{"id":48314,"date":"2026-06-26T15:33:19","date_gmt":"2026-06-26T07:33:19","guid":{"rendered":"https:\/\/www.qiusir.com\/?p=48314"},"modified":"2026-07-01T07:46:02","modified_gmt":"2026-06-30T23:46:02","slug":"qiutopia-2","status":"publish","type":"post","link":"https:\/\/www.qiusir.com\/?p=48314","title":{"rendered":"qiutopIA:\u7403\u9759\u6b62\u6469\u64e6\u7cfb\u6570\u7684\u6761\u4ef6"},"content":{"rendered":"<img decoding=\"async\" alt=\"\" src=\"https:\/\/www.qiusir.com\/wp-content\/gallery\/physics\/4blai.jpeg\" width=\"580\" class=\"framed\" \/>\r\n\u56db\u4e2a\u5b8c\u5168\u76f8\u540c\u7684\u5c0f\u7403\uff0c\u4e09\u4e2a\u5728\u684c\u9762\uff0c\u4e24\u4e24\u63a5\u89e6\uff08\u7528\u6a61\u76ae\u7b4b\u6346\u4f4f\uff09\uff0c\u7b2c\u56db\u4e2a\u5c0f\u7403\u653e\u5728\u6b63\u4e0a\u65b9\uff08\u5f53\u6210\u5149\u6ed1\uff1b\u4e0d\u5149\u6ed1\uff0c\u4ece\u5bf9\u79f0\u6027\u4e0a\u770b\u4e0d\u53d7\u6469\u64e6\uff1f\uff09\uff0c\u6c42\u5e95\u4e0b\u6bcf\u4e2a\u5c0f\u7403\u5bf9\u9876\u90e8\u5c0f\u7403\u7684\u652f\u6301\u529b\u3002\uff08$\\frac{\\sqrt{6}}{6}mg$\uff09\r\n\u5bf9\u8fd9\u9053\u5f88\u7ecf\u5178\u7684\u7acb\u4f53\u53d7\u529b\u5206\u6790\u7684\u9898\u76ee\u5370\u8c61\u5f88\u6df1\uff0c\u81ea\u7136\u60f3\u5230\u6469\u64e6\u7cfb\u6570\u7684\u95ee\u9898\uff0c\u6b63\u597d\u7ec3\u4e60\u518c\u5c31\u6709\u8fd9\u4e2a\u9898\u76ee\uff0c\u548c\u5f53\u5e74\u7684\u60f3\u6cd5\u4e00\u81f4\u3002\u4fdd\u6301\u9759\u6b62\u7684\u6469\u64e6\u7cfb\u6570\u6700\u4e0b\u4e3a$\\frac{\\sqrt{2}}{8}$\u3002\u8fd9\u4e2a\u9898\u76ee\u7684\u95ee\u9898\u5c31\u662f\u201c\u4e0d\u5149\u6ed1\uff0c\u4ece\u5bf9\u79f0\u6027\u4e0a\u770b\u4e0d\u53d7\u6469\u64e6\uff1f\u201d\u758f\u5ffd\u4e86\u5e95\u90e8\u5c0f\u7403\u7684\u5171\u70b9\u529b\u5e73\u8861\u7684\u6761\u4ef6\uff0c\u9876\u90e8\u5c0f\u7403\u5fc5\u7136\u548c\u5e95\u90e8\u5c0f\u7403\u6709\u9759\u6469\u64e6\u3002\r\n<br>\r\n\r\n<div>\r\n<div>\r\n<script>\r\n    (function() {\r\n        var containerId = 'wp-spheres-physics-container-8848';\r\n        var loadingId = 'wp-loading-text-8848';\r\n\r\n        \/\/ \u6838\u5fc3\u6e32\u67d3\u903b\u8f91\r\n        function startApp() {\r\n            var container = document.getElementById(containerId);\r\n            if (!container) {\r\n                return; \r\n            }\r\n\r\n            \/\/ \u79fb\u9664\u52a0\u8f7d\u63d0\u793a\r\n            var loadingText = document.getElementById(loadingId);\r\n            if (loadingText) {\r\n                loadingText.style.display = 'none';\r\n            }\r\n\r\n            var width = container.clientWidth;\r\n            var height = container.clientHeight;\r\n\r\n            \/\/ WordPress \u5bb9\u5668\u9632\u584c\u9677\u56de\u9000\u673a\u5236\r\n            if (width === 0) {\r\n                width = 800; \/\/ \u9ed8\u8ba4\u5907\u7528\u5bbd\u5ea6\r\n            }\r\n            if (height === 0) {\r\n                height = 500; \/\/ \u9ed8\u8ba4\u5907\u7528\u9ad8\u5ea6\r\n            }\r\n\r\n            \/\/ --- \u521d\u59cb\u5316\u573a\u666f ---\r\n            var scene = new THREE.Scene();\r\n            scene.background = new THREE.Color('#1e293b');\r\n\r\n            var camera = new THREE.PerspectiveCamera(45, width \/ height, 0.1, 100);\r\n            camera.position.set(5, 3, 8); \r\n\r\n            var renderer = new THREE.WebGLRenderer({ antialias: true });\r\n            renderer.setSize(width, height);\r\n            renderer.setPixelRatio(window.devicePixelRatio);\r\n            container.appendChild(renderer.domElement);\r\n\r\n            var controls = new THREE.OrbitControls(camera, renderer.domElement);\r\n            controls.enableDamping = true;\r\n            controls.dampingFactor = 0.05;\r\n            controls.target.set(0, -0.2, 0);\r\n\r\n            \/\/ --- \u5149\u7167 ---\r\n            var ambientLight = new THREE.AmbientLight(0xffffff, 0.6);\r\n            scene.add(ambientLight);\r\n            \r\n            var dirLight = new THREE.DirectionalLight(0xffffff, 1.5);\r\n            dirLight.position.set(5, 10, 5);\r\n            scene.add(dirLight);\r\n\r\n            \/\/ --- \u51e0\u4f55\u4e0e\u6750\u8d28 ---\r\n            var R = 1; \r\n            var sphereGeo = new THREE.SphereGeometry(R, 64, 64);\r\n            \r\n            var materialBottom = new THREE.MeshLambertMaterial({ \r\n                color: 0x94a3b8, transparent: true, opacity: 0.6, side: THREE.DoubleSide\r\n            });\r\n            var materialBottomActive = new THREE.MeshLambertMaterial({ \r\n                color: 0x22c55e, transparent: true, opacity: 0.6, side: THREE.DoubleSide\r\n            });\r\n            var materialTop = new THREE.MeshLambertMaterial({ \r\n                color: 0x3b82f6, transparent: true, opacity: 0.4, emissive: 0x1e3a8a, emissiveIntensity: 0.8, side: THREE.DoubleSide\r\n            });\r\n\r\n            \/\/ --- \u7403\u5fc3\u5750\u6807\u8ba1\u7b97 ---\r\n            var zDist = (2 * Math.sqrt(3)) \/ 3 * R;\r\n            var posA = new THREE.Vector3(0, 0, zDist);\r\n            var posB = new THREE.Vector3(R, 0, -Math.sqrt(3)\/3 * R);\r\n            var posC = new THREE.Vector3(-R, 0, -Math.sqrt(3)\/3 * R);\r\n            \r\n            var h = (2 * Math.sqrt(6)) \/ 3 * R;\r\n            var posD = new THREE.Vector3(0, h, 0);\r\n\r\n            \/\/ --- \u521b\u5efa\u7403\u4f53 ---\r\n            var sphereA = new THREE.Mesh(sphereGeo, materialBottomActive);\r\n            sphereA.position.copy(posA);\r\n            scene.add(sphereA);\r\n\r\n            var sphereB = new THREE.Mesh(sphereGeo, materialBottom);\r\n            sphereB.position.copy(posB);\r\n            scene.add(sphereB);\r\n\r\n            var sphereC = new THREE.Mesh(sphereGeo, materialBottom);\r\n            sphereC.position.copy(posC);\r\n            scene.add(sphereC);\r\n\r\n            var sphereD = new THREE.Mesh(sphereGeo, materialTop);\r\n            sphereD.position.copy(posD);\r\n            scene.add(sphereD);\r\n\r\n            \/\/ --- \u663e\u793a\u7403\u5fc3 ---\r\n            var centerGeo = new THREE.SphereGeometry(0.04, 16, 16);\r\n            var centerGeoActive = new THREE.SphereGeometry(0.08, 16, 16);\r\n            \r\n            var centerMat = new THREE.MeshBasicMaterial({ color: 0xaaaaaa, depthTest: false }); \r\n            var centerMatActive = new THREE.MeshBasicMaterial({ color: 0xffffff, depthTest: false }); \r\n            \r\n            var centerA = new THREE.Mesh(centerGeoActive, centerMatActive);\r\n            centerA.position.copy(posA);\r\n            centerA.renderOrder = 1; \r\n            scene.add(centerA);\r\n\r\n            var centerB = new THREE.Mesh(centerGeo, centerMat);\r\n            centerB.position.copy(posB);\r\n            centerB.renderOrder = 1;\r\n            scene.add(centerB);\r\n\r\n            var centerC = new THREE.Mesh(centerGeo, centerMat);\r\n            centerC.position.copy(posC);\r\n            centerC.renderOrder = 1;\r\n            scene.add(centerC);\r\n\r\n            var centerD = new THREE.Mesh(centerGeo, centerMat);\r\n            centerD.position.copy(posD);\r\n            centerD.renderOrder = 1;\r\n            scene.add(centerD);\r\n\r\n            \/\/ --- \u5730\u9762 ---\r\n            var gridHelper = new THREE.GridHelper(10, 10, 0x475569, 0x334155);\r\n            gridHelper.position.y = -R;\r\n            scene.add(gridHelper);\r\n\r\n            \/\/ --- \u81ea\u5b9a\u4e49\u53d7\u529b\u7bad\u5934\u751f\u6210\u51fd\u6570 (\u7981\u7528\u4e00\u5207\u903b\u8f91\u8fd0\u7b97\u7b26) ---\r\n            function createArrow(dir, origin, length, color, headLength, headWidth, lineThickness, opacity) {\r\n                var group = new THREE.Group();\r\n                group.position.copy(origin);\r\n                \r\n                var axis = new THREE.Vector3(0, 1, 0);\r\n                group.quaternion.setFromUnitVectors(axis, dir.clone().normalize());\r\n\r\n                \/\/ \u66ff\u4ee3\u903b\u8f91\u8fd0\u7b97\u7b26\r\n                var isTransparent = false;\r\n                if (opacity < 1.0) {\r\n                    isTransparent = true;\r\n                }\r\n                \r\n                var doDepthWrite = false;\r\n                if (opacity >= 1.0) {\r\n                    doDepthWrite = true;\r\n                }\r\n\r\n                var mat = new THREE.MeshBasicMaterial({ \r\n                    color: color, \r\n                    transparent: isTransparent, \r\n                    opacity: opacity,\r\n                    depthWrite: doDepthWrite\r\n                });\r\n                \r\n                var hLen = headLength;\r\n                if (length < hLen) {\r\n                    hLen = length;\r\n                }\r\n                var shaftLength = length - hLen;\r\n                \r\n                if (shaftLength > 0) {\r\n                    var shaftGeo = new THREE.CylinderGeometry(lineThickness, lineThickness, shaftLength, 12);\r\n                    shaftGeo.translate(0, shaftLength \/ 2, 0);\r\n                    var shaft = new THREE.Mesh(shaftGeo, mat);\r\n                    group.add(shaft);\r\n                }\r\n\r\n                if (hLen > 0) {\r\n                    var headGeo = new THREE.ConeGeometry(headWidth \/ 2, hLen, 16);\r\n                    headGeo.translate(0, hLen \/ 2, 0);\r\n                    var head = new THREE.Mesh(headGeo, mat);\r\n                    head.position.y = shaftLength;\r\n                    group.add(head);\r\n                }\r\n\r\n                scene.add(group);\r\n                return group;\r\n            }\r\n\r\n            \/\/ --- \u5173\u952e\u70b9\u5b9a\u4e49 ---\r\n            var contactGroundA = new THREE.Vector3(posA.x, -R, posA.z); \r\n            var dirDA = new THREE.Vector3().subVectors(posA, posD).normalize(); \r\n            var contactAD = new THREE.Vector3().copy(posD).addScaledVector(dirDA, R); \r\n\r\n            \/\/ 1. \u91cd\u529b\r\n            createArrow(new THREE.Vector3(0, -1, 0), posA, 1.5, 0xff3333, 0.35, 0.15, 0.04, 1.0);\r\n            createArrow(new THREE.Vector3(0, -1, 0), posD, 1.5, 0xff3333, 0.35, 0.15, 0.012, 1.0);\r\n\r\n            \/\/ 2. \u5730\u9762\u652f\u6301\u529b\r\n            createArrow(new THREE.Vector3(0, 1, 0), contactGroundA, 1.8, 0x33ff33, 0.35, 0.15, 0.04, 1.0);\r\n\r\n            \/\/ 3. \u5730\u9762\u9759\u6469\u64e6\u529b (\u8bbe\u5b9a\u57fa\u51c6\u957f\u5ea6)\r\n            var visualFrictionLength = 1.2; \/\/ \u8bbe\u5b9a\u4e00\u4e2a\u56fa\u5b9a\u7684\u89c6\u89c9\u957f\u5ea6\r\n            var dirFrictionGnd = new THREE.Vector3(0, -R, 0).sub(contactGroundA).normalize();\r\n            createArrow(dirFrictionGnd, contactGroundA, visualFrictionLength, 0xffaa00, 0.35, 0.15, 0.04, 1.0);\r\n\r\n            \/\/ --- \u6838\u5fc3\u4fee\u6b63\uff1a\u57fa\u4e8e\u529b\u77e9\u5e73\u8861\u4e25\u683c\u63a7\u5236\u957f\u5ea6\u76f8\u7b49 ---\r\n            var dirTrueResultant = new THREE.Vector3().subVectors(contactGroundA, contactAD).normalize();\r\n            \r\n            \/\/ \u4e3a\u4e86\u4fdd\u8bc1\u7403\u95f4\u6469\u64e6\u529b(\u7d2b)\u4e0e\u5730\u9762\u6469\u64e6\u529b(\u6a59)\u5728\u753b\u9762\u4e0a100%\u7b49\u957f\uff0c\u53cd\u63a8\u5408\u529b\u7684\u7f29\u653e\u6bd4\u4f8b\r\n            var proj = dirTrueResultant.dot(dirDA);\r\n            var vecNormalUnit = dirDA.clone().multiplyScalar(proj);\r\n            var vecFrictionUnit = new THREE.Vector3().subVectors(dirTrueResultant, vecNormalUnit);\r\n            \r\n            \/\/ \u6838\u5fc3\u63a8\u7b97\uff1a\u6240\u9700\u7684\u5408\u529b\u7f29\u653e\u500d\u6570 = \u76ee\u6807\u6469\u64e6\u529b\u957f\u5ea6 \/ \u5355\u4f4d\u6469\u64e6\u529b\u5411\u91cf\u957f\u5ea6\r\n            var totalForceScale = visualFrictionLength \/ vecFrictionUnit.length();\r\n            \r\n            var vecTotal = dirTrueResultant.clone().multiplyScalar(totalForceScale);\r\n            var vecNormal = dirDA.clone().multiplyScalar(totalForceScale * proj);\r\n            var vecFriction = new THREE.Vector3().subVectors(vecTotal, vecNormal);\r\n\r\n            \/\/ 4. \u7403\u95f4\u6cd5\u5411\u538b\u529b\r\n            createArrow(vecNormal, contactAD, vecNormal.length(), 0x00ffff, 0.35, 0.15, 0.04, 1.0);\r\n\r\n            \/\/ 5. \u7403\u95f4\u9759\u6469\u64e6\u529b (\u73b0\u5728\u5b83\u7684\u89c6\u89c9\u957f\u5ea6\u4e25\u683c\u7b49\u4e8e visualFrictionLength\uff0c\u5373\u7b49\u4e8e\u5730\u9762\u6469\u64e6\u529b)\r\n            createArrow(vecFriction, contactAD, vecFriction.length(), 0xd884ff, 0.25, 0.12, 0.04, 1.0);\r\n\r\n            \/\/ --- \u5e73\u884c\u56db\u8fb9\u5f62\u5b9a\u5219\u5c55\u793a ---\r\n            createArrow(vecTotal, contactAD, vecTotal.length(), 0xffffff, 0.15, 0.06, 0.006, 1.0);\r\n\r\n            var lineMaterial = new THREE.LineDashedMaterial({\r\n                color: 0xffffff, linewidth: 1, scale: 1, dashSize: 0.1, gapSize: 0.05, opacity: 0.5, transparent: true\r\n            });\r\n\r\n            var points1 = [contactAD.clone().add(vecNormal), contactAD.clone().add(vecTotal)];\r\n            var line1 = new THREE.Line(new THREE.BufferGeometry().setFromPoints(points1), lineMaterial);\r\n            line1.computeLineDistances();\r\n            scene.add(line1);\r\n\r\n            var points2 = [contactAD.clone().add(vecFriction), contactAD.clone().add(vecTotal)];\r\n            var line2 = new THREE.Line(new THREE.BufferGeometry().setFromPoints(points2), lineMaterial);\r\n            line2.computeLineDistances();\r\n            scene.add(line2);\r\n\r\n            \/\/ --- \u5207\u5e73\u9762\u4e0e\u76f4\u89d2\u6807\u8bb0 ---\r\n            var planeGeo = new THREE.PlaneGeometry(1.5, 1.5);\r\n            var planeMat = new THREE.MeshBasicMaterial({ \r\n                color: 0x94a3b8, transparent: true, opacity: 0.15, side: THREE.DoubleSide, depthWrite: false \r\n            });\r\n            var tangentPlane = new THREE.Mesh(planeGeo, planeMat);\r\n            tangentPlane.position.copy(contactAD);\r\n            tangentPlane.lookAt(contactAD.clone().add(dirDA)); \r\n            scene.add(tangentPlane);\r\n\r\n            var rSize = 0.15;\r\n            var dirN = vecNormal.clone().normalize();\r\n            var dirF = vecFriction.clone().normalize();\r\n            var pN = contactAD.clone().add(dirN.clone().multiplyScalar(rSize));\r\n            var pF = contactAD.clone().add(dirF.clone().multiplyScalar(rSize));\r\n            var pCorner = contactAD.clone().add(dirN.clone().multiplyScalar(rSize)).add(dirF.clone().multiplyScalar(rSize));\r\n            \r\n            var rightAngleGeo = new THREE.BufferGeometry().setFromPoints([pN, pCorner, pF]);\r\n            var rightAngleMat = new THREE.LineBasicMaterial({color: 0xffffff, linewidth: 2});\r\n            var rightAngleLine = new THREE.Line(rightAngleGeo, rightAngleMat);\r\n            scene.add(rightAngleLine);\r\n\r\n            \/\/ --- \u9876\u90e8\u7403\u4f53 (D) \u7684\u53d7\u529b\u5206\u6790 ---\r\n            var contactGroundB = new THREE.Vector3(posB.x, -R, posB.z);\r\n            var contactGroundC = new THREE.Vector3(posC.x, -R, posC.z);\r\n            var dirDB = new THREE.Vector3().subVectors(posB, posD).normalize();\r\n            var dirDC = new THREE.Vector3().subVectors(posC, posD).normalize();\r\n            var contactBD = new THREE.Vector3().copy(posD).addScaledVector(dirDB, R);\r\n            var contactCD = new THREE.Vector3().copy(posD).addScaledVector(dirDC, R);\r\n\r\n            var dirTrueResultantB = new THREE.Vector3().subVectors(contactGroundB, contactBD).normalize();\r\n            var dirTrueResultantC = new THREE.Vector3().subVectors(contactGroundC, contactCD).normalize();\r\n            \r\n            var vecTotalB = dirTrueResultantB.clone().multiplyScalar(totalForceScale);\r\n            var vecTotalC = dirTrueResultantC.clone().multiplyScalar(totalForceScale);\r\n\r\n            var vecNormalB = dirDB.clone().multiplyScalar(vecTotalB.dot(dirDB));\r\n            var vecFrictionB = new THREE.Vector3().subVectors(vecTotalB, vecNormalB);\r\n\r\n            var vecNormalC = dirDC.clone().multiplyScalar(vecTotalC.dot(dirDC));\r\n            var vecFrictionC = new THREE.Vector3().subVectors(vecTotalC, vecNormalC);\r\n\r\n            var topNormalColor = 0x00ffff; \r\n            var topFrictionColor = 0xd884ff; \r\n            \r\n            createArrow(vecTotal.clone().negate(), contactAD, vecTotal.length(), 0xffffff, 0.15, 0.06, 0.005, 1.0);\r\n            createArrow(vecNormal.clone().negate(), contactAD, vecNormal.length(), topNormalColor, 0.35, 0.15, 0.012, 1.0);\r\n            createArrow(vecFriction.clone().negate(), contactAD, vecFriction.length(), topFrictionColor, 0.25, 0.12, 0.012, 1.0);\r\n\r\n            createArrow(vecTotalB.clone().negate(), contactBD, vecTotalB.length(), 0xffffff, 0.15, 0.06, 0.005, 1.0);\r\n            createArrow(vecNormalB.clone().negate(), contactBD, vecNormalB.length(), topNormalColor, 0.35, 0.15, 0.012, 1.0);\r\n            createArrow(vecFrictionB.clone().negate(), contactBD, vecFrictionB.length(), topFrictionColor, 0.25, 0.12, 0.012, 1.0);\r\n\r\n            createArrow(vecTotalC.clone().negate(), contactCD, vecTotalC.length(), 0xffffff, 0.15, 0.06, 0.005, 1.0);\r\n            createArrow(vecNormalC.clone().negate(), contactCD, vecNormalC.length(), topNormalColor, 0.35, 0.15, 0.012, 1.0);\r\n            createArrow(vecFrictionC.clone().negate(), contactCD, vecFrictionC.length(), topFrictionColor, 0.25, 0.12, 0.012, 1.0);\r\n\r\n            \/\/ --- \u663e\u793a\u6240\u6709\u63a5\u89e6\u70b9 ---\r\n            var contactPointGeo = new THREE.SphereGeometry(0.06, 16, 16);\r\n            var contactPointMat = new THREE.MeshBasicMaterial({ color: 0xffff00, depthTest: false }); \r\n\r\n            function createContactPoint(position) {\r\n                var point = new THREE.Mesh(contactPointGeo, contactPointMat);\r\n                point.position.copy(position);\r\n                point.renderOrder = 2; \r\n                scene.add(point);\r\n            }\r\n\r\n            createContactPoint(contactGroundA);\r\n            createContactPoint(contactGroundB);\r\n            createContactPoint(contactGroundC);\r\n\r\n            createContactPoint(contactAD);\r\n            createContactPoint(contactBD);\r\n            createContactPoint(contactCD);\r\n\r\n            var contactAB = new THREE.Vector3().addVectors(posA, posB).multiplyScalar(0.5);\r\n            var contactBC = new THREE.Vector3().addVectors(posB, posC).multiplyScalar(0.5);\r\n            var contactCA = new THREE.Vector3().addVectors(posC, posA).multiplyScalar(0.5);\r\n            createContactPoint(contactAB);\r\n            createContactPoint(contactBC);\r\n            createContactPoint(contactCA);\r\n\r\n            \/\/ --- \u52a8\u6001\u81ea\u9002\u5e94\u8c03\u6574 (\u54cd\u5e94\u5bb9\u5668\u5927\u5c0f) ---\r\n            window.addEventListener('resize', function() {\r\n                var currentContainer = document.getElementById(containerId);\r\n                if (currentContainer) {\r\n                    var newWidth = currentContainer.clientWidth;\r\n                    var newHeight = currentContainer.clientHeight;\r\n                    if (newWidth > 0) {\r\n                        if (newHeight > 0) {\r\n                            camera.aspect = newWidth \/ newHeight;\r\n                            camera.updateProjectionMatrix();\r\n                            renderer.setSize(newWidth, newHeight);\r\n                        }\r\n                    }\r\n                }\r\n            }, false);\r\n\r\n            \/\/ --- \u6e32\u67d3\u5faa\u73af ---\r\n            function animate() {\r\n                requestAnimationFrame(animate);\r\n                controls.update();\r\n                renderer.render(scene, camera);\r\n            }\r\n            animate();\r\n        }\r\n\r\n        \/\/ ==========================================\r\n        \/\/ \u52a8\u6001\u811a\u672c\u52a0\u8f7d\u5668 (\u89e3\u51b3 WordPress \u7ade\u6001\u6761\u4ef6\u9ed1\u5c4f\u95ee\u9898)\r\n        \/\/ ==========================================\r\n        \r\n        function checkOrbitControls() {\r\n            if (typeof window.THREE === 'undefined') {\r\n                return false;\r\n            }\r\n            if (typeof window.THREE.OrbitControls === 'undefined') {\r\n                return false;\r\n            }\r\n            return true;\r\n        }\r\n\r\n        function loadOrbitControlsAndStart() {\r\n            if (checkOrbitControls()) {\r\n                startApp();\r\n            } else {\r\n                var orbitScript = document.createElement('script');\r\n                orbitScript.src = 'https:\/\/cdn.jsdelivr.net\/npm\/three@0.128.0\/examples\/js\/controls\/OrbitControls.js';\r\n                orbitScript.onload = startApp;\r\n                document.head.appendChild(orbitScript);\r\n            }\r\n        }\r\n\r\n        function initLoader() {\r\n            if (typeof window.THREE !== 'undefined') {\r\n                loadOrbitControlsAndStart();\r\n            } else {\r\n                var threeScript = document.createElement('script');\r\n                threeScript.src = 'https:\/\/cdnjs.cloudflare.com\/ajax\/libs\/three.js\/r128\/three.min.js';\r\n                threeScript.onload = loadOrbitControlsAndStart;\r\n                document.head.appendChild(threeScript);\r\n            }\r\n        }\r\n\r\n        \/\/ \u542f\u52a8\u52a0\u8f7d\u5668\r\n        initLoader();\r\n\r\n    })();\r\n<\/script>\r\n\r\n\r\n\r\n<!-- 1. \u73af\u5883\u9694\u79bb\u7684\u552f\u4e00 ID \u5bb9\u5668\uff0c\u91c7\u7528\u7eaf\u5185\u8054 CSS \u63a7\u5236\u6837\u5f0f\u548c\u5c3a\u5bf8 -->\r\n<div id=\"wp-spheres-physics-container-8848\" style=\"position: relative; width: 100%; max-width: 900px; height: 60vh; min-height: 500px; background-color: #1e293b; border-radius: 12px; overflow: hidden; box-shadow: 0 10px 25px -5px rgba(0, 0, 0, 0.2); margin: 0 auto;\">\r\n    <div id=\"wp-loading-text-8848\" style=\"position: absolute; top: 50%; left: 50%; transform: translate(-50%, -50%); color: #94a3b8; font-family: sans-serif; font-size: 14px;\">\u6b63\u5728\u52a0\u8f7d 3D \u7269\u7406\u5f15\u64ce&#8230;<\/div>\r\n<\/div>\r\n\r\n<!-- 2. \u6838\u5fc3\u4e1a\u52a1\u903b\u8f91\uff0c\u81ea\u5e26\u52a8\u6001\u811a\u672c\u52a0\u8f7d\u5668\uff0c\u7edd\u4e0d\u6c61\u67d3\u5168\u5c40\uff0c\u4e25\u7981\u4f7f\u7528 && \u548c || -->\r\n<script>\r\n    (function() {\r\n        var containerId = 'wp-spheres-physics-container-8848';\r\n        var loadingId = 'wp-loading-text-8848';\r\n\r\n        \/\/ \u6838\u5fc3\u6e32\u67d3\u903b\u8f91\r\n        function startApp() {\r\n            var container = document.getElementById(containerId);\r\n            if (!container) {\r\n                return; \r\n            }\r\n\r\n            \/\/ \u79fb\u9664\u52a0\u8f7d\u63d0\u793a\r\n            var loadingText = document.getElementById(loadingId);\r\n            if (loadingText) {\r\n                loadingText.style.display = 'none';\r\n            }\r\n\r\n            var width = container.clientWidth;\r\n            var height = container.clientHeight;\r\n\r\n            \/\/ WordPress \u5bb9\u5668\u9632\u584c\u9677\u56de\u9000\u673a\u5236\r\n            if (width === 0) {\r\n                width = 800; \/\/ \u9ed8\u8ba4\u5907\u7528\u5bbd\u5ea6\r\n            }\r\n            if (height === 0) {\r\n                height = 500; \/\/ \u9ed8\u8ba4\u5907\u7528\u9ad8\u5ea6\r\n            }\r\n\r\n            \/\/ --- \u521d\u59cb\u5316\u573a\u666f ---\r\n            var scene = new THREE.Scene();\r\n            scene.background = new THREE.Color('#1e293b');\r\n\r\n            var camera = new THREE.PerspectiveCamera(45, width \/ height, 0.1, 100);\r\n            camera.position.set(5, 3, 8); \r\n\r\n            var renderer = new THREE.WebGLRenderer({ antialias: true });\r\n            renderer.setSize(width, height);\r\n            renderer.setPixelRatio(window.devicePixelRatio);\r\n            container.appendChild(renderer.domElement);\r\n\r\n            var controls = new THREE.OrbitControls(camera, renderer.domElement);\r\n            controls.enableDamping = true;\r\n            controls.dampingFactor = 0.05;\r\n            controls.target.set(0, -0.2, 0);\r\n\r\n            \/\/ --- \u5149\u7167 ---\r\n            var ambientLight = new THREE.AmbientLight(0xffffff, 0.6);\r\n            scene.add(ambientLight);\r\n            \r\n            var dirLight = new THREE.DirectionalLight(0xffffff, 1.5);\r\n            dirLight.position.set(5, 10, 5);\r\n            scene.add(dirLight);\r\n\r\n            \/\/ --- \u51e0\u4f55\u4e0e\u6750\u8d28 ---\r\n            var R = 1; \r\n            var sphereGeo = new THREE.SphereGeometry(R, 64, 64);\r\n            \r\n            var materialBottom = new THREE.MeshLambertMaterial({ \r\n                color: 0x94a3b8, transparent: true, opacity: 0.6, side: THREE.DoubleSide\r\n            });\r\n            var materialBottomActive = new THREE.MeshLambertMaterial({ \r\n                color: 0x22c55e, transparent: true, opacity: 0.6, side: THREE.DoubleSide\r\n            });\r\n            var materialTop = new THREE.MeshLambertMaterial({ \r\n                color: 0x3b82f6, transparent: true, opacity: 0.4, emissive: 0x1e3a8a, emissiveIntensity: 0.8, side: THREE.DoubleSide\r\n            });\r\n\r\n            \/\/ --- \u7403\u5fc3\u5750\u6807\u8ba1\u7b97 ---\r\n            var zDist = (2 * Math.sqrt(3)) \/ 3 * R;\r\n            var posA = new THREE.Vector3(0, 0, zDist);\r\n            var posB = new THREE.Vector3(R, 0, -Math.sqrt(3)\/3 * R);\r\n            var posC = new THREE.Vector3(-R, 0, -Math.sqrt(3)\/3 * R);\r\n            \r\n            var h = (2 * Math.sqrt(6)) \/ 3 * R;\r\n            var posD = new THREE.Vector3(0, h, 0);\r\n\r\n            \/\/ --- \u521b\u5efa\u7403\u4f53 ---\r\n            var sphereA = new THREE.Mesh(sphereGeo, materialBottomActive);\r\n            sphereA.position.copy(posA);\r\n            scene.add(sphereA);\r\n\r\n            var sphereB = new THREE.Mesh(sphereGeo, materialBottom);\r\n            sphereB.position.copy(posB);\r\n            scene.add(sphereB);\r\n\r\n            var sphereC = new THREE.Mesh(sphereGeo, materialBottom);\r\n            sphereC.position.copy(posC);\r\n            scene.add(sphereC);\r\n\r\n            var sphereD = new THREE.Mesh(sphereGeo, materialTop);\r\n            sphereD.position.copy(posD);\r\n            scene.add(sphereD);\r\n\r\n            \/\/ --- \u663e\u793a\u7403\u5fc3 ---\r\n            var centerGeo = new THREE.SphereGeometry(0.04, 16, 16);\r\n            var centerGeoActive = new THREE.SphereGeometry(0.08, 16, 16);\r\n            \r\n            var centerMat = new THREE.MeshBasicMaterial({ color: 0xaaaaaa, depthTest: false }); \r\n            var centerMatActive = new THREE.MeshBasicMaterial({ color: 0xffffff, depthTest: false }); \r\n            \r\n            var centerA = new THREE.Mesh(centerGeoActive, centerMatActive);\r\n            centerA.position.copy(posA);\r\n            centerA.renderOrder = 1; \r\n            scene.add(centerA);\r\n\r\n            var centerB = new THREE.Mesh(centerGeo, centerMat);\r\n            centerB.position.copy(posB);\r\n            centerB.renderOrder = 1;\r\n            scene.add(centerB);\r\n\r\n            var centerC = new THREE.Mesh(centerGeo, centerMat);\r\n            centerC.position.copy(posC);\r\n            centerC.renderOrder = 1;\r\n            scene.add(centerC);\r\n\r\n            var centerD = new THREE.Mesh(centerGeo, centerMat);\r\n            centerD.position.copy(posD);\r\n            centerD.renderOrder = 1;\r\n            scene.add(centerD);\r\n\r\n            \/\/ --- \u5730\u9762 ---\r\n            var gridHelper = new THREE.GridHelper(10, 10, 0x475569, 0x334155);\r\n            gridHelper.position.y = -R;\r\n            scene.add(gridHelper);\r\n\r\n            \/\/ --- \u81ea\u5b9a\u4e49\u53d7\u529b\u7bad\u5934\u751f\u6210\u51fd\u6570 (\u7981\u7528\u4e00\u5207\u903b\u8f91\u8fd0\u7b97\u7b26) ---\r\n            function createArrow(dir, origin, length, color, headLength, headWidth, lineThickness, opacity) {\r\n                var group = new THREE.Group();\r\n                group.position.copy(origin);\r\n                \r\n                var axis = new THREE.Vector3(0, 1, 0);\r\n                group.quaternion.setFromUnitVectors(axis, dir.clone().normalize());\r\n\r\n                \/\/ \u66ff\u4ee3\u903b\u8f91\u8fd0\u7b97\u7b26\r\n                var isTransparent = false;\r\n                if (opacity < 1.0) {\r\n                    isTransparent = true;\r\n                }\r\n                \r\n                var doDepthWrite = false;\r\n                if (opacity >= 1.0) {\r\n                    doDepthWrite = true;\r\n                }\r\n\r\n                var mat = new THREE.MeshBasicMaterial({ \r\n                    color: color, \r\n                    transparent: isTransparent, \r\n                    opacity: opacity,\r\n                    depthWrite: doDepthWrite\r\n                });\r\n                \r\n                var hLen = headLength;\r\n                if (length < hLen) {\r\n                    hLen = length;\r\n                }\r\n                var shaftLength = length - hLen;\r\n                \r\n                if (shaftLength > 0) {\r\n                    var shaftGeo = new THREE.CylinderGeometry(lineThickness, lineThickness, shaftLength, 12);\r\n                    shaftGeo.translate(0, shaftLength \/ 2, 0);\r\n                    var shaft = new THREE.Mesh(shaftGeo, mat);\r\n                    group.add(shaft);\r\n                }\r\n\r\n                if (hLen > 0) {\r\n                    var headGeo = new THREE.ConeGeometry(headWidth \/ 2, hLen, 16);\r\n                    headGeo.translate(0, hLen \/ 2, 0);\r\n                    var head = new THREE.Mesh(headGeo, mat);\r\n                    head.position.y = shaftLength;\r\n                    group.add(head);\r\n                }\r\n\r\n                scene.add(group);\r\n                return group;\r\n            }\r\n\r\n            \/\/ --- \u5173\u952e\u70b9\u5b9a\u4e49 ---\r\n            var contactGroundA = new THREE.Vector3(posA.x, -R, posA.z); \r\n            var dirDA = new THREE.Vector3().subVectors(posA, posD).normalize(); \r\n            var contactAD = new THREE.Vector3().copy(posD).addScaledVector(dirDA, R); \r\n\r\n            \/\/ 1. \u91cd\u529b\r\n            createArrow(new THREE.Vector3(0, -1, 0), posA, 1.5, 0xff3333, 0.35, 0.15, 0.04, 1.0);\r\n            createArrow(new THREE.Vector3(0, -1, 0), posD, 1.5, 0xff3333, 0.35, 0.15, 0.012, 1.0);\r\n\r\n            \/\/ 2. \u5730\u9762\u652f\u6301\u529b\r\n            createArrow(new THREE.Vector3(0, 1, 0), contactGroundA, 1.8, 0x33ff33, 0.35, 0.15, 0.04, 1.0);\r\n\r\n            \/\/ 3. \u5730\u9762\u9759\u6469\u64e6\u529b (\u8bbe\u5b9a\u57fa\u51c6\u957f\u5ea6)\r\n            var visualFrictionLength = 1.2; \/\/ \u8bbe\u5b9a\u4e00\u4e2a\u56fa\u5b9a\u7684\u89c6\u89c9\u957f\u5ea6\r\n            var dirFrictionGnd = new THREE.Vector3(0, -R, 0).sub(contactGroundA).normalize();\r\n            createArrow(dirFrictionGnd, contactGroundA, visualFrictionLength, 0xffaa00, 0.35, 0.15, 0.04, 1.0);\r\n\r\n            \/\/ --- \u6838\u5fc3\u4fee\u6b63\uff1a\u57fa\u4e8e\u529b\u77e9\u5e73\u8861\u4e25\u683c\u63a7\u5236\u957f\u5ea6\u76f8\u7b49 ---\r\n            var dirTrueResultant = new THREE.Vector3().subVectors(contactGroundA, contactAD).normalize();\r\n            \r\n            \/\/ \u4e3a\u4e86\u4fdd\u8bc1\u7403\u95f4\u6469\u64e6\u529b(\u7d2b)\u4e0e\u5730\u9762\u6469\u64e6\u529b(\u6a59)\u5728\u753b\u9762\u4e0a100%\u7b49\u957f\uff0c\u53cd\u63a8\u5408\u529b\u7684\u7f29\u653e\u6bd4\u4f8b\r\n            var proj = dirTrueResultant.dot(dirDA);\r\n            var vecNormalUnit = dirDA.clone().multiplyScalar(proj);\r\n            var vecFrictionUnit = new THREE.Vector3().subVectors(dirTrueResultant, vecNormalUnit);\r\n            \r\n            \/\/ \u6838\u5fc3\u63a8\u7b97\uff1a\u6240\u9700\u7684\u5408\u529b\u7f29\u653e\u500d\u6570 = \u76ee\u6807\u6469\u64e6\u529b\u957f\u5ea6 \/ \u5355\u4f4d\u6469\u64e6\u529b\u5411\u91cf\u957f\u5ea6\r\n            var totalForceScale = visualFrictionLength \/ vecFrictionUnit.length();\r\n            \r\n            var vecTotal = dirTrueResultant.clone().multiplyScalar(totalForceScale);\r\n            var vecNormal = dirDA.clone().multiplyScalar(totalForceScale * proj);\r\n            var vecFriction = new THREE.Vector3().subVectors(vecTotal, vecNormal);\r\n\r\n            \/\/ 4. \u7403\u95f4\u6cd5\u5411\u538b\u529b\r\n            createArrow(vecNormal, contactAD, vecNormal.length(), 0x00ffff, 0.35, 0.15, 0.04, 1.0);\r\n\r\n            \/\/ 5. \u7403\u95f4\u9759\u6469\u64e6\u529b (\u73b0\u5728\u5b83\u7684\u89c6\u89c9\u957f\u5ea6\u4e25\u683c\u7b49\u4e8e visualFrictionLength\uff0c\u5373\u7b49\u4e8e\u5730\u9762\u6469\u64e6\u529b)\r\n            createArrow(vecFriction, contactAD, vecFriction.length(), 0xd884ff, 0.25, 0.12, 0.04, 1.0);\r\n\r\n            \/\/ --- \u5e73\u884c\u56db\u8fb9\u5f62\u5b9a\u5219\u5c55\u793a ---\r\n            createArrow(vecTotal, contactAD, vecTotal.length(), 0xffffff, 0.15, 0.06, 0.006, 1.0);\r\n\r\n            var lineMaterial = new THREE.LineDashedMaterial({\r\n                color: 0xffffff, linewidth: 1, scale: 1, dashSize: 0.1, gapSize: 0.05, opacity: 0.5, transparent: true\r\n            });\r\n\r\n            var points1 = [contactAD.clone().add(vecNormal), contactAD.clone().add(vecTotal)];\r\n            var line1 = new THREE.Line(new THREE.BufferGeometry().setFromPoints(points1), lineMaterial);\r\n            line1.computeLineDistances();\r\n            scene.add(line1);\r\n\r\n            var points2 = [contactAD.clone().add(vecFriction), contactAD.clone().add(vecTotal)];\r\n            var line2 = new THREE.Line(new THREE.BufferGeometry().setFromPoints(points2), lineMaterial);\r\n            line2.computeLineDistances();\r\n            scene.add(line2);\r\n\r\n            \/\/ --- \u5207\u5e73\u9762\u4e0e\u76f4\u89d2\u6807\u8bb0 ---\r\n            var planeGeo = new THREE.PlaneGeometry(1.5, 1.5);\r\n            var planeMat = new THREE.MeshBasicMaterial({ \r\n                color: 0x94a3b8, transparent: true, opacity: 0.15, side: THREE.DoubleSide, depthWrite: false \r\n            });\r\n            var tangentPlane = new THREE.Mesh(planeGeo, planeMat);\r\n            tangentPlane.position.copy(contactAD);\r\n            tangentPlane.lookAt(contactAD.clone().add(dirDA)); \r\n            scene.add(tangentPlane);\r\n\r\n            var rSize = 0.15;\r\n            var dirN = vecNormal.clone().normalize();\r\n            var dirF = vecFriction.clone().normalize();\r\n            var pN = contactAD.clone().add(dirN.clone().multiplyScalar(rSize));\r\n            var pF = contactAD.clone().add(dirF.clone().multiplyScalar(rSize));\r\n            var pCorner = contactAD.clone().add(dirN.clone().multiplyScalar(rSize)).add(dirF.clone().multiplyScalar(rSize));\r\n            \r\n            var rightAngleGeo = new THREE.BufferGeometry().setFromPoints([pN, pCorner, pF]);\r\n            var rightAngleMat = new THREE.LineBasicMaterial({color: 0xffffff, linewidth: 2});\r\n            var rightAngleLine = new THREE.Line(rightAngleGeo, rightAngleMat);\r\n            scene.add(rightAngleLine);\r\n\r\n            \/\/ --- \u9876\u90e8\u7403\u4f53 (D) \u7684\u53d7\u529b\u5206\u6790 ---\r\n            var contactGroundB = new THREE.Vector3(posB.x, -R, posB.z);\r\n            var contactGroundC = new THREE.Vector3(posC.x, -R, posC.z);\r\n            var dirDB = new THREE.Vector3().subVectors(posB, posD).normalize();\r\n            var dirDC = new THREE.Vector3().subVectors(posC, posD).normalize();\r\n            var contactBD = new THREE.Vector3().copy(posD).addScaledVector(dirDB, R);\r\n            var contactCD = new THREE.Vector3().copy(posD).addScaledVector(dirDC, R);\r\n\r\n            var dirTrueResultantB = new THREE.Vector3().subVectors(contactGroundB, contactBD).normalize();\r\n            var dirTrueResultantC = new THREE.Vector3().subVectors(contactGroundC, contactCD).normalize();\r\n            \r\n            var vecTotalB = dirTrueResultantB.clone().multiplyScalar(totalForceScale);\r\n            var vecTotalC = dirTrueResultantC.clone().multiplyScalar(totalForceScale);\r\n\r\n            var vecNormalB = dirDB.clone().multiplyScalar(vecTotalB.dot(dirDB));\r\n            var vecFrictionB = new THREE.Vector3().subVectors(vecTotalB, vecNormalB);\r\n\r\n            var vecNormalC = dirDC.clone().multiplyScalar(vecTotalC.dot(dirDC));\r\n            var vecFrictionC = new THREE.Vector3().subVectors(vecTotalC, vecNormalC);\r\n\r\n            var topNormalColor = 0x00ffff; \r\n            var topFrictionColor = 0xd884ff; \r\n            \r\n            createArrow(vecTotal.clone().negate(), contactAD, vecTotal.length(), 0xffffff, 0.15, 0.06, 0.005, 1.0);\r\n            createArrow(vecNormal.clone().negate(), contactAD, vecNormal.length(), topNormalColor, 0.35, 0.15, 0.012, 1.0);\r\n            createArrow(vecFriction.clone().negate(), contactAD, vecFriction.length(), topFrictionColor, 0.25, 0.12, 0.012, 1.0);\r\n\r\n            createArrow(vecTotalB.clone().negate(), contactBD, vecTotalB.length(), 0xffffff, 0.15, 0.06, 0.005, 1.0);\r\n            createArrow(vecNormalB.clone().negate(), contactBD, vecNormalB.length(), topNormalColor, 0.35, 0.15, 0.012, 1.0);\r\n            createArrow(vecFrictionB.clone().negate(), contactBD, vecFrictionB.length(), topFrictionColor, 0.25, 0.12, 0.012, 1.0);\r\n\r\n            createArrow(vecTotalC.clone().negate(), contactCD, vecTotalC.length(), 0xffffff, 0.15, 0.06, 0.005, 1.0);\r\n            createArrow(vecNormalC.clone().negate(), contactCD, vecNormalC.length(), topNormalColor, 0.35, 0.15, 0.012, 1.0);\r\n            createArrow(vecFrictionC.clone().negate(), contactCD, vecFrictionC.length(), topFrictionColor, 0.25, 0.12, 0.012, 1.0);\r\n\r\n            \/\/ --- \u663e\u793a\u6240\u6709\u63a5\u89e6\u70b9 ---\r\n            var contactPointGeo = new THREE.SphereGeometry(0.06, 16, 16);\r\n            var contactPointMat = new THREE.MeshBasicMaterial({ color: 0xffff00, depthTest: false }); \r\n\r\n            function createContactPoint(position) {\r\n                var point = new THREE.Mesh(contactPointGeo, contactPointMat);\r\n                point.position.copy(position);\r\n                point.renderOrder = 2; \r\n                scene.add(point);\r\n            }\r\n\r\n            createContactPoint(contactGroundA);\r\n            createContactPoint(contactGroundB);\r\n            createContactPoint(contactGroundC);\r\n\r\n            createContactPoint(contactAD);\r\n            createContactPoint(contactBD);\r\n            createContactPoint(contactCD);\r\n\r\n            var contactAB = new THREE.Vector3().addVectors(posA, posB).multiplyScalar(0.5);\r\n            var contactBC = new THREE.Vector3().addVectors(posB, posC).multiplyScalar(0.5);\r\n            var contactCA = new THREE.Vector3().addVectors(posC, posA).multiplyScalar(0.5);\r\n            createContactPoint(contactAB);\r\n            createContactPoint(contactBC);\r\n            createContactPoint(contactCA);\r\n\r\n            \/\/ --- \u52a8\u6001\u81ea\u9002\u5e94\u8c03\u6574 (\u54cd\u5e94\u5bb9\u5668\u5927\u5c0f) ---\r\n            window.addEventListener('resize', function() {\r\n                var currentContainer = document.getElementById(containerId);\r\n                if (currentContainer) {\r\n                    var newWidth = currentContainer.clientWidth;\r\n                    var newHeight = currentContainer.clientHeight;\r\n                    if (newWidth > 0) {\r\n                        if (newHeight > 0) {\r\n                            camera.aspect = newWidth \/ newHeight;\r\n                            camera.updateProjectionMatrix();\r\n                            renderer.setSize(newWidth, newHeight);\r\n                        }\r\n                    }\r\n                }\r\n            }, false);\r\n\r\n            \/\/ --- \u6e32\u67d3\u5faa\u73af ---\r\n            function animate() {\r\n                requestAnimationFrame(animate);\r\n                controls.update();\r\n                renderer.render(scene, camera);\r\n            }\r\n            animate();\r\n        }\r\n\r\n 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