alles afgemaakt

3 files changed, 96 insertions, 34 deletions

diff --git a/circrect.py b/circrect.py
index e85ed77..a1dd27c 100644
--- a/circrect.py
+++ b/circrect.py
@@ -1,40 +1,43 @@
 from numbers import Number
+from math import sqrt, asin
 
 def clamp(low: Number, high: Number, num: Number) -> Number:
     """Limit range of ``num`` between ``low`` and ``high``"""
     return max(low, min(high, num))
 
-def circrect(a: Number, b: tuple[Number, Number], c: tuple[Number, Number]) -> Number:
+def circrect(a: tuple[Number, Number], b: tuple[Number, Number], c: Number = 1) -> Number:
     """
-    Calculate the area of the overlap between a circle with radius ``a`` and a
-    rectangle defined by opposing corner points ``b`` and ``c``.
+    Calculate the area of the overlap between a rectangle defined by opposing
+    corner points ``a`` and ``b`` and a circle with radius ``c``.
 
     Args:
-        a: circle radius
-        b: first corner coordinate of rectangle
-        c: second corner coordinate of rectangle (opposite of b)
+        a: first corner coordinate of rectangle
+        b: second corner coordinate of rectangle (opposite of b)
+        c: circle radius (optional, with default of 1)
 
     Returns:
         Overlap area (square units)
     """
-
-    # normalize input variables
-    a = abs(a)
-    b, c = (
-      (min(b[0], c[0]), min(b[1], c[1])),
-      (max(b[0], c[0]), max(b[1], c[1])),
-    )
-    b = (clamp(-a, a, b[0]), clamp(-a, a, b[1]))
-    c = (clamp(-a, a, c[0]), clamp(-a, a, c[1]))
-
-    return 0
-
-if __name__ == "__main__":
-    print(
-        circrect(1, (-1, -1), (1, 1)), # pi
-        circrect(1, (1, 1), (-1, -1)), # pi
-        circrect(-1, (-1, -1), (1, 1)), # pi
-        circrect(2, (10, 10), (11, 11)), # 0
-        circrect(10, (0, 0), (1, 1)), # 1
+    # input variable normalization
+    a, b = (
+      (min(a[0], b[0]), min(a[1], b[1])),
+      (max(a[0], b[0]), max(a[1], b[1])),
     )
+    c = abs(c)
+    a = (clamp(-1, 1, a[0] / c), clamp(-1, 1, a[1] / c))
+    b = (clamp(-1, 1, b[0] / c), clamp(-1, 1, b[1] / c))
+
+    # can not have shared area if ab already has no area
+    if a == b: return 0.0
+
+    f = lambda x: sqrt(1 - x**2)
+    F = lambda x, n: 0.5 * (2*n*x + sqrt(1 - x**2)*x + asin(x))
+    g = lambda a, b: F(clamp(a[0], b[0],  f(a[1])), -a[1]) \
+                   - F(clamp(a[0], b[0], -f(a[1])), -a[1]) \
+                   - F(clamp(a[0], b[0],  f(b[1])), -b[1]) \
+                   + F(clamp(a[0], b[0], -f(b[1])), -b[1])
+    h = lambda a, b: g((a[0], max(0,  a[1])), (b[0], max(0,  b[1]))) \
+                   + g((a[0], max(0, -b[1])), (b[0], max(0, -a[1])))
+    o = h(a, b) * (c ** 2)
+    return o
 
diff --git a/circrect.tex b/circrect.tex
index 8411fe8..d64e32e 100644
--- a/circrect.tex
+++ b/circrect.tex
@@ -13,6 +13,7 @@
 	bibencoding=utf8,
 ]{biblatex}
 \usepackage{tikz}
+\usepackage{needspace}
 
 \usetikzlibrary{patterns}
 \lstset{
@@ -233,25 +234,64 @@ F\left(x,n\right)&=\int\left(f\left(x\right) + n\right) dx\\
 	\end{aligned}
 \end{equation}
 
-Nu kan een complete functie $g$ beschreven worden:
-\begin{equation}\label{eq:fn:g}
-g\left(a,b\right) = \text{TODO}
-\end{equation}
+\needspace{3cm}
+Nu kan een nieuwe functie $g$ beschreven worden die alleen afhankelijk is van
+ingevoerde variabelen:
+\begin{align}
+\begin{split}
+o ={}& o_a - o_b
+\end{split}\\
+\begin{split}
+={} &\int_{x_{a_1}}^{x_{a_2}}f_a\left(x\right)dx\\
+  - &\int_{x_{b_1}}^{x_{b_2}}f_b\left(x\right)dx
+\end{split}\\
+\begin{split}
+={} &\int_{x_{a_1}}^{x_{a_2}}\left(f\left(x\right)-a_y\right)dx\\
+  - &\int_{x_{b_1}}^{x_{b_2}}\left(f\left(x\right)-b_y\right)dx
+\end{split}\\
+\begin{split}
+={} &\left(F\left(x_{a_2}, -a_y\right) - F\left(x_{a_1}, -a_y\right)\right)\\
+  - &\left(F\left(x_{b_2}, -b_y\right) - F\left(x_{b_1}, -b_y\right)\right)
+\end{split}\\
+\begin{split}
+={} &F\left(x_{a_2}, -a_y\right) - F\left(x_{a_1}, -a_y\right)\\
+  - &F\left(x_{b_2}, -b_y\right) + F\left(x_{b_1}, -b_y\right)
+\end{split}\\
+\begin{split}\label{eq:fn:g}
+g\left(a,b\right)
+={} &F\left(\clamp\left(a_x,b_x, f\left(a_y\right)\right), -a_y\right)\\
+  - &F\left(\clamp\left(a_x,b_x,-f\left(a_y\right)\right), -a_y\right)\\
+  - &F\left(\clamp\left(a_x,b_x, f\left(b_y\right)\right), -b_y\right)\\
+  + &F\left(\clamp\left(a_x,b_x,-f\left(b_y\right)\right), -b_y\right)
+\end{split}
+\end{align}
 
 De complete oplossing bestaat uit de functie $g$ \'e\'en keer normaal berekenen, en
 nog een keer met de rechthoek $ab$ gespiegeld om de $x$-as (vermenigvuldig $a_y$ en
 $b_y$ met $-1$). De som van deze twee waarden vormt oppervlak $o$.
+
+Let op dat bij de gespiegelde aanroep van functie $g$, de $y$-co\"ordinaten van de
+punten $a$ en $b$ omgedraaid zijn, om de eisen uit
+\cref{eq:constrain:x,eq:constrain:y} te waarborgen.
 \begin{equation}\label{eq:fn:h}
 	\begin{aligned}[c]
 		h\left(a,b\right)
-		= g\,(&\left(a_x,\max\left(0,a_y\right)\right),\\
-			    &\left(b_x, \max\left(0,b_y\right)\right))\\
-		+ g\,(&\left(a_x,\max\left(0,-a_y\right)\right),\\
-			    &\left(b_x, \max\left(0,-b_y\right)\right)
+		= g\,(&\left(a_x, \max\left(0, a_y\right)\right),\\
+			    &\left(b_x, \max\left(0, b_y\right)\right))\\
+		+ g\,(&\left(a_x, \max\left(0,-b_y\right)\right),\\
+			    &\left(b_x, \max\left(0,-a_y\right)\right)
 		)
 	\end{aligned}
 \end{equation}
 
+De Python implementatie in bijlage \ref{attach:python-impl} heeft enkele handige
+toevoegingen:
+\begin{itemize}
+	\item Invoergetallen worden automatisch genormaliseerd
+	\item De formule schaalt het oppervlak $o$ automatisch voor elke willekeurige
+		straal van cirkel $c$
+\end{itemize}
+
 \section*{Colofon}
 
 Ik heb geen wiskunde gestudeerd, en kan geen bewijs geven dat de methode die
@@ -266,4 +306,7 @@ beschreven is in dit document foutloos is.
 \fontsize{8.5}{12}\selectfont
 \lstinputlisting{circrect.py}
 
+Dit document \autocite{self}, en de bovenstaande broncode \autocite{selfsrc} zijn ook
+online beschikbaar.
+
 \end{document}
diff --git a/refs.bib b/refs.bib
index 9535627..83ff217 100644
--- a/refs.bib
+++ b/refs.bib
@@ -8,9 +8,25 @@
 
 @online{circrect2,
 	title = {circrect2},
-	url = {https://www.desmos.com/calculator/zudzhuj9yi},
+	url = {https://www.desmos.com/calculator/e8u2i24zza},
 	author = {Loek Le Blansch},
 	date = {2024-01-08},
 	urldate = {2024-01-08},
 }
 
+@online{self,
+	title = {Exacte berekening van het gedeelde oppervlak van een cirkel en een rechthoek},
+	url = {https://media.pipeframe.xyz/circrect.pdf},
+	author = {Loek Le Blansch},
+	date = {2024-01-09},
+	urldate = {2024-01-09},
+}
+
+@online{selfsrc,
+	title = {circrect repository},
+	url = {https://git.pipeframe.xyz/lonkaars/circrect},
+	author = {Loek Le Blansch},
+	date = {2024-01-09},
+	urldate = {2024-01-09},
+}
+