Ratio interval half way 3.

docs/Airbrush/Airbrush.mdx

@@ -40,7 +40,7 @@ its alpha value $\Alpha(\pmb p)$ is equal to blend all the alpha values $\Alpha_
 The $\pmb d_i$ is the vector from stamp $i$ and the current pixel.
 
 $$
-\Alpha(\pmb p) = 1-\prod_{i=1}^n \left(1-\Alpha_s(\pmb d_i)\right)
+\Alpha(\Game) = 1-\prod_{i=1}^n \left(1-\Alpha_s(\pmb d_i)\right)
 $$
 
 We define "alpha density" value, denoted with small alpha $\alpha$.
@@ -51,14 +51,14 @@ Hopefully, the notations remind you of the probability density and probability v
 Replace the $\Alpha_s(\pmb d_i)$ and we get:
 
 $$
-\Alpha(\pmb p) = 1-\prod_{i=1}^n (1-\alpha_s(\pmb d_i) \Delta L)
+\Alpha(\bm p) = 1-\prod_{i=1}^n (1-\alpha_s(\bm d_i) \Delta L)
 $$
 
 So, given any $\alpha_s(\pmb d_i)$ function,
 we can calculate the stamp strokes' continuous form by substituting the function into the formula.
 
 We reuse the old local coordinate, originating at $p_0$, and X and Y axes align to the tangent and normal direction.
-So, $\pmb p = (x, y)$ and $\pmb d_i = (x - l_i, y)$ in the coordinate.
+So, $\bm p = (x, y)$ and $\bm d_i = (x - l_i, y)$ in the coordinate.
 The $l_i$ is the X position of stamp i.
 As $n\rightarrow \infty$ and $\Delta L \rightarrow 0$, and apply [product integral](https://en.wikipedia.org/wiki/Product_integral) (Volterra Integral) on the formula.
 
@@ -72,7 +72,7 @@ But the dot is transparent, and we need to know the alpha value associated with
 ## Special Alpha Density
 To get a clearer comprehension of the theory, let's examine a special case.
 Consider the alpha density value is a constant,
-indicating that a stamp stroke's footprint is a consistently transparent dot, defined by the function
+indicating that a stamp stroke's footprint is a transparent dot, defined by the function
 $$
 \alpha_s(d) =
 \begin{cases}
@@ -89,7 +89,7 @@ $$
 \Alpha(x, y) = 1 - \exp\left(-\int_{d \le R} \alpha_c dl -\int_{d > R} 0 dl \right)
 $$
 
-The second integral does not contribute to the expression and can be omitted for simplicity.
+The second integral does not contribute to the expression and can be omitted.
 The first integral $\int_{d \le R}$ represents the integral over the segment of the edge that stamps can cover the current pixel.
 The segment is marked with thick solid line in the figure below.
 

docs/Basics/Stamp/Stamp.mdx

@@ -59,7 +59,7 @@ The distances represent the cumulative length of the edges, and we calculated th
   By using one of them, the whole rendering process is GPU-accelerated.
 
   The best tutorial about parallel prefix sum I've ever found is this free course
-  [High Performance Computing](https://www.udacity.com/course/high-performance-computing--ud281) by Prof. Rich Vuduc.
+  [High Performance Computing](https://www.udacity.com/catalog/all/any-price/any-school/any-skill/any-difficulty/any-duration/any-type/relevance/page-1?searchValue=High%20Performance%20Computing) by Prof. Rich Vuduc.
   The course introduces prefix sum in Lesson 6.
 
 </details>
@@ -100,16 +100,18 @@ $$
 
 It's not hard to derive $r(x)$ from the figure below.
 $$
-r(x) = \frac{x}{L}r_1 + \left(1-\frac{x}{L}\right)r_0
+r(x) = \left(1-\frac{x}{L}\right)r_0 + \frac{x}{L}r_1 = r_0 - \cos\theta x
 $$
 $L$ is the length of the current edge.
 ![xradius](./stamp-xradius.png)
 By replacing $r(x)$, we get a quadratic equation $ax^2 + bx + c = 0$:
 
 $$
-a = 1 - \cos^2\theta;
-b = - 2 * (r_0\cos\theta + x_p);
-c = x_p^2 + y_p^2 - r_0^2
+\begin{align}
+a &= 1 - \cos^2\theta \\
+b &= - 2(r_0\cos\theta + x_p) \\
+c &= x_p^2 + y_p^2 - r_0^2
+\end{align}
 $$
 
 Remind that $\cos\theta = (r_0 - r_1)/L$.

docs/Basics/Stamp/stampRound.frag

@@ -68,7 +68,7 @@ void main() {
         vec2 textureCoordinate = (pToCurrStamp/currStampRadius + 1.0)/2.0; // uv coordinate
         vec4 sampledColor = texture(footprint, textureCoordinate);
 
-        // The alpha compositing function, https://en.wikipedia.org/wiki/Alpha_compositing
+        // The alpha compositing function, wiki: https://en.wikipedia.org/wiki/Alpha_compositing
         vec4 color;
         color.a = sampledColor.a + currColor.a * (1.0 - sampledColor.a);
         color.rgb = (sampledColor.rgb * sampledColor.a + currColor.rgb * currColor.a * (1.0 - sampledColor.a))/color.a;

docs/Basics/Stamp/stampRound.vert

@@ -31,6 +31,7 @@ void main(){
     l0 = length0;
     l1 = length1;
 
+    // ----------There's nothing new underneath-------------
     float cosTheta = (r0 - r1)/distance(p0, p1);
     if(abs(cosTheta) >= 1.0) return;
 

docs/Proportional-Interval-Stamp/Proportional-Interval-Stamp.mdx

@@ -19,10 +19,9 @@ This content is under construction and not peer-reviewed. Learn it under your ow
 
 </details>
 
-
 <figcaption>
 
-© 2024 Shen Ciao. All rights reserved on this webpage.
+© 2024 Shen Ciao. All rights reserved on this webpage (until Shen publishes the paper).
 
 Unauthorized use and/or duplication of this material without express and written permission from this site’s author and/or owner is strictly prohibited.
 Excerpts and links may be used, provided that full and clear credit is given to Shen Ciao with appropriate and specific direction to the original content.
@@ -31,7 +30,7 @@ Excerpts and links may be used, provided that full and clear credit is given to
 
 :::
 
-## Introduction to stamp patterns
+## Stamp patterns
 
 ### Fixed interval
 In the [Stamp](../Basics/Stamp) section, we learned stamp brush and methods to render it.
@@ -168,14 +167,59 @@ To overcome this, we need to eliminate this dependency while maintaining the app
 It's a significant challenge and will require some calculus to solve it.
 
 ## Theory
-We want the interval to be proportional to radius.
-Let's define this in a calculus way!
-
-Given a polyline and pick one of its edges, cut the edge into infinite small segments with length $\Delta L$.
-Each segment has its local coordinate $x$, and we know its radius $r(x) = r_0 - \cos\theta x$.
-Within a small segment, define the stamp interval as $\eta r(x)$.
-So the number of stamps $\Delta n$ in this segment is $\frac{\Delta L}{\eta r(x)}$.
-Sum $\Delta n$ over the edge, we get the number of stamps $n(x)$ on this edge;
-sum the number on all edges, we get the count of stamps on the polyline.
-So we have the stamp number $n$ grows along the polyline.
-Every time this $n$ value hits an integer, we stamp on that point.
+Given a polyline, pick one of its edges and divide it into infinitely small segments of length $\Delta x$.
+We reuse the old local coordinate system, so each segment has its local coordinate $x$,
+and we already know the radius at a segment $r(x) = r_0 - \cos\theta x$.
+
+![small segment](./small-segment.png)
+
+We need the stamp interval to be proportional to radius.
+Therefore, for each segment, define the stamp interval as $\eta r(x)$.
+To find the number of stamps $\Delta N$ in a segment, divide the length by the interval:
+
+$$
+\Delta n = \frac{\Delta x}{\eta r(x)}
+$$
+
+To find the number of stamps $\Delta n$ over the edge, we need to integrate $\Delta n$ from 0 to x.
+
+$$
+n(x) = \int_0^{x} \frac{dx}{\eta r(x)}
+$$
+
+Given $r(x) = r_0 - \cos\theta x$, we substitute this into the integral and get:
+
+$$
+\tag{1} n(x) = \frac{1}{\eta \cos\theta} \ln(r_0 - \cos\theta x) \bigg|_0^x = \frac{1}{\eta \cos\theta} \ln (1 - \frac{\cos\theta x}{r_0})
+$$
+
+Exchange the dependent and independent variable in formula (1) and get:
+
+$$
+\tag{2} x(n) = \frac{r_0}{\cos\theta}(1 - e^{\eta\cos\theta n})
+$$
+
+As $x = L$, we know the total stamp number on the edge, remind that $\cos\theta L = r_0-r_1$:
+
+$$
+\tag{3} n(L) = \frac{1}{\eta \cos\theta}\ln \frac{r_0}{r_1}
+$$
+
+Soon we will use the formula (1)(2)(3) in our code.
+
+![store](store-n.png)
+
+How do we place footprint on the polyline?
+Imagine a point starts from the first vertex and moves along the polyline.
+The stamp number to the first vertex grows with its movement.
+Every time the number hits an integer, we place a footprint at this point.
+
+Remind the figure below in the Stamp section.
+Given a point or pixel, we calculate a range on the edge that can cover the point.
+We label the two roots with $x_1$ and $x_2$.
+
+![locate stamp](./locate-stamp.png)
+
+With the stamp number of each edge $n(L)$, we can prefix sum $n(L)$ the number of stamps from each vertex to the first vertex.
+We will store the number into each vertex.
+For the current edge, we use $n_0$ and $n_1$ to indicate the stamp number at vertex0 and vertex1.

docusaurus.config.ts

@@ -96,18 +96,18 @@ export default {
         hideable: true,
       },
     },
-    stylesheets: [
-      {
-        href: 'https://cdn.jsdelivr.net/npm/[email protected]/dist/katex.min.css',
-        type: 'text/css',
-        integrity:
-          'sha384-odtC+0UGzzFL/6PNoE8rX/SPcQDXBJ+uRepguP4QkPCm2LBxH3FA3y+fKSiJ+AmM',
-        crossorigin: 'anonymous',
-      },
-    ],
   },
   plugins: [
     'raw-loaders',
   ],
+  stylesheets: [
+    {
+      href: 'https://cdn.jsdelivr.net/npm/[email protected]/dist/katex.min.css',
+      type: 'text/css',
+      integrity:
+          'sha384-odtC+0UGzzFL/6PNoE8rX/SPcQDXBJ+uRepguP4QkPCm2LBxH3FA3y+fKSiJ+AmM',
+      crossorigin: 'anonymous',
+    },
+  ],
   trailingSlash : true
 } satisfies Config;

src/css/custom.css

@@ -31,18 +31,10 @@
   --docusaurus-highlighted-code-line-bg: rgba(0, 0, 0, 0.3);
 }
 
-.katex-html{
-  display: none
-}
-
 span.katex-mathml{
   font-size: 18px;
 }
 
-span.katex-display math{
-  padding-bottom: 0.7em;
-}
-
 figcaption{
   text-after-overflow: 0;
   padding-bottom: 0.0em;