From 8ea52962bfbc9a3acb1d111b0f76fd7f68ee617b Mon Sep 17 00:00:00 2001 From: Kenneth Date: Sun, 5 Jan 2025 00:41:05 +0000 Subject: [PATCH] new poast: selection detection --- astro.config.mjs | 4 + package.json | 4 +- pnpm-lock.yaml | 5576 ++++++++++------- public/selection-detection-find-a-light.png | Bin 0 -> 51930 bytes public/selection-detection-find-a.png | Bin 0 -> 51469 bytes ...tion-detection-find-intersection-light.png | Bin 0 -> 55647 bytes .../selection-detection-find-intersection.png | Bin 0 -> 55250 bytes public/selection-detection-problem-light.png | Bin 0 -> 25671 bytes public/selection-detection-problem.png | Bin 0 -> 25904 bytes ...election-detection-special-cases-light.png | Bin 0 -> 34703 bytes public/selection-detection-special-cases.png | Bin 0 -> 34417 bytes public/selection-projection-light.png | Bin 0 -> 27514 bytes public/selection-projection.png | Bin 0 -> 27387 bytes .../selection-detection-with-vector-math.md | 134 + src/content/config.ts | 5 +- src/layouts/BlogPost.astro | 19 +- 16 files changed, 3342 insertions(+), 2400 deletions(-) create mode 100644 public/selection-detection-find-a-light.png create mode 100644 public/selection-detection-find-a.png create mode 100644 public/selection-detection-find-intersection-light.png create mode 100644 public/selection-detection-find-intersection.png create mode 100644 public/selection-detection-problem-light.png create mode 100644 public/selection-detection-problem.png create mode 100644 public/selection-detection-special-cases-light.png create mode 100644 public/selection-detection-special-cases.png create mode 100644 public/selection-projection-light.png create mode 100644 public/selection-projection.png create mode 100644 src/content/blog/selection-detection-with-vector-math.md diff --git a/astro.config.mjs b/astro.config.mjs index 5e3d91d..8c490ff 100644 --- a/astro.config.mjs +++ b/astro.config.mjs @@ -1,6 +1,8 @@ import { defineConfig } from "astro/config"; import mdx from "@astrojs/mdx"; import sitemap from "@astrojs/sitemap"; +import remarkMath from "remark-math"; +import rehypeKatex from "rehype-katex"; import tailwind from "@astrojs/tailwind"; @@ -14,5 +16,7 @@ export default defineConfig({ // https://shiki.style/themes theme: "catppuccin-mocha", }, + remarkPlugins: [remarkMath], + rehypePlugins: [rehypeKatex], }, }); diff --git a/package.json b/package.json index c1a95b9..de050a5 100644 --- a/package.json +++ b/package.json @@ -24,6 +24,8 @@ "@flydotio/dockerfile": "latest", "@tailwindcss/typography": "^0.5.10", "prettier": "^3.2.5", - "prettier-plugin-astro": "^0.13.0" + "prettier-plugin-astro": "^0.13.0", + "rehype-katex": "^7.0.1", + "remark-math": "^6.0.0" } } \ No newline at end of file diff --git a/pnpm-lock.yaml b/pnpm-lock.yaml index e444229..aef4543 100644 --- a/pnpm-lock.yaml +++ b/pnpm-lock.yaml @@ -1,95 +1,86 @@ -lockfileVersion: '6.0' +lockfileVersion: '9.0' settings: autoInstallPeers: true excludeLinksFromLockfile: false -dependencies: - '@astrojs/check': - specifier: ^0.5.6 - version: 0.5.10(prettier-plugin-astro@0.13.0)(prettier@3.3.3)(typescript@5.5.4) - '@astrojs/mdx': - specifier: ^2.1.1 - version: 2.3.1(astro@4.15.4) - '@astrojs/rss': - specifier: ^4.0.5 - version: 4.0.7 - '@astrojs/sitemap': - specifier: ^3.1.1 - version: 3.1.6 - '@astrojs/tailwind': - specifier: ^5.1.0 - version: 5.1.0(astro@4.15.4)(tailwindcss@3.4.10) - astro: - specifier: ^4.4.15 - version: 4.15.4(typescript@5.5.4) - tailwindcss: - specifier: ^3.4.1 - version: 3.4.10 - typescript: - specifier: ^5.4.2 - version: 5.5.4 +importers: -devDependencies: - '@catppuccin/tailwindcss': - specifier: ^0.1.6 - version: 0.1.6(tailwindcss@3.4.10) - '@flydotio/dockerfile': - specifier: latest - version: 0.5.8 - '@tailwindcss/typography': - specifier: ^0.5.10 - version: 0.5.15(tailwindcss@3.4.10) - prettier: - specifier: ^3.2.5 - version: 3.3.3 - prettier-plugin-astro: - specifier: ^0.13.0 - version: 0.13.0 + .: + dependencies: + '@astrojs/check': + 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zt99O-1L1N)*Qeg>9{BZ_VSov7DY<*EhBF+J{O99DzTyQSF4e!C+$M6RA_o&ynOP;3P zq%Iu!1;P11PH>*O)PlvpsQ6yiiKGk_@=4|YCj^Hd|4#^xl>XnAh>7ny8=v_z;$y4- g{O9Nl8kd;cis@g*?%)qsadE9MU1d^a?6voQ00r^S-2eap literal 0 HcmV?d00001 diff --git a/src/content/blog/selection-detection-with-vector-math.md b/src/content/blog/selection-detection-with-vector-math.md new file mode 100644 index 0000000..b895488 --- /dev/null +++ b/src/content/blog/selection-detection-with-vector-math.md @@ -0,0 +1,134 @@ +--- +title: Selection Detection with Vector Math +description: 'a guide in how to implement object selection detection in canvas with vector math' +pubDate: '5 Jan 2025' +useKatex: true +heroImage: /selection-projection.png +image: /selection-projection.png +--- + +I am currently working on making my own whiteboard program using [Dear ImGui](https://github.com/ocornut/imgui), and one of the features I had to implement was the ability to select objects on the whiteboard. Since ImGui does not offer click detections for custom shapes, I had to implement my own selection detection algorithm. With the help of some vector math and the Internet, it was actually quite straightforward. + +## Framing the Problem + +Whether an object should be selected can be framed as asking how close a mouse click is to an object. If the distance is below a threshold, then we can consider the object to be selected. + +Let's reframe this question again in math. Consider a line (segment) $\textbf{b}$ with two points, $v_1$ and $v_2$, and point $P$, where the mouse click happened. The problem is to find a point on $\textbf{b}$ that is closest to $P$. Intuitively, that point can be found by drawing a perpendicular line from $P$ to $\textbf{b}$, and the intersection point will be the closest point on $\textbf{b}$ to $P$. The distance between the two points then becomes the closest distance between $P$ and $\textbf{b}$. Below is an illustration of the problem: + + + + an illustration of the selection detection problem + + +where: + +- $P$ is the point where the mouse click occurred; +- $\textbf{b}$ is the line that can be selected; and +- $d$ is the shortest distance between $P$ and $\textbf{b}$. + +## Applying Vector Math + +Notice how the letter $\textbf{b}$ is bolded. This means that the line is treated as a vector. Now, we introduce another vector $\textbf{a}$ from the beginning of $\textbf{b}$ to $P$. Observe that the projection of $\textbf{a}$ onto $\textbf{b}$ $proj_{\textbf{b}}\textbf{a}$ points to the shortest point on $b$ from $P$. + + + + vector projection illustration + + +The projection is written as: + +$$ +proj_{\textbf{b}}\textbf{a} = \frac{\textbf{a}\cdot\textbf{b}}{||\textbf{b}||} \hat{\textbf{b}} +$$ + +where + +- $||\textbf{b}||$ is the magnitude of $\textbf{b}$ +- $\hat{\textbf{b}}$ is the unit vector of $\textbf{b}$ + +Recall that the magnitude of a vector $\textbf{a}$ is defined as: + +$$ +||\textbf{a}|| = \sqrt{\sum_{i=1}^{n}x_i^2} +$$ + +and its unit vector as: + +$$ +\hat{\textbf{a}} = \frac{\textbf{a}}{||\textbf{a}||} +$$ + +for any finite n-dimensional vector $\textbf{a} \in \mathbb{R}^n$. + +Substituting into the projection formula, we get + +$$ +proj_{\textbf{b}}\textbf{a} = \frac{\textbf{a}\cdot\textbf{b}}{\textbf{b}\cdot\textbf{b}}\textbf{b} +$$ + +### Finding $\textbf{a}$ and $\textbf{b}$ + +To find $\textbf{a}$, we treat $v_1$ and $P$ as two vectors: + + + + vector from starting point of b to P + + +Then, we subtract $\textbf{v}_1$ from $\textbf{P}$ using vector subtraction: + +$$ +\textbf{a} = \textbf{P} - \textbf{v}_1 +$$ + +$\textbf{b}$ can be found in a similar fashion. + +### Finding the Intersection Point + +Once the projection vector is found, we add it to $\textbf{v}_1$ to obtain the coordinates of the intersection point: + + + + find intersection point + + +### Computing the Shortest Distance + +Finally, to find the shortest distance between $P$ and $\textbf{b}$, we now have two choices: + +- Find the euclidean distance between $P$ and the intersection point +- Perform vector subtraction between the two points, then find its magnitude. + +In either case, we have successfully found the shortest distance between a point and a line segment! + +## Special Cases + +There are two special cases that we have not considered: + +1. The projection of the point falls outside of $\textbf{b}$ to the left of it +2. The projection of the point falls outside of $\textbf{b}$ to the right of it + + + + special cases of the selection detection problem + + +In the first case, the dot product $\textbf{a} \cdot \textbf{b}$ is less than zero. Therefore, if we find that the dot product is less than zero, we know that the closest point from $P$ to $\textbf{b}$ is the starting point of $\textbf{b}$. + +In the second case, the projection of $\textbf{a}$ onto $\textbf{b}$ is longer than $\textbf{b}$. We can write this relation as: + +$$ +\begin{align} +||proj_{\textbf{b}}\textbf{a}|| &> ||\textbf{b}|| \nonumber \\ +\frac{\textbf{a}\cdot\textbf{b}}{||\textbf{b}||} &> ||\textbf{b}|| \nonumber \\ +\textbf{a}\cdot\textbf{b} &> ||\textbf{b}||^2 \nonumber\\ +\textbf{a}\cdot\textbf{b} &> \textbf{b}\cdot\textbf{b} \nonumber\\ +\end{align} +$$ + +Therefore, when $\textbf{b}\cdot\textbf{b}$ is greater than $\textbf{a}\cdot\textbf{b}$, we know that we are in the second case, and we can infer that the end point of $\textbf{b}$ is the closest to $P$. + +## Implementation Note + +At the final step of finding the distance between a point and the intersection point, a square root is involved using either method. Since we only care about whether the distance meets a threshold and not the actual number, we can skip the square root calculation and compare it with the square of the threshold we want. For example, if we want the threshold to be 5px for an object to be considered selected, we compare the computation with $5^2 = 25$. + diff --git a/src/content/config.ts b/src/content/config.ts index 667a31c..32b905a 100644 --- a/src/content/config.ts +++ b/src/content/config.ts @@ -1,7 +1,7 @@ -import { defineCollection, z } from 'astro:content'; +import { defineCollection, z } from "astro:content"; const blog = defineCollection({ - type: 'content', + type: "content", // Type-check frontmatter using a schema schema: z.object({ title: z.string(), @@ -10,6 +10,7 @@ const blog = defineCollection({ pubDate: z.coerce.date(), updatedDate: z.coerce.date().optional(), heroImage: z.string().optional(), + useKatex: z.boolean().optional(), }), }); diff --git a/src/layouts/BlogPost.astro b/src/layouts/BlogPost.astro index 2692e2b..bdda330 100644 --- a/src/layouts/BlogPost.astro +++ b/src/layouts/BlogPost.astro @@ -5,15 +5,30 @@ import Header from "../components/Header.astro"; import Footer from "../components/Footer.astro"; import FormattedDate from "../components/FormattedDate.astro"; -type Props = CollectionEntry<"blog">["data"]; +type Props = CollectionEntry<"blog">["data"] & { useKatex?: boolean }; -const { title, description, pubDate, updatedDate, heroImage } = Astro.props; +const { + title, + description, + pubDate, + updatedDate, + heroImage, + useKatex = false, +} = Astro.props; --- + { + useKatex ? ( + + ) : null + }