], Wk[0][p], 0);
+                sum += prefactor(p, b, j, k, T, &Wk[0]) * basis(&Wk[k], b, p - k, 0, o - k, Wk[0][p]);
             }
             KW[(k, j)] = sum;
         }
@@ -229,7 +230,7 @@ fn calculateIV(a: &Curve) -> MatD {
             let mut sum = 0.;
 
             for a in m - p..=m - k {
-                sum += prefactor(p, a, i, k, S, &Vk[0]) * evaluate(a, p - k, m - k, &Vk[k], Vk[0][m + 1], 0);
+                sum += prefactor(p, a, i, k, S, &Vk[0]) * basis(&Vk[k], a, p - k, 0, m - k, Vk[0][m + 1]);
             }
             IV[(i - (m + 1 - p), k)] = sum;
         }
@@ -252,7 +253,7 @@ fn calculateJW(b: &Curve) -> MatD {
             let mut sum = 0.;
 
             for b in 0..=p - k {
-                sum += prefactor(p, b, j, k, T, &Wk[0]) * evaluate(b, p - k, o - k, &Wk[k], Wk[0][p], 0);
+                sum += prefactor(p, b, j, k, T, &Wk[0]) * basis(&Wk[k], b, p - k, 0, o - k, Wk[0][p]);
             }
             JW[(j, k)] = sum;
         }
@@ -273,7 +274,7 @@ fn calculateGV(a: &ConstrainedCurve) -> MatD {
 
     for g in 0..=mg {
         for i in m - p + 1..=m {
-            GV[(g, i - (m - p + 1))] = evaluate(i, p, m, Vk0, a.constraints.params[g], 0);
+            GV[(g, i - (m - p + 1))] = basis(Vk0, i, p, 0, m, a.constraints.params[g]);
         }
     }
     GV
@@ -289,7 +290,7 @@ fn calculateHW(b: &ConstrainedCurve) -> MatD {
 
     for h in 0..=oh {
         for i in 0..=p - 1 {
-            HW[(h, i)] = evaluate(i, p, o, Wk0, b.constraints.params[h], 0);
+            HW[(h, i)] = basis(Wk0, i, p, 0, o, b.constraints.params[h]);
         }
     }
 
@@ -315,7 +316,7 @@ fn calculateIpV(a: &ConstrainedCurve) -> MatD {
 
     for i in m - p + 1..=m {
         for g in 0..=mg {
-            IpV[(i - (m + 1 - p), g)] = evaluate(i, p, m, Vk0, a.constraints.params[g], 0);
+            IpV[(i - (m + 1 - p), g)] = basis(Vk0, i, p, 0, m, a.constraints.params[g]);
         }
     }
     IpV *= -0.5;
@@ -333,7 +334,7 @@ fn calculateJppW(b: &ConstrainedCurve) -> MatD {
 
     for j in 0..=p_ - 1 {
         for h in 0..=oh_ {
-            JppW[(j, h)] = evaluate(j, p_, o_, Wk0, b.constraints.params[h], 0);
+            JppW[(j, h)] = basis(Wk0, j, p_, 0, o_, b.constraints.params[h]);
         }
     }
     JppW *= -0.5;
@@ -362,8 +363,7 @@ fn calculateKconst(a: &Curve, b: &Curve) -> MatD {
         for i in m - p..=m {
             for a in m - p..=m - k {
                 // TODO use point getter instead of .column(i)
-                sum +=
-                    prefactor(p, a, i, k, S, &Vk[0]) * evaluate(a, p - k, m - k, &Vk[k], Vk[0][m + 1], 0) * S.column(i);
+                sum += prefactor(p, a, i, k, S, &Vk[0]) * basis(&Vk[k], a, p - k, 0, m - k, Vk[0][m + 1]) * S.column(i);
             }
         }
 
@@ -371,7 +371,7 @@ fn calculateKconst(a: &Curve, b: &Curve) -> MatD {
             for b in 0..=p - k {
                 //sumB += -prefactor(p, b, j, k, &T, &Wk[0]) * evaluate(b, p - k, o - k, &Wk[k], Wk[0][p]) * T.row(j);
                 // TODO use point getter instead of .column(j)
-                sum -= prefactor(p, b, j, k, T, &Wk[0]) * evaluate(b, p - k, o - k, &Wk[k], Wk[0][p], 0) * T.column(j);
+                sum -= prefactor(p, b, j, k, T, &Wk[0]) * basis(&Wk[k], b, p - k, 0, o - k, Wk[0][p]) * T.column(j);
             }
         }
         Kconst.column_mut(k).sub_assign(&sum);
@@ -581,8 +581,8 @@ fn prefactor(p: usize, i: usize, i0: usize, k: usize, P0: &MatD, U0: &VecD) -> f
         } else if U0[i + p + 1] == U0[i + k] {
             0.
         } else {
-            (p + 1 - k) as f64 / (U0[i + p + 1] - U0[i + k]) *
-                (prefactor(p, i + 1, i0, k - 1, P0, U0) - prefactor(p, i, i0, k - 1, P0, U0))
+            (p + 1 - k) as f64 / (U0[i + p + 1] - U0[i + k])
+                * (prefactor(p, i + 1, i0, k - 1, P0, U0) - prefactor(p, i, i0, k - 1, P0, U0))
         }
     } else {
         0.
diff --git a/src/manipulation/mod.rs b/src/manipulation/mod.rs
index 9bac688..91ba75a 100644
--- a/src/manipulation/mod.rs
+++ b/src/manipulation/mod.rs
@@ -1,3 +1,5 @@
+//! Implements different curve manipulation methods.
+
 pub mod insert;
 pub mod merge;
 pub mod reverse;

From 0f7302bb3df8a8b8d15e7ca2fb98b403a3a8675f Mon Sep 17 00:00:00 2001
From: Michael Heuer <michael.heuer@rwth-aachen.de>
Date: Thu, 8 Feb 2024 22:47:57 +0100
Subject: [PATCH 2/2] chore: bump version

---
 Cargo.toml | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/Cargo.toml b/Cargo.toml
index 9097d0c..b173722 100644
--- a/Cargo.toml
+++ b/Cargo.toml
@@ -1,7 +1,7 @@
 [package]
 name = "bsplines"
 license = "Apache-2.0"
-version = "0.0.1-alpha.5"
+version = "0.0.1-alpha.6"
 authors = [
     "Michael A. Heuer <decentralised.systems@gmail.com>",
 ]