Slide
Buffer geometry
CG

Leonhard Euler (1707–1783) was a Swiss mathematician, physicist, astronomer, logician, and engineer who made significant contributions to numerous fields of mathematics and science. Born in Basel, Switzerland, Euler's prolific output, totaling over 800 publications, has earned him a reputation as one of the greatest mathematicians of all time.

Euler's contributions to mathematics are vast and varied, encompassing areas such as calculus, number theory, graph theory, geometry, and mathematical analysis. He developed important concepts and notation that are still used today, including the notation for functions (f(x)), the symbol "e" for the base of the natural logarithm, and the Greek letter sigma (Σ) for summation.

Among Euler's many achievements are his solutions to the famous Seven Bridges of Königsberg problem, his work on the Basel problem leading to the discovery of the value of the Riemann zeta function at even positive integers (π²/6), and his groundbreaking work in graph theory, laying the foundations for the field.

Euler also made significant contributions to physics, fluid dynamics, astronomy, and engineering. His studies in mechanics and celestial mechanics helped advance understanding in these fields, while his work on optics and acoustics furthered knowledge in the natural sciences.

Beyond his mathematical and scientific accomplishments, Euler was known for his exceptional productivity, working tirelessly despite losing his vision in one eye later in life. His works continue to influence mathematics and science today, and his legacy remains celebrated in the scientific community.

Polygon mesh based on Three.js & WebGL
00:00 / 00:00

The Cartesian took the floor and said: "The soul is a pure spirit that has received in the belly of its mother all metaphysical ideas, and which, leaving that place, is obliged to go to school, and to learn all over again what it already knew, and will not know again."

"It is not worth the trouble," responded the animal with the height of eight leagues, "for your soul to be so knowledgeable in its mother's stomach, only to be so ignorant when you have hair on your chin. But what do you understand by the mind?"

"You are asking me?" said the reasoner. "I have no idea. We say that it is not matter."

"But do you at least know what matter is?"

"Certainly," replied the man. "For example this stone is grey, has such and such a form, has three dimensions, is heavy and divisible."

"Well!" said the Sirian, "this thing that appears to you to be divisible, heavy, and grey, will you tell me what it is? You see some attributes, but behind those, are you familiar with that?

"No," said the other.

"So you do not know what matter is."

“Die Mathematik ist es, die uns vor dem Trug der Sinne schützt und uns den Unterschied zwischen Schein und Wahrheit kennen lehrt.” ― Leonhard Euler
“Nothing takes place in the world whose meaning is not that of some maximum or minimum.” ― Leonhard Euler
“Logic is the foundation of the certainty of all the knowledge we acquire.” ― Leonhard Euler
“Madam, I have just come from a country where people are hanged if they talk.” ― Leonard Euler
“Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers,
and we have reason to believe that it is a mystery into which the human mind will never penetrate.” ― Leonhard Euler
“The kind of knowledge which is supported only by observations and is not yet proved must be carefully distinguished from the truth;
it is gained by induction, as we usually say. Yet we have seen cases in which mere induction led to error.” ― Leonhard Euler
Slide
Buffer geometry
CG

Leonhard Euler (1707–1783) was a Swiss mathematician, physicist, astronomer, logician, and engineer who made significant contributions to numerous fields of mathematics and science. Born in Basel, Switzerland, Euler's prolific output, totaling over 800 publications, has earned him a reputation as one of the greatest mathematicians of all time.

Euler's contributions to mathematics are vast and varied, encompassing areas such as calculus, number theory, graph theory, geometry, and mathematical analysis. He developed important concepts and notation that are still used today, including the notation for functions (f(x)), the symbol "e" for the base of the natural logarithm, and the Greek letter sigma (Σ) for summation.

Among Euler's many achievements are his solutions to the famous Seven Bridges of Königsberg problem, his work on the Basel problem leading to the discovery of the value of the Riemann zeta function at even positive integers (π²/6), and his groundbreaking work in graph theory, laying the foundations for the field.

Euler also made significant contributions to physics, fluid dynamics, astronomy, and engineering. His studies in mechanics and celestial mechanics helped advance understanding in these fields, while his work on optics and acoustics furthered knowledge in the natural sciences.

Beyond his mathematical and scientific accomplishments, Euler was known for his exceptional productivity, working tirelessly despite losing his vision in one eye later in life. His works continue to influence mathematics and science today, and his legacy remains celebrated in the scientific community.

Polygon mesh based on Three.js & WebGL

The Cartesian took the floor and said: "The soul is a pure spirit that has received in the belly of its mother all metaphysical ideas, and which, leaving that place, is obliged to go to school, and to learn all over again what it already knew, and will not know again."

"It is not worth the trouble," responded the animal with the height of eight leagues, "for your soul to be so knowledgeable in its mother's stomach, only to be so ignorant when you have hair on your chin. But what do you understand by the mind?"

"You are asking me?" said the reasoner. "I have no idea. We say that it is not matter."

"But do you at least know what matter is?"

"Certainly," replied the man. "For example this stone is grey, has such and such a form, has three dimensions, is heavy and divisible."

"Well!" said the Sirian, "this thing that appears to you to be divisible, heavy, and grey, will you tell me what it is? You see some attributes, but behind those, are you familiar with that?

"No," said the other.

"So you do not know what matter is."

“Die Mathematik ist es, die uns vor dem Trug der Sinne schützt und uns den Unterschied zwischen Schein und Wahrheit kennen lehrt.” ― Leonhard Euler
“Nothing takes place in the world whose meaning is not that of some maximum or minimum.” ― Leonhard Euler
“Logic is the foundation of the certainty of all the knowledge we acquire.” ― Leonhard Euler
“Madam, I have just come from a country where people are hanged if they talk.” ― Leonard Euler
“Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers,
and we have reason to believe that it is a mystery into which the human mind will never penetrate.” ― Leonhard Euler
“The kind of knowledge which is supported only by observations and is not yet proved must be carefully distinguished from the truth;
it is gained by induction, as we usually say. Yet we have seen cases in which mere induction led to error.” ― Leonhard Euler
00:00 / 00:00
Slide

Leonhard Euler (1707–1783) was a Swiss mathematician, physicist, astronomer, logician, and engineer who made significant contributions to numerous fields of mathematics and science. Born in Basel, Switzerland, Euler's prolific output, totaling over 800 publications, has earned him a reputation as one of the greatest mathematicians of all time.

Euler's contributions to mathematics are vast and varied, encompassing areas such as calculus, number theory, graph theory, geometry, and mathematical analysis. He developed important concepts and notation that are still used today, including the notation for functions (f(x)), the symbol "e" for the base of the natural logarithm, and the Greek letter sigma (Σ) for summation.

Among Euler's many achievements are his solutions to the famous Seven Bridges of Königsberg problem, his work on the Basel problem leading to the discovery of the value of the Riemann zeta function at even positive integers (π²/6), and his groundbreaking work in graph theory, laying the foundations for the field.

Euler also made significant contributions to physics, fluid dynamics, astronomy, and engineering. His studies in mechanics and celestial mechanics helped advance understanding in these fields, while his work on optics and acoustics furthered knowledge in the natural sciences.

Beyond his mathematical and scientific accomplishments, Euler was known for his exceptional productivity, working tirelessly despite losing his vision in one eye later in life. His works continue to influence mathematics and science today, and his legacy remains celebrated in the scientific community.

Polygon mesh based on Three.js & WebGL
00:00 / 00:00

The Cartesian took the floor and said: "The soul is a pure spirit that has received in the belly of its mother all metaphysical ideas, and which, leaving that place, is obliged to go to school, and to learn all over again what it already knew, and will not know again."

"It is not worth the trouble," responded the animal with the height of eight leagues, "for your soul to be so knowledgeable in its mother's stomach, only to be so ignorant when you have hair on your chin. But what do you understand by the mind?"

"You are asking me?" said the reasoner. "I have no idea. We say that it is not matter."

"But do you at least know what matter is?"

"Certainly," replied the man. "For example this stone is grey, has such and such a form, has three dimensions, is heavy and divisible."

"Well!" said the Sirian, "this thing that appears to you to be divisible, heavy, and grey, will you tell me what it is? You see some attributes, but behind those, are you familiar with that?

"No," said the other.

"So you do not know what matter is."

“Die Mathematik ist es, die uns vor dem Trug der Sinne schützt und uns den Unterschied zwischen Schein und Wahrheit kennen lehrt.” ― Leonhard Euler
“Nothing takes place in the world whose meaning is not that of some maximum or minimum.” ― Leonhard Euler
“Logic is the foundation of the certainty of all the knowledge we acquire.” ― Leonhard Euler
“Madam, I have just come from a country where people are hanged if they talk.” ― Leonard Euler
“Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers,
and we have reason to believe that it is a mystery into which the human mind will never penetrate.” ― Leonhard Euler
“The kind of knowledge which is supported only by observations and is not yet proved must be carefully distinguished from the truth;
it is gained by induction, as we usually say. Yet we have seen cases in which mere induction led to error.” ― Leonhard Euler
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***
The fusiform face area (FFA) is a region of the human brain, located in the fusiform gyrus of the temporal lobe. It is primarily associated with the processing and recognition of faces.

Research using techniques such as functional magnetic resonance imaging (fMRI) has consistently shown increased activity in the fusiform face area when individuals are presented with images of faces compared to other objects or stimuli. This specialization suggests that the FFA plays a critical role in the neural circuitry underlying face perception.

While the exact mechanisms of how the FFA processes facial information are not fully understood, it is believed to be involved in various aspects of face processing, including holistic face perception (viewing faces as integrated wholes rather than individual features), face identification, discrimination between different faces, and emotional expressions.

Additionally, studies have shown that damage or dysfunction in the FFA “module” can lead to impairments in face recognition abilities, a condition known as prosopagnosia (cf. localisation of function). Individuals with prosopagnosia may struggle to recognize familiar faces, including those of friends, family members, and even themselves, despite having intact vision and cognitive functions in other domains.

Overall, the fusiform face area represents a crucial neural substrate for the specialized processing of facial information in the human brain, contributing significantly to our ability to perceive, recognize, and interact with others based on their facial features.
***
face-mesh
Source:

Fabry, T., Smeets, D., & Vandermeule, D.. (2010). Surface representations for 3D face recognition. In Face Recognition

. InTech
Plain numerical DOI: 10.5772/8951
DOI URL
directSciHub download

***
The “Golden Ratio” in 3D human face modeling
Human_head_proportions
Source: http://de.wikipedia.org/w/index.php?title=Datei:Human_head_proportions.svg&filetimestamp=20100503192018
***
References

Gibbon, J. D.. (2008). The three-dimensional Euler equations: Where do we stand?. Physica D: Nonlinear Phenomena

, 237(14–17), 1894–1904.
Plain numerical DOI: 10.1016/j.physd.2007.10.014
DOI URL
directSciHub download

***

Boutsi, A.-M., Ioannidis, C., & Soile, S.. (2019). Interactive Online Visualization of Complex 3D Geometries. The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences

, XLII-2/W9, 173–180.
Plain numerical DOI: 10.5194/isprs-archives-XLII-2-W9-173-2019
DOI URL
directSciHub download

***

Almansoury, F., Kpodjedo, S., & Boussaidi, G. El. (2020). Investigating Web3D topics on StackOverflow: a preliminary study of WebGL and Three.js. In The 25th International Conference on 3D Web Technology

(pp. 1–2). New York, NY, USA: ACM
Plain numerical DOI: 10.1145/3424616.3424726
DOI URL
directSciHub download

***

Yun, W. J., & Kim, J.. (2020). 3D Modeling and WebVR Implementation using Azure Kinect, Open3D, and Three.js. In International Conference on ICT Convergence

Plain numerical DOI: 10.1109/ICTC49870.2020.9289518
DOI URL
directSciHub download

***

Min, Q., Wang, Z., & Liu, N.. (2018). An Evaluation of HTML5 and WebGL for Medical Imaging Applications. Journal of Healthcare Engineering

, 2018, 1–11.
Plain numerical DOI: 10.1155/2018/1592821
DOI URL
directSciHub download

***

Xu, Z., Zhang, Y., & Xu, X.. (2016). 3D visualization for building information models based upon IFC and WebGL integration. Multimedia Tools and Applications

, 75(24), 17421–17441.
Plain numerical DOI: 10.1007/s11042-016-4104-9
DOI URL
directSciHub download

***
Topically related:

Nair, P., & Cavallaro, A.. (2009). 3-D Face Detection, Landmark Localization, and Registration Using a Point Distribution Model. IEEE Transactions on Multimedia

, 11(4), 611–623.
Plain numerical DOI: 10.1109/TMM.2009.2017629
DOI URL
directSciHub download

***

Hangaragi, S., Singh, T., & N, N.. (2023). Face Detection and Recognition Using Face Mesh and Deep Neural Network. Procedia Computer Science

, 218, 741–749.
Plain numerical DOI: 10.1016/j.procs.2023.01.054
DOI URL
directSciHub download

***
Source code

<!DOCTYPE html>
<html lang="en">
<head>
    <title>three.js webgl</title>
    <meta charset="utf-8">
    <meta name="viewport" content="width=device-width, user-scalable=no, minimum-scale=1.0, maximum-scale=1.0">
    <link type="text/css" rel="stylesheet" href="main.css">
    <style>
        /* CSS */
        body{
            cursor: crosshair !important;
        }
    </style>
  
  
</head>
<body>
<script type="importmap">
    {
        "imports": {
            "three": "https://unpkg.com/three/build/three.module.js",
            "three/addons/": "https://unpkg.com/three@latest/examples/jsm/"
        }
    }
</script>
<script type="module">

    import * as THREE from 'three';
    import { GLTFLoader } from '/threejs/GLTFLoader.js';
    import { VertexNormalsHelper } from '/threejs/VertexNormalsHelper.js';
    import { VertexTangentsHelper } from '/threejs/VertexTangentsHelper.js';
    import { OrbitControls } from '/threejs/OrbitControls.js'; // Import OrbitControls

    let scene, renderer;
    let camera, light;
    let vnh;
    let vth;
    let controls; // Declare controls variable

    init();
    animate();

    function init() {

        renderer = new THREE.WebGLRenderer({ alpha: true }); // Set alpha to true for transparent background
        renderer.setPixelRatio( window.devicePixelRatio );
        renderer.setSize( window.innerWidth, window.innerHeight );
        document.body.appendChild( renderer.domElement );

        //

        camera = new THREE.PerspectiveCamera( 70, window.innerWidth / window.innerHeight, 1, 1000 );
        camera.position.z = 400;

        scene = new THREE.Scene();

        light = new THREE.PointLight();
        light.position.set( 200, 100, 150 );
        scene.add( light );

        scene.add( new THREE.PointLightHelper( light, 15 ) );

        const gridHelper = new THREE.GridHelper( 400, 40, 0x0000ff, 0x808080 );
        gridHelper.position.y = - 150;
        gridHelper.position.x = - 150;
        scene.add( gridHelper );

        const polarGridHelper = new THREE.PolarGridHelper( 200, 16, 8, 64, 0x0000ff, 0x808080 );
        polarGridHelper.position.y = - 150;
        polarGridHelper.position.x = 200;
        scene.add( polarGridHelper );

        const loader = new GLTFLoader();
        loader.load( 'models/gltf/LeePerrySmith/LeePerrySmith.glb', function ( gltf ) {

            const mesh = gltf.scene.children[ 0 ];

            mesh.geometry.computeTangents(); // generates bad data due to degenerate UVs

            const group = new THREE.Group();
            group.scale.multiplyScalar( 50 );
            scene.add( group );

            // To make sure that the matrixWorld is up to date for the boxhelpers
            group.updateMatrixWorld( true );

            group.add( mesh );

            vnh = new VertexNormalsHelper( mesh, 5 );
            scene.add( vnh );

            vth = new VertexTangentsHelper( mesh, 5 );
            scene.add( vth );

            scene.add( new THREE.BoxHelper( mesh ) );

            const wireframe = new THREE.WireframeGeometry( mesh.geometry );
            let line = new THREE.LineSegments( wireframe );
            line.material.depthTest = false;
            line.material.opacity = 0.25;
            line.material.transparent = true;
            line.position.x = 4;
            group.add( line );
            scene.add( new THREE.BoxHelper( line ) );

            const edges = new THREE.EdgesGeometry( mesh.geometry );
            line = new THREE.LineSegments( edges );
            line.material.depthTest = false;
            line.material.opacity = 0.25;
            line.material.transparent = true;
            line.position.x = - 4;
            group.add( line );
            scene.add( new THREE.BoxHelper( line ) );

            scene.add( new THREE.BoxHelper( group ) );
            scene.add( new THREE.BoxHelper( scene ) );

        } );

        // Initialize controls
        controls = new OrbitControls( camera, renderer.domElement );
        controls.enableDamping = true; // an animation loop is required when either damping or auto-rotation are enabled
        controls.dampingFactor = 0.25; // set to 0.25 for smoother rotation
        controls.screenSpacePanning = false;
        controls.minDistance = 100;
        controls.maxDistance = 500;

        //

        window.addEventListener( 'resize', onWindowResize );

    }

    function onWindowResize() {

        camera.aspect = window.innerWidth / window.innerHeight;
        camera.updateProjectionMatrix();

        renderer.setSize( window.innerWidth, window.innerHeight );

    }

    function animate() {

        requestAnimationFrame( animate );

        const time = - performance.now() * 0.0003;

        camera.position.x = 400 * Math.cos( time );
        camera.position.z = 400 * Math.sin( time );
        camera.lookAt( scene.position );

        light.position.x = Math.sin( time * 1.7 ) * 300;
        light.position.y = Math.cos( time * 1.5 ) * 400;
        light.position.z = Math.cos( time * 1.3 ) * 300;

        if ( vnh ) vnh.update();
        if ( vth ) vth.update();

        // Update controls
        controls.update();

        renderer.render( scene, camera );

    }

</script>

</body>
</html>

 
***
Sources:

3D Head Scan by Lee Perry-Smith is licensed under a Creative Commons Attribution 3.0 Unported License.: https://github.com/mrdoob/three.js/tree/dev/examples/models/gltf/LeePerrySmith

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