Leslie Leigh (李的序) shrinktofit

Let $N$ to be normal of plane decided by $a$ and $b$, where $N$ is at the side so that it has the minimal angle between the input normal. Let $angle$ be the unsigned angle between $a$ and $b$. Around $N$, following the right hand rule, if $a$ can rotate to $b$ in positive $angle$ angle, then the result is $angle$. Otherwise, the result is $-angle$.

Let $b$ be the inverse of $a$, then for the translation term $T$(Vector), rotation term $Q$(Quaternion), scale term $S$(Vector):

$$\begin{equation} \begin{split} T_(a * b) & = T_b + (Q_b \times (S_b \times T_a) \times Q_b^{-1}) = 0 \\\ T(b) & = -(Q_b \times S_b \times T_a \times Q_b^{-1}) \\\ & = Q_b \times (S_b \times -T_a) \times Q_b^{-1} \end{split} \end{equation}$$

Abstract

In our engine, bones exist in form of scene node. We can't tell if a node denotes a bone or not. This sometimes becomes obstacle in animation system.

Problem Statement

Obstacles

Why do we need to distinguish bone nodes from normal? I'll list some cases.

也就是说，下面两种算法是等价的，我们不需要遍历两遍：

// 预先累加权重和
function blend(samples: ReadonlyArray<readonly [value: number, weight: number]>) {
  let result = 0.0;
  let sumWeights = 0.0;
  for (const [v, w] of samples) {

elementary
rudimentary
feasible
viable
impair
interconnect

	function trigger(el, eventType) {
	if (typeof eventType === 'string' && typeof el[eventType] === 'function') {
	el[eventType]();
	} else {
	const event =
	eventType === 'string'
	? new Event(eventType, {bubbles: true})
	: eventType;
	el.dispatchEvent(event);
	}

	import bpy

	def print(data):
	for window in bpy.context.window_manager.windows:
	screen = window.screen
	for area in screen.areas:
	if area.type == 'CONSOLE':
	override = {'window': window, 'screen': screen, 'area': area}
	bpy.ops.console.scrollback_append(override, text=str(data), type="OUTPUT")

	import { clamp, Node, NodeSpace, Quat, Vec3 } from "cc";

	/**
	* 解析双骨骼（三关节）的 IK 问题。
	* 三关节分别称为根关节、中间关节和末端关节。例如，分别对应于大腿、膝盖和脚关节。
	* @param root 根关节。
	* @param middle 中间关节。
	* @param end 末端关节。
	* @param target 末端关节要抵达的目标位置（世界空间）。
	* @param hint 中间关节的提示位置（世界空间），用于决定中间关节的朝向。

	export function solveTwoBoneIK(
	a: Node,
	b: Node,
	c: Node,
	target: Vec3,
	) {
	const sanityChecker = new TwoBoneIKNodeSanityChecker(a, b, c);

	const pA = Vec3.clone(a.worldPosition);
	const pB = Vec3.clone(b.worldPosition);

	\documentclass{article}
	\usepackage[utf8]{inputenc}
	\usepackage{amsmath}

	\begin{document}

	Let RS(denoted as $R$) be "rotation then scale". \\

	RS then translation(denoted as $t$):