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- title: Home
url: /
sublinks:
- title: Meshedit
url: /meshedit
- title: Pathtracer
url: /pathtracer
- title: Animation
url: /animation
docs/_site/animation/index.html 0 → 100644
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<!DOCTYPE html> <html lang="en-US"> <head> <meta charset="UTF-8"> <meta http-equiv="X-UA-Compatible" content="IE=Edge"> <title>A4: Animation - </title> <link rel="shortcut icon" href="/favicon.ico" type="image/x-icon"> <link rel="stylesheet" href="/assets/css/just-the-docs-default.css"> <script type="text/javascript" src="/assets/js/vendor/lunr.min.js"></script> <script type="text/javascript" src="/assets/js/just-the-docs.js"></script> <meta name="viewport" content="width=device-width, initial-scale=1"> <!-- Begin Jekyll SEO tag v2.7.1 --> <title>A4: Animation</title> <meta name="generator" content="Jekyll v4.2.0" /> <meta property="og:title" content="A4: Animation" /> <meta property="og:locale" content="en_US" /> <meta name="twitter:card" content="summary" /> <meta property="twitter:title" content="A4: Animation" /> <script type="application/ld+json"> {"headline":"A4: Animation","@type":"WebPage","url":"/animation/","@context":"https://schema.org"}</script> <!-- End Jekyll SEO tag --> </head> <body> <svg xmlns="http://www.w3.org/2000/svg" style="display: none;"> <symbol id="svg-link" viewBox="0 0 24 24"> <title>Link</title> <svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="feather feather-link"> <path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path> </svg> </symbol> <symbol id="svg-search" viewBox="0 0 24 24"> <title>Search</title> <svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="feather feather-search"> <circle cx="11" cy="11" r="8"></circle><line x1="21" y1="21" x2="16.65" y2="16.65"></line> </svg> </symbol> <symbol id="svg-menu" viewBox="0 0 24 24"> <title>Menu</title> <svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="feather feather-menu"> <line x1="3" y1="12" x2="21" y2="12"></line><line x1="3" y1="6" x2="21" y2="6"></line><line x1="3" y1="18" x2="21" y2="18"></line> </svg> </symbol> <symbol id="svg-arrow-right" viewBox="0 0 24 24"> <title>Expand</title> <svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="feather feather-chevron-right"> <polyline points="9 18 15 12 9 6"></polyline> </svg> </symbol> <symbol id="svg-doc" viewBox="0 0 24 24"> <title>Document</title> <svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="feather feather-file"> <path d="M13 2H6a2 2 0 0 0-2 2v16a2 2 0 0 0 2 2h12a2 2 0 0 0 2-2V9z"></path><polyline points="13 2 13 9 20 9"></polyline> </svg> </symbol> </svg> <div class="side-bar"> <div class="site-header"> <a href="/" class="site-title lh-tight"> </a> <a href="#" id="menu-button" class="site-button"> <svg viewBox="0 0 24 24" class="icon"><use xlink:href="#svg-menu"></use></svg> </a> </div> <nav role="navigation" aria-label="Main" id="site-nav" class="site-nav"> <ul class="nav-list"><li class="nav-list-item"><a href="/" class="nav-list-link">Home</a></li><li class="nav-list-item"><a href="/git/" class="nav-list-link">GitHub Setup</a></li><li class="nav-list-item"><a href="/build/" class="nav-list-link">Building Scotty3D</a></li><li class="nav-list-item"><a href="#" class="nav-list-expander"><svg viewBox="0 0 24 24"><use xlink:href="#svg-arrow-right"></use></svg></a><a href="/guide/" class="nav-list-link">User Guide</a><ul class="nav-list "><li class="nav-list-item "><a href="/guide/animate_mode/" class="nav-list-link">Animate</a></li><li class="nav-list-item "><a href="/guide/layout_mode/" class="nav-list-link">Layout</a></li><li class="nav-list-item "><a href="/guide/model_mode/" class="nav-list-link">Model</a></li><li class="nav-list-item "><a href="/guide/render_mode/" class="nav-list-link">Render</a></li><li class="nav-list-item "><a href="/guide/rigging_mode/" class="nav-list-link">Rig</a></li><li class="nav-list-item "><a href="/guide/simulate_mode/" class="nav-list-link">Simulate</a></li></ul></li><li class="nav-list-item"><a href="#" class="nav-list-expander"><svg viewBox="0 0 24 24"><use xlink:href="#svg-arrow-right"></use></svg></a><a href="/meshedit/" class="nav-list-link">A2: MeshEdit</a><ul class="nav-list "><li class="nav-list-item "><a href="/meshedit/halfedge" class="nav-list-link">Halfedge Mesh</a></li><li class="nav-list-item "><a href="#" class="nav-list-expander"><svg viewBox="0 0 24 24"><use xlink:href="#svg-arrow-right"></use></svg></a><a href="/meshedit/local/" class="nav-list-link">Local Operations</a><ul class="nav-list"><li class="nav-list-item "> <a href="/meshedit/local/edge_flip" class="nav-list-link">Edge Flip Tutorial</a> </li><li class="nav-list-item "> <a href="/meshedit/local/bevel/" class="nav-list-link">Bevelling</a> </li></ul></li><li class="nav-list-item "><a href="#" class="nav-list-expander"><svg viewBox="0 0 24 24"><use xlink:href="#svg-arrow-right"></use></svg></a><a href="/meshedit/global/" class="nav-list-link">Global Operations</a><ul class="nav-list"><li class="nav-list-item "> <a href="/meshedit/global/catmull/" class="nav-list-link">Catmull-Clark Subdivision</a> </li><li class="nav-list-item "> <a href="/meshedit/global/remesh/" class="nav-list-link">Isotropic Remeshing</a> </li><li class="nav-list-item "> <a href="/meshedit/global/linear/" class="nav-list-link">Linear Subdivision</a> </li><li class="nav-list-item "> <a href="/meshedit/global/loop/" class="nav-list-link">Loop Subdivision</a> </li><li class="nav-list-item "> <a href="/meshedit/global/simplify/" class="nav-list-link">Simplification</a> </li><li class="nav-list-item "> <a href="/meshedit/global/triangulate/" class="nav-list-link">Triangulation</a> </li></ul></li></ul></li><li class="nav-list-item"><a href="#" class="nav-list-expander"><svg viewBox="0 0 24 24"><use xlink:href="#svg-arrow-right"></use></svg></a><a href="/pathtracer/" class="nav-list-link">A3: Pathtracer</a><ul class="nav-list "><li class="nav-list-item "><a href="/pathtracer/camera_rays" class="nav-list-link">(Task 1) Camera Rays</a></li><li class="nav-list-item "><a href="#" class="nav-list-expander"><svg viewBox="0 0 24 24"><use xlink:href="#svg-arrow-right"></use></svg></a><a href="/pathtracer/intersecting_objects" class="nav-list-link">(Task 2) Intersections</a><ul class="nav-list"><li class="nav-list-item "> <a href="/pathtracer/ray_triangle_intersection" class="nav-list-link">Ray Triangle Intersection</a> </li><li class="nav-list-item "> <a href="/pathtracer/ray_sphere_intersection" class="nav-list-link">Ray Sphere Intersection</a> </li></ul></li><li class="nav-list-item "><a href="/pathtracer/bounding_volume_hierarchy" class="nav-list-link">(Task 3) BVH</a></li><li class="nav-list-item "><a href="/pathtracer/shadow_rays" class="nav-list-link">(Task 4) Shadow Rays</a></li><li class="nav-list-item "><a href="/pathtracer/path_tracing" class="nav-list-link">(Task 5) Path Tracing</a></li><li class="nav-list-item "><a href="#" class="nav-list-expander"><svg viewBox="0 0 24 24"><use xlink:href="#svg-arrow-right"></use></svg></a><a href="/pathtracer/materials" class="nav-list-link">(Task 6) Materials</a><ul class="nav-list"><li class="nav-list-item "> <a href="/pathtracer/dielectrics_and_transmission" class="nav-list-link">Dielectrics and Transmission</a> </li></ul></li><li class="nav-list-item "><a href="#" class="nav-list-expander"><svg viewBox="0 0 24 24"><use xlink:href="#svg-arrow-right"></use></svg></a><a href="/pathtracer/environment_lighting" class="nav-list-link">(Task 7) Environment Lighting</a><ul class="nav-list"><li class="nav-list-item "> <a href="/pathtracer/importance_sampling" class="nav-list-link">Environment Light Importance Sampling</a> </li></ul></li><li class="nav-list-item "><a href="/pathtracer/visualization_of_normals" class="nav-list-link">Visualization of normals</a></li></ul></li><li class="nav-list-item active"><a href="#" class="nav-list-expander"><svg viewBox="0 0 24 24"><use xlink:href="#svg-arrow-right"></use></svg></a><a href="/animation/" class="nav-list-link active">A4: Animation</a><ul class="nav-list "><li class="nav-list-item "><a href="/animation/splines" class="nav-list-link">Splines</a></li><li class="nav-list-item "><a href="/animation/skeleton_kinematics" class="nav-list-link">Skeleton Kinematics</a></li><li class="nav-list-item "><a href="/animation/skinning" class="nav-list-link">Skinning</a></li><li class="nav-list-item "><a href="/animation/particles" class="nav-list-link">Particles</a></li></ul></li></ul> </nav> <footer class="site-footer"> This site uses <a href="https://github.com/pmarsceill/just-the-docs">Just the Docs</a>, a documentation theme for Jekyll. </footer> </div> <div class="main" id="top"> <div id="main-header" class="main-header"> <div class="search"> <div class="search-input-wrap"> <input type="text" id="search-input" class="search-input" tabindex="0" placeholder="Search " aria-label="Search " autocomplete="off"> <label for="search-input" class="search-label"><svg viewBox="0 0 24 24" class="search-icon"><use xlink:href="#svg-search"></use></svg></label> </div> <div id="search-results" class="search-results"></div> </div> </div> <div id="main-content-wrap" class="main-content-wrap"> <div id="main-content" class="main-content" role="main"> <h1 id="animation-overview"> <a href="#animation-overview" class="anchor-heading" aria-labelledby="animation-overview"><svg viewBox="0 0 16 16" aria-hidden="true"><use xlink:href="#svg-link"></use></svg></a> Animation Overview </h1> <p>There are four primary components that must be implemented to support Animation functionality.</p> <p><strong>A4.0</strong></p> <ul> <li><a href="splines">(Task 1) Spline Interpolation</a></li> <li><a href="skeleton_kinematics">(Task 2) Skeleton Kinematics</a></li> </ul> <p><strong>A4.5</strong></p> <ul> <li><a href="skinning">(Task 3) Linear Blend Skinning</a></li> <li><a href="particles">(Task 4) Particle Simulation</a></li> </ul> <p>Each task is described at the linked page.</p> <h2 id="converting-frames-to-video"> <a href="#converting-frames-to-video" class="anchor-heading" aria-labelledby="converting-frames-to-video"><svg viewBox="0 0 16 16" aria-hidden="true"><use xlink:href="#svg-link"></use></svg></a> Converting Frames to Video </h2> <p>Additionally, we will ask you to create your own animation. Once you’ve rendered out each frame of your animation, you can combine them into a video by using:</p> <p><code class="language-plaintext highlighter-rouge">ffmpeg -r 30 -f image2 -s 640x360 -pix_fmt yuv420p -i ./%4d.png -vcodec libx264 out.mp4</code></p> <p>You may want to change the default <code class="language-plaintext highlighter-rouge">30</code> and <code class="language-plaintext highlighter-rouge">640x360</code> to the frame rate and resolution you chose to render at.</p> <p>If you don’t have ffmpeg installed on your system, you can get it through most package managers, or you can <a href="https://ffmpeg.org/download.html">download it directly</a>. Alternatively, you may use your preferred video editing tool.</p> </div> </div> <div class="search-overlay"></div> </div> </body> </html>
docs/_site/animation/particles.html 0 → 100644
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<!DOCTYPE html> <html lang="en-US"> <head> <meta charset="UTF-8"> <meta http-equiv="X-UA-Compatible" content="IE=Edge"> <title>Particles - </title> <link rel="shortcut icon" href="/favicon.ico" type="image/x-icon"> <link rel="stylesheet" href="/assets/css/just-the-docs-default.css"> <script type="text/javascript" src="/assets/js/vendor/lunr.min.js"></script> <script type="text/javascript" src="/assets/js/just-the-docs.js"></script> <meta name="viewport" content="width=device-width, initial-scale=1"> <!-- Begin Jekyll SEO tag v2.7.1 --> <title>Particles</title> <meta name="generator" content="Jekyll v4.2.0" /> <meta property="og:title" content="Particles" /> <meta property="og:locale" content="en_US" /> <meta name="twitter:card" content="summary" /> <meta property="twitter:title" content="Particles" /> <script type="application/ld+json"> {"headline":"Particles","@type":"WebPage","url":"/animation/particles","@context":"https://schema.org"}</script> <!-- End Jekyll SEO tag --> </head> <body> <svg xmlns="http://www.w3.org/2000/svg" style="display: none;"> <symbol id="svg-link" viewBox="0 0 24 24"> <title>Link</title> <svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="feather feather-link"> <path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path> </svg> </symbol> <symbol id="svg-search" viewBox="0 0 24 24"> <title>Search</title> <svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="feather feather-search"> <circle cx="11" cy="11" r="8"></circle><line x1="21" y1="21" x2="16.65" y2="16.65"></line> </svg> </symbol> <symbol id="svg-menu" viewBox="0 0 24 24"> <title>Menu</title> <svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="feather feather-menu"> <line x1="3" y1="12" x2="21" y2="12"></line><line x1="3" y1="6" x2="21" y2="6"></line><line x1="3" y1="18" x2="21" y2="18"></line> </svg> </symbol> <symbol id="svg-arrow-right" viewBox="0 0 24 24"> <title>Expand</title> <svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="feather feather-chevron-right"> <polyline points="9 18 15 12 9 6"></polyline> </svg> </symbol> <symbol id="svg-doc" viewBox="0 0 24 24"> <title>Document</title> <svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="feather feather-file"> <path d="M13 2H6a2 2 0 0 0-2 2v16a2 2 0 0 0 2 2h12a2 2 0 0 0 2-2V9z"></path><polyline points="13 2 13 9 20 9"></polyline> </svg> </symbol> </svg> <div class="side-bar"> <div class="site-header"> <a href="/" class="site-title lh-tight"> </a> <a href="#" id="menu-button" class="site-button"> <svg viewBox="0 0 24 24" class="icon"><use xlink:href="#svg-menu"></use></svg> </a> </div> <nav role="navigation" aria-label="Main" id="site-nav" class="site-nav"> <ul class="nav-list"><li class="nav-list-item"><a href="/" class="nav-list-link">Home</a></li><li class="nav-list-item"><a href="/git/" class="nav-list-link">GitHub Setup</a></li><li class="nav-list-item"><a href="/build/" class="nav-list-link">Building Scotty3D</a></li><li class="nav-list-item"><a href="#" class="nav-list-expander"><svg viewBox="0 0 24 24"><use xlink:href="#svg-arrow-right"></use></svg></a><a href="/guide/" class="nav-list-link">User Guide</a><ul class="nav-list "><li class="nav-list-item "><a href="/guide/animate_mode/" class="nav-list-link">Animate</a></li><li class="nav-list-item "><a href="/guide/layout_mode/" class="nav-list-link">Layout</a></li><li class="nav-list-item "><a href="/guide/model_mode/" class="nav-list-link">Model</a></li><li class="nav-list-item "><a href="/guide/render_mode/" class="nav-list-link">Render</a></li><li class="nav-list-item "><a href="/guide/rigging_mode/" class="nav-list-link">Rig</a></li><li class="nav-list-item "><a href="/guide/simulate_mode/" class="nav-list-link">Simulate</a></li></ul></li><li class="nav-list-item"><a href="#" class="nav-list-expander"><svg viewBox="0 0 24 24"><use xlink:href="#svg-arrow-right"></use></svg></a><a href="/meshedit/" class="nav-list-link">A2: MeshEdit</a><ul class="nav-list "><li class="nav-list-item "><a href="/meshedit/halfedge" class="nav-list-link">Halfedge Mesh</a></li><li class="nav-list-item "><a href="#" class="nav-list-expander"><svg viewBox="0 0 24 24"><use xlink:href="#svg-arrow-right"></use></svg></a><a href="/meshedit/local/" class="nav-list-link">Local Operations</a><ul class="nav-list"><li class="nav-list-item "> <a href="/meshedit/local/edge_flip" class="nav-list-link">Edge Flip Tutorial</a> </li><li class="nav-list-item "> <a href="/meshedit/local/bevel/" class="nav-list-link">Bevelling</a> </li></ul></li><li class="nav-list-item "><a href="#" class="nav-list-expander"><svg viewBox="0 0 24 24"><use xlink:href="#svg-arrow-right"></use></svg></a><a href="/meshedit/global/" class="nav-list-link">Global Operations</a><ul class="nav-list"><li class="nav-list-item "> <a href="/meshedit/global/catmull/" class="nav-list-link">Catmull-Clark Subdivision</a> </li><li class="nav-list-item "> <a href="/meshedit/global/remesh/" class="nav-list-link">Isotropic Remeshing</a> </li><li class="nav-list-item "> <a href="/meshedit/global/linear/" class="nav-list-link">Linear Subdivision</a> </li><li class="nav-list-item "> <a href="/meshedit/global/loop/" class="nav-list-link">Loop Subdivision</a> </li><li class="nav-list-item "> <a href="/meshedit/global/simplify/" class="nav-list-link">Simplification</a> </li><li class="nav-list-item "> <a href="/meshedit/global/triangulate/" class="nav-list-link">Triangulation</a> </li></ul></li></ul></li><li class="nav-list-item"><a href="#" class="nav-list-expander"><svg viewBox="0 0 24 24"><use xlink:href="#svg-arrow-right"></use></svg></a><a href="/pathtracer/" class="nav-list-link">A3: Pathtracer</a><ul class="nav-list "><li class="nav-list-item "><a href="/pathtracer/camera_rays" class="nav-list-link">(Task 1) Camera Rays</a></li><li class="nav-list-item "><a href="#" class="nav-list-expander"><svg viewBox="0 0 24 24"><use xlink:href="#svg-arrow-right"></use></svg></a><a href="/pathtracer/intersecting_objects" class="nav-list-link">(Task 2) Intersections</a><ul class="nav-list"><li class="nav-list-item "> <a href="/pathtracer/ray_triangle_intersection" class="nav-list-link">Ray Triangle Intersection</a> </li><li class="nav-list-item "> <a href="/pathtracer/ray_sphere_intersection" class="nav-list-link">Ray Sphere Intersection</a> </li></ul></li><li class="nav-list-item "><a href="/pathtracer/bounding_volume_hierarchy" class="nav-list-link">(Task 3) BVH</a></li><li class="nav-list-item "><a href="/pathtracer/shadow_rays" class="nav-list-link">(Task 4) Shadow Rays</a></li><li class="nav-list-item "><a href="/pathtracer/path_tracing" class="nav-list-link">(Task 5) Path Tracing</a></li><li class="nav-list-item "><a href="#" class="nav-list-expander"><svg viewBox="0 0 24 24"><use xlink:href="#svg-arrow-right"></use></svg></a><a href="/pathtracer/materials" class="nav-list-link">(Task 6) Materials</a><ul class="nav-list"><li class="nav-list-item "> <a href="/pathtracer/dielectrics_and_transmission" class="nav-list-link">Dielectrics and Transmission</a> </li></ul></li><li class="nav-list-item "><a href="#" class="nav-list-expander"><svg viewBox="0 0 24 24"><use xlink:href="#svg-arrow-right"></use></svg></a><a href="/pathtracer/environment_lighting" class="nav-list-link">(Task 7) Environment Lighting</a><ul class="nav-list"><li class="nav-list-item "> <a href="/pathtracer/importance_sampling" class="nav-list-link">Environment Light Importance Sampling</a> </li></ul></li><li class="nav-list-item "><a href="/pathtracer/visualization_of_normals" class="nav-list-link">Visualization of normals</a></li></ul></li><li class="nav-list-item active"><a href="#" class="nav-list-expander"><svg viewBox="0 0 24 24"><use xlink:href="#svg-arrow-right"></use></svg></a><a href="/animation/" class="nav-list-link">A4: Animation</a><ul class="nav-list "><li class="nav-list-item "><a href="/animation/splines" class="nav-list-link">Splines</a></li><li class="nav-list-item "><a href="/animation/skeleton_kinematics" class="nav-list-link">Skeleton Kinematics</a></li><li class="nav-list-item "><a href="/animation/skinning" class="nav-list-link">Skinning</a></li><li class="nav-list-item active"><a href="/animation/particles" class="nav-list-link active">Particles</a></li></ul></li></ul> </nav> <footer class="site-footer"> This site uses <a href="https://github.com/pmarsceill/just-the-docs">Just the Docs</a>, a documentation theme for Jekyll. </footer> </div> <div class="main" id="top"> <div id="main-header" class="main-header"> <div class="search"> <div class="search-input-wrap"> <input type="text" id="search-input" class="search-input" tabindex="0" placeholder="Search " aria-label="Search " autocomplete="off"> <label for="search-input" class="search-label"><svg viewBox="0 0 24 24" class="search-icon"><use xlink:href="#svg-search"></use></svg></label> </div> <div id="search-results" class="search-results"></div> </div> </div> <div id="main-content-wrap" class="main-content-wrap"> <nav aria-label="Breadcrumb" class="breadcrumb-nav"> <ol class="breadcrumb-nav-list"> <li class="breadcrumb-nav-list-item"><a href="/animation/">A4: Animation</a></li> <li class="breadcrumb-nav-list-item"><span>Particles</span></li> </ol> </nav> <div id="main-content" class="main-content" role="main"> <h1 id="particle-simulation"> <a href="#particle-simulation" class="anchor-heading" aria-labelledby="particle-simulation"><svg viewBox="0 0 16 16" aria-hidden="true"><use xlink:href="#svg-link"></use></svg></a> Particle Simulation </h1> <p>And now for something completely different: physics simulation for particles.</p> <h2 id="ray-traced-physics"> <a href="#ray-traced-physics" class="anchor-heading" aria-labelledby="ray-traced-physics"><svg viewBox="0 0 16 16" aria-hidden="true"><use xlink:href="#svg-link"></use></svg></a> Ray traced physics </h2> </div> </div> <div class="search-overlay"></div> </div> </body> </html>
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<li class="breadcrumb-nav-list-item"><a href="/animation/">A4: Animation</a></li> <li class="breadcrumb-nav-list-item"><span>Skeleton Kinematics</span></li> </ol> </nav> <div id="main-content" class="main-content" role="main"> <h1 id="skeleton-kinematics"> <a href="#skeleton-kinematics" class="anchor-heading" aria-labelledby="skeleton-kinematics"><svg viewBox="0 0 16 16" aria-hidden="true"><use xlink:href="#svg-link"></use></svg></a> Skeleton Kinematics </h1> <p>A <code class="language-plaintext highlighter-rouge">Skeleton</code>(defined in <code class="language-plaintext highlighter-rouge">scene/skeleton.h</code>) is what we use to drive our animation. You can think of them like the set of bones we have in our own bodies and joints that connect these bones. For convenience, we have merged the bones and joints into the <code class="language-plaintext highlighter-rouge">Joint</code> class which holds the orientation of the joint relative to its parent as euler angle in its <code class="language-plaintext highlighter-rouge">pose</code>, and <code class="language-plaintext highlighter-rouge">extent</code> representing the direction and length of the bone with respect to its parent <code class="language-plaintext highlighter-rouge">Joint</code>. Each <code class="language-plaintext highlighter-rouge">Mesh</code> has an associated <code class="language-plaintext highlighter-rouge">Skeleton</code> class which holds a rooted tree of <code class="language-plaintext highlighter-rouge">Joint</code>s, where each <code class="language-plaintext highlighter-rouge">Joint</code> can have an arbitrary number of children.</p> <p>All of our joints are ball <code class="language-plaintext highlighter-rouge">Joint</code>s which have a set of 3 rotations around the <img src="task2_media/0027.png" style="height:14px" />, <img src="task2_media/0028.png" style="height: 16px" />, and <img src="task2_media/0029.png" style="height: 16px" /> axes, called <em>Euler angles</em>. Whenever you deal with angles in this way, a fixed order of operations must be enforced, otherwise the same set of angles will not represent the same rotation. In order to get the full rotational transformation matrix, <img src="task2_media/0030.png" style="height:16px" />, we can create individual rotation matrices around the <img src="task2_media/0031.png" style="height:16px" />, <img src="task2_media/0032.png" style="height:16px" />, and <img src="task2_media/0033.png" style="height:16px" /> axes, which we call <img src="task2_media/0034.png" style="height:20px" />, <img src="task2_media/0035.png" style="height:20px" />, and <img src="task2_media/0036.png" style="height:20px" /> respectively. The particular order of operations that we adopted for this assignment is that <img src="task2_media/0037.png" style="height:20px" />.</p> <h3 id="forward-kinematics"> <a href="#forward-kinematics" class="anchor-heading" aria-labelledby="forward-kinematics"><svg viewBox="0 0 16 16" aria-hidden="true"><use xlink:href="#svg-link"></use></svg></a> Forward Kinematics </h3> <p><em>Note: These diagrams are in 2D for visual clarity, but we will work with a 3D kinematic skeleton.</em></p> <p>When a joint’s parent is rotated, that transformation should be propagated down to all of its children. In the diagram below, <img src="task2_media/0038.png" style="height:18px" /> is the parent of <img src="task2_media/0039.png" style="height:18px" /> and <img src="task2_media/0040.png" style="height:18px" /> is the parent of <img src="task2_media/0041.png" style="height:18px" />. When a translation of <img src="task2_media/0042.png" style="height:18px" /> and rotation of <img src="task2_media/0043.png" style="height:18px" /> is applied to <img src="task2_media/0044.png" style="height:18px" />, all of the descendants are affected by this transformation as well. Then, <img src="task2_media/0045.png" style="height:18px" /> is rotated by <img src="task2_media/0046.png" style="height:18px" /> which affects itself and <img src="task2_media/0047.png" style="height:18px" />. Finally, when rotation of <img src="task2_media/0048.png" style="height:18px" /> is applied to <img src="task2_media/0049.png" style="height:18px" />, it only affects itself because it has no children.</p> <center><img src="task2_media/forward_kinematic_diagram.jpg" style="height:480px" /></center> <p>You need to implement these routines in <code class="language-plaintext highlighter-rouge">student/skeleton.cpp</code> for forward kinematics.</p> <ul> <li><code class="language-plaintext highlighter-rouge">Joint::joint_to_bind</code> Rreturn a matrix transforming points in the space of this joint to points in mesh space in bind position up to the base of this joint (end of its parent joint). You should traverse upwards from this joint’s parent all the way up to the root joint and accumulate their transformations.</li> <li><code class="language-plaintext highlighter-rouge">Joint::joint_to_posed</code> Return a matrix transforming points in the space of this joint to points in mesh space, taking into account joint poses. Again, you should traverse upwards from this joint’s parent to the root joint.</li> <li><code class="language-plaintext highlighter-rouge">Skeleton::end_of</code> Returns the end position of the joint in world coordinate frame, and you should take into account the base position of the skeleton (<code class="language-plaintext highlighter-rouge">Skeleton::base_pos</code>).</li> <li><code class="language-plaintext highlighter-rouge">Skeleton::posed_end_of</code> Returns the end position of the joint in world coordinate frame with poses, and you should take into account <code class="language-plaintext highlighter-rouge">Skeleton::base_pos</code>.</li> <li><code class="language-plaintext highlighter-rouge">Skeleton::joint_to_bind</code> Rreturn a matrix transforming points in the space of this joint to points in mesh space in bind position but with the base position of the skeleton taken in to account. Hint: use some function that you have implemented wisely!</li> <li><code class="language-plaintext highlighter-rouge">Skeleton::joint_to_posed</code> Return a matrix transforming points in the space of this joint to points in mesh space, taking into account joint poses but with the base position of the skeleton taken in to account. Hint: use some function that you have implemented wisely!</li> </ul> <p>Once you have implemented these basic kinematics, you should be able to define skeletons, set their positions at a collection of keyframes, and watch the skeleton smoothly interpolate the motion (see the <a href="../guide/animate.md">user guide</a> for an explanation of the interface). The gif below shows a very hasty demo defining a few joints and interpolating their motion.</p> <center><img src="task2_media/gif1.gif" /></center> <center><img src="task2_media/gif2.gif" /></center> <p>Note that the skeleton does not yet influence the geometry of the cube in this scene – that will come in Task 3!</p> <h3 id="task-2b---inverse-kinematics"> <a href="#task-2b---inverse-kinematics" class="anchor-heading" aria-labelledby="task-2b---inverse-kinematics"><svg viewBox="0 0 16 16" aria-hidden="true"><use xlink:href="#svg-link"></use></svg></a> Task 2b - Inverse Kinematics </h3> <h3 id="single-target-ik"> <a href="#single-target-ik" class="anchor-heading" aria-labelledby="single-target-ik"><svg viewBox="0 0 16 16" aria-hidden="true"><use xlink:href="#svg-link"></use></svg></a> Single Target IK </h3> <p>Now that we have a logical way to move joints around, we can implement Inverse Kinematics, which will move the joints around in order to reach a target point. There are a few different ways we can do this, but for this assignment we’ll implement an iterative method called gradient descent in order to find the minimum of a function. For a function <img src="task2_media/0050.png" style="height:18px" />, we’ll have the update scheme:</p> <center><img src="task2_media/0051.png" style="height:26px" /></center> <p>Where <img src="task2_media/0052.png" style="height:14px" /> is a small timestep. For this task, we’ll be using gradient descent to find the minimum of the cost function:</p> <center><img src="task2_media/0053.png" style="height:50px" /></center> <p>Where <img src="task2_media/0054.png" style="height:24px" /> is the position in world space of the target joint, and <img src="task2_media/0055.png" style="height:18px" /> is the position in world space of the target point. More specifically, we’ll be using a technique called Jacobian Transpose, which relies on the assumption:</p> <center><img src="task2_media/0056.png" style="height:30px" /></center> <p>Where:</p> <ul> <li><img src="task2_media/0057.png" style="height:20px" /> (n x 1) is the function <img src="task2_media/0058.png" style="height:22px" />, where <img src="task2_media/0059.png" style="height:22px" /> is the angle of joint <img src="task2_media/0060.png" style="height:18px" /> around the axis of rotation</li> <li><img src="task2_media/0061.png" style="height:16px" /> is a constant</li> <li><img src="task2_media/0062.png" style="height:22px" /> (3 x n) is the Jacobian of <img src="task2_media/0063.png" style="height:18px" /></li> </ul> <p>Note that here <img src="task2_media/0064.png" style="height:14px" /> refers to the number of joints in the skeleton. Although in reality this can be reduced to just the number of joints between the target joint and the root, inclusive, because all joints not on that path should stay where they are, so their columns in <img src="task2_media/0065.png" style="height:20px" /> will be 0. So <img src="task2_media/0066.png" style="height:14px" /> can just be the number of joints between the target and the root, inclusive. Additionally note that since this will get multiplied by <img src="task2_media/0067.png" style="height:16px" /> anyways, you can ignore the value of <img src="task2_media/0068.png" style="height:14px" />, and just consider the timestep as <img src="task2_media/0069.png" style="height:16px" />.</p> <p>Now we just need a way to calcluate the Jacobian of <img src="task2_media/0070.png" style="height:16px" />. For this, we can use the fact that:</p> <center><img src="task2_media/0071.png" style="height:34px" /></center> <p>Where:</p> <ul> <li><img src="task2_media/0072.png" style="height:24px" /> is the <img src="task2_media/0073.png" style="height:24px" /> column of <img src="task2_media/0074.png" style="height:24px" /></li> <li><img src="task2_media/0075.png" style="height:24px" /> is the axis of rotation</li> <li><img src="task2_media/0076.png" style="height:24px" /> is the vector from the base of joint <img src="task2_media/0077.png" style="height:24px" /> to the end point of the target joint</li> </ul> <p>For a more in-depth derivation of Jacobian transpose (and a look into other inverse kinematics algorithms), please check out <a href="https://web.archive.org/web/20190501035728/https://autorob.org/lectures/autorob_11_ik_jacobian.pdf">this presentation</a>. (Pages 45-56 in particular)</p> <p>Now, all of this will work for updating the angle along a single axis, but we have 3 axes to deal with. Luckily, extending it to 3 dimensions isn’t very difficult, we just need to update the angle along each axis independently.</p> <h3 id="multi-target"> <a href="#multi-target" class="anchor-heading" aria-labelledby="multi-target"><svg viewBox="0 0 16 16" aria-hidden="true"><use xlink:href="#svg-link"></use></svg></a> Multi-Target </h3> <p>We’ll extend this so we can have multiple targets, which will then use the function to minimize:</p> <center><img src="task2_media/0078.png" /></center> <p>which is a simple extension actually. Since each term is independent and added together, we can get the gradient of this new cost function just by summing the gradients of each of the constituent cost functions!</p> <p>You should implement multi-target IK, which will take a <code class="language-plaintext highlighter-rouge">vector</code> of <code class="language-plaintext highlighter-rouge">IK_Handle*</code>s called <code class="language-plaintext highlighter-rouge">active_handles</code> which stores the information a target point for a joint. See <code class="language-plaintext highlighter-rouge">scene/skeleton.h</code> for the definition of <code class="language-plaintext highlighter-rouge">IK_Handle</code> structure.</p> <p>In order to implement this, you should update <code class="language-plaintext highlighter-rouge">Joint::compute_gradient</code> and <code class="language-plaintext highlighter-rouge">Skeleton::step_ik</code>. <code class="language-plaintext highlighter-rouge">Joint::compute_gradient</code> should calculate the gradient of <img src="task2_media/0079.png" style="height:18px" /> in the x,y, and z directions, and add them to <code class="language-plaintext highlighter-rouge">Joint::angle_gradient</code> for all relevant joints. <code class="language-plaintext highlighter-rouge">Skeleton::step_ik</code> should actually do the gradient descent calculations and update the <code class="language-plaintext highlighter-rouge">pose</code> of each joint. In this function, you should probably use a very small timestep, but do several iterations (say, 10s to 100s) of gradient descent in order to speed things up. For even faster and better results, you can also implement a variable timestep instead of just using a fixed one. Note also that the root joint should never be updated.</p> <p>A key thing for this part is to <em>remember what coordinate frame you’re in</em>, because if you calculate the gradients in the wrong coordinate frame or use the axis of rotation in the wrong coordinate frame your answers will come out very wrong!</p> <h3 id="using-your-ik"> <a href="#using-your-ik" class="anchor-heading" aria-labelledby="using-your-ik"><svg viewBox="0 0 16 16" aria-hidden="true"><use xlink:href="#svg-link"></use></svg></a> Using your IK! </h3> <p>Once you have IK implemented, you should be able to create a series of joints, and get a particular joint to move to the desired final position you have selected.</p> </div> </div> <div class="search-overlay"></div> </div> </body> </html>
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href="/animation/particles" class="nav-list-link">Particles</a></li></ul></li></ul> </nav> <footer class="site-footer"> This site uses <a href="https://github.com/pmarsceill/just-the-docs">Just the Docs</a>, a documentation theme for Jekyll. </footer> </div> <div class="main" id="top"> <div id="main-header" class="main-header"> <div class="search"> <div class="search-input-wrap"> <input type="text" id="search-input" class="search-input" tabindex="0" placeholder="Search " aria-label="Search " autocomplete="off"> <label for="search-input" class="search-label"><svg viewBox="0 0 24 24" class="search-icon"><use xlink:href="#svg-search"></use></svg></label> </div> <div id="search-results" class="search-results"></div> </div> </div> <div id="main-content-wrap" class="main-content-wrap"> <nav aria-label="Breadcrumb" class="breadcrumb-nav"> <ol class="breadcrumb-nav-list"> <li class="breadcrumb-nav-list-item"><a href="/animation/">A4: Animation</a></li> <li class="breadcrumb-nav-list-item"><span>Skinning</span></li> </ol> </nav> <div id="main-content" class="main-content" role="main"> <h1 id="linear-blend-skinning"> <a href="#linear-blend-skinning" class="anchor-heading" aria-labelledby="linear-blend-skinning"><svg viewBox="0 0 16 16" aria-hidden="true"><use xlink:href="#svg-link"></use></svg></a> Linear Blend Skinning </h1> <p>Now that we have a skeleton set up, we need to link the skeleton to the mesh in order to get the mesh to follow the movements of the skeleton. We will implement linear blend skinning using the following functions: <code class="language-plaintext highlighter-rouge">Skeleton::skin()</code>, <code class="language-plaintext highlighter-rouge">Skeleton::find_joints()</code>, and <code class="language-plaintext highlighter-rouge">closest_on_line_segment</code>.</p> <p>The easiest way to do this is to update each of mesh vertices’ positions in relation to the bones (Joints) in the skeleton. There are 3 types of coordinate spaces: bind, joint, and pose. Bind is the initial coordinate frame of the vertices of where they are bound to relative to the mesh. Joint is the position of the vertex relative to a given joint. Pose is the world-space position after the joint transforms have been applied. You’ll want to compute transforms that take vertices in bind space and convert them to posed space (Hint: <code class="language-plaintext highlighter-rouge">joint_to_bind</code>, <code class="language-plaintext highlighter-rouge">joint_to_posed</code>, and <code class="language-plaintext highlighter-rouge">inverse()</code> will come in handy.)</p> <p>Your implementation should have the following basic steps for each vertex:</p> <ul> <li>Compute the vertex’s position with respect to each joint j in the skeleton in j’s coordinate frame when no transformations have been applied to the skeleton (bind pose, vertex bind position).</li> <li>Find where this vertex would end up (in world coordinates) if it were transformed along with bone j.</li> <li>Find the closest point on joint j’s bone segment (axis) and compute the distance to this closest point (Hint: <code class="language-plaintext highlighter-rouge">closest_on_line_segment</code> might come in handy). <ul> <li>Diagram of <code class="language-plaintext highlighter-rouge">closest_on_line_segment</code>:</li> </ul> </li> </ul> <center><img src="task3_media/closest_on_line_segment.png" style="height:280px" /></center> <ul> <li>Compute the resulting position of the vertex by doing a weighted average of the bind-to-posed transforms from each bone and applying it to the vertex. The weights for the weighted average should be the inverse distance to the joint, so closer bones have a stronger influence.</li> </ul> <p>Below we have an equation representation. The ith vertex v is the new vertex position. The weight w is the weight metric computed as the inverse of distance between the ith vertex and the closest point on joint j. We multiply this term with the position of the ith vertex v with respect to joint j after joint’s transformations has been applied.</p> <!--# TODO: fix this--> <!--![skinnning_equations](task3_media/skinning_equations.png)--> <center><img src="task3_media/skinning_eqn1.png" style="height:100px" /> <img src="task3_media/skinning_eqn2.png" style="height:120px" /></center> <p>In Scotty3D, the <code class="language-plaintext highlighter-rouge">Skeleton::skin()</code> function gets called on every frame draw iteration, recomputing all skinning related quantities. In this function, you should read vertices from <code class="language-plaintext highlighter-rouge">input.verts()</code> and indices from <code class="language-plaintext highlighter-rouge">input.indices()</code>, and write the resulting positions and norms to <code class="language-plaintext highlighter-rouge">v.pos</code> and <code class="language-plaintext highlighter-rouge">v.norm</code> for every vertex in the input vertices list.</p> <p>You will be implementing a Capsule-Radius Linear Blend Skin method, which only moves vertices with a joint if they lie in the joint’s radius. The <code class="language-plaintext highlighter-rouge">Skeleton::skin()</code> function also takes in a <code class="language-plaintext highlighter-rouge">map</code> of vertex index to relevant joints that you must compute the above distance/transformation metrics on. You are also responsible for creating this <code class="language-plaintext highlighter-rouge">map</code>, which is done so in <code class="language-plaintext highlighter-rouge">Skeleton::find_joints()</code>. Don’t worry about calling this function, it is called automatically before skin is called, populating the <code class="language-plaintext highlighter-rouge">map</code> field and sending it over to the <code class="language-plaintext highlighter-rouge">skin()</code> function. Your <code class="language-plaintext highlighter-rouge">Skeleton::find_joints()</code> implementation should iterate over all the vertices and add joint j to vertex index i in the map if the distance between the vertex and joint is less than <code class="language-plaintext highlighter-rouge">j-&gt;radius</code> (remember make sure they’re both in the same coordinate frame.)</p> <center><img src="task3_media/skinning.gif" style="height:240px" /></center> </div> </div> <div class="search-overlay"></div> </div> </body> </html>
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Operations</a><ul class="nav-list"><li class="nav-list-item "> <a href="/meshedit/local/edge_flip" class="nav-list-link">Edge Flip Tutorial</a> </li><li class="nav-list-item "> <a href="/meshedit/local/bevel/" class="nav-list-link">Bevelling</a> </li></ul></li><li class="nav-list-item "><a href="#" class="nav-list-expander"><svg viewBox="0 0 24 24"><use xlink:href="#svg-arrow-right"></use></svg></a><a href="/meshedit/global/" class="nav-list-link">Global Operations</a><ul class="nav-list"><li class="nav-list-item "> <a href="/meshedit/global/catmull/" class="nav-list-link">Catmull-Clark Subdivision</a> </li><li class="nav-list-item "> <a href="/meshedit/global/remesh/" class="nav-list-link">Isotropic Remeshing</a> </li><li class="nav-list-item "> <a href="/meshedit/global/linear/" class="nav-list-link">Linear Subdivision</a> </li><li class="nav-list-item "> <a href="/meshedit/global/loop/" class="nav-list-link">Loop Subdivision</a> </li><li class="nav-list-item "> <a 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href="/pathtracer/ray_sphere_intersection" class="nav-list-link">Ray Sphere Intersection</a> </li></ul></li><li class="nav-list-item "><a href="/pathtracer/bounding_volume_hierarchy" class="nav-list-link">(Task 3) BVH</a></li><li class="nav-list-item "><a href="/pathtracer/shadow_rays" class="nav-list-link">(Task 4) Shadow Rays</a></li><li class="nav-list-item "><a href="/pathtracer/path_tracing" class="nav-list-link">(Task 5) Path Tracing</a></li><li class="nav-list-item "><a href="#" class="nav-list-expander"><svg viewBox="0 0 24 24"><use xlink:href="#svg-arrow-right"></use></svg></a><a href="/pathtracer/materials" class="nav-list-link">(Task 6) Materials</a><ul class="nav-list"><li class="nav-list-item "> <a href="/pathtracer/dielectrics_and_transmission" class="nav-list-link">Dielectrics and Transmission</a> </li></ul></li><li class="nav-list-item "><a href="#" class="nav-list-expander"><svg viewBox="0 0 24 24"><use xlink:href="#svg-arrow-right"></use></svg></a><a href="/pathtracer/environment_lighting" class="nav-list-link">(Task 7) Environment Lighting</a><ul class="nav-list"><li class="nav-list-item "> <a href="/pathtracer/importance_sampling" class="nav-list-link">Environment Light Importance Sampling</a> </li></ul></li><li class="nav-list-item "><a href="/pathtracer/visualization_of_normals" class="nav-list-link">Visualization of normals</a></li></ul></li><li class="nav-list-item active"><a href="#" class="nav-list-expander"><svg viewBox="0 0 24 24"><use xlink:href="#svg-arrow-right"></use></svg></a><a href="/animation/" class="nav-list-link">A4: Animation</a><ul class="nav-list "><li class="nav-list-item active"><a href="/animation/splines" class="nav-list-link active">Splines</a></li><li class="nav-list-item "><a href="/animation/skeleton_kinematics" class="nav-list-link">Skeleton Kinematics</a></li><li class="nav-list-item "><a href="/animation/skinning" class="nav-list-link">Skinning</a></li><li class="nav-list-item "><a href="/animation/particles" class="nav-list-link">Particles</a></li></ul></li></ul> </nav> <footer class="site-footer"> This site uses <a href="https://github.com/pmarsceill/just-the-docs">Just the Docs</a>, a documentation theme for Jekyll. </footer> </div> <div class="main" id="top"> <div id="main-header" class="main-header"> <div class="search"> <div class="search-input-wrap"> <input type="text" id="search-input" class="search-input" tabindex="0" placeholder="Search " aria-label="Search " autocomplete="off"> <label for="search-input" class="search-label"><svg viewBox="0 0 24 24" class="search-icon"><use xlink:href="#svg-search"></use></svg></label> </div> <div id="search-results" class="search-results"></div> </div> </div> <div id="main-content-wrap" class="main-content-wrap"> <nav aria-label="Breadcrumb" class="breadcrumb-nav"> <ol class="breadcrumb-nav-list"> <li class="breadcrumb-nav-list-item"><a href="/animation/">A4: Animation</a></li> <li class="breadcrumb-nav-list-item"><span>Splines</span></li> </ol> </nav> <div id="main-content" class="main-content" role="main"> <h1 id="spline-interpolation"> <a href="#spline-interpolation" class="anchor-heading" aria-labelledby="spline-interpolation"><svg viewBox="0 0 16 16" aria-hidden="true"><use xlink:href="#svg-link"></use></svg></a> Spline Interpolation </h1> <p>As we discussed in class, data points in time can be interpolated by constructing an approximating piecewise polynomial or spline. In this assignment you will implement a particular kind of spline, called a Catmull-Rom spline. A Catmull-Rom spline is a piecewise cubic spline defined purely in terms of the points it interpolates. It is a popular choice in real animation systems, because the animator does not need to define additional data like tangents, etc. (However, your code may still need to numerically evaluate these tangents after the fact; more on this point later.) All of the methods relevant to spline interpolation can be found in <code class="language-plaintext highlighter-rouge">spline.h</code> with implementations in <code class="language-plaintext highlighter-rouge">spline.inl</code>.</p> <h3 id="task-1a---hermite-curve-over-the-unit-interval"> <a href="#task-1a---hermite-curve-over-the-unit-interval" class="anchor-heading" aria-labelledby="task-1a---hermite-curve-over-the-unit-interval"><svg viewBox="0 0 16 16" aria-hidden="true"><use xlink:href="#svg-link"></use></svg></a> Task 1a - Hermite Curve over the Unit Interval </h3> <p>Recall that a cubic polynomial is a function of the form:</p> <center><img src="task1_media/0000.png" style="height:30px" /></center> <p>where <img src="task1_media/0001.png" style="height:24px" />, and <img src="task1_media/0002.png" style="height:20px" /> are fixed coefficients. However, there are many different ways of specifying a cubic polynomial. In particular, rather than specifying the coefficients directly, we can specify the endpoints and tangents we wish to interpolate. This construction is called the “Hermite form” of the polynomial. In particular, the Hermite form is given by</p> <center><img src="task1_media/0003.png" style="height:28px" /></center> <p>where <img src="task1_media/0004.png" style="height:16px" /> are endpoint positions, <img src="task1_media/0005.png" style="height:16px" /> are endpoint tangents, and <img src="task1_media/0006.png" style="height:24px" /> are the Hermite bases</p> <center><img src="task1_media/0007.png" style="height:30px" /> <br /></center> <center><img src="task1_media/0008.png" style="height:30px" /> <br /></center> <center><img src="task1_media/0009.png" style="height:30px" /> <br /></center> <center><img src="task1_media/0010.png" style="height:30px" /> <br /></center> <p>Your first task is to implement the method <code class="language-plaintext highlighter-rouge">Spline::cubic_unit_spline()</code>, which evaluates a spline defined over the time interval <img src="task1_media/0011.png" style="height:24px" /> given a pair of endpoints and tangents at endpoints.</p> <p>Your basic strategy for implementing this routine should be:</p> <ul> <li>Evaluate the time, its square, and its cube (for readability, you may want to make a local copy).</li> <li>Using these values, as well as the position and tangent values, compute the four basis functions <img src="task1_media/0012.png" style="height:20px" /><img src="task1_media/0013.png" style="height:20px" /> of a cubic polynomial in Hermite form.</li> <li>Finally, combine the endpoint and tangent data using the evaluated bases, and return the result.</li> </ul> <p>Notice that this function is templated on a type T. In C++, a templated class can operate on data of a variable type. In the case of a spline, for instance, we want to be able to interpolate all sorts of data: angles, vectors, colors, etc. So it wouldn’t make sense to rewrite our spline class once for each of these types; instead, we use templates. In terms of implementation, your code will look no different than if you were operating on a basic type (e.g., doubles). However, the compiler will complain if you try to interpolate a type for which interpolation doesn’t make sense! For instance, if you tried to interpolate <code class="language-plaintext highlighter-rouge">Skeleton</code> objects, the compiler would likely complain that there is no definition for the sum of two skeletons (via a + operator). In general, our spline interpolation will only make sense for data that comes from a vector space, since we need to add T values and take scalar multiples.</p> <h3 id="task-1b-evaluation-of-a-catmull-rom-spline"> <a href="#task-1b-evaluation-of-a-catmull-rom-spline" class="anchor-heading" aria-labelledby="task-1b-evaluation-of-a-catmull-rom-spline"><svg viewBox="0 0 16 16" aria-hidden="true"><use xlink:href="#svg-link"></use></svg></a> Task 1B: Evaluation of a Catmull-Rom spline </h3> <p>The routine from part 1A just defines the interpolated spline between two points, but in general we will want smooth splines between a long sequence of points. You will now use your solution from part 1A to implement the method <code class="language-plaintext highlighter-rouge">Spline::at()</code> which evaluates a general Catmull-Romspline at the specified time in a sequence of points (called “knots”). Since we now know how to interpolate a pair of endpoints and tangents, the only task remaining is to find the interval closest to the query time, and evaluate its endpoints and tangents.</p> <p>The basic idea behind Catmull-Rom is that for a given time t, we first find the four closest knots at times</p> <center><img src="task1_media/0014.png" style="height:20px" /></center> <p>We then use t1 and t2 as the endpoints of our cubic “piece,” and for tangents we use the values</p> <center><img src="task1_media/0015.png" style="height:24px" /> <br /></center> <center><img src="task1_media/0016.png" style="height:24px" /> <br /></center> <p>In other words, a reasonable guess for the tangent is given by the difference between neighboring points. (See the Wikipedia and our course slides for more details.)</p> <center><img src="task1_media/spline_diagram.jpg" style="height:240px" /> <br /></center> <p>This scheme works great if we have two well-defined knots on either side of the query time t. But what happens if we get a query time near the beginning or end of the spline? Or what if the spline contains fewer than four knots? We still have to somehow come up with a reasonable definition for the positions and tangents of the curve at these times. For this assignment, your Catmull-Rom spline interpolation should satisfy the following properties:</p> <ul> <li>If there are no knots at all in the spline, interpolation should return the default value for the interpolated type. This value can be computed by simply calling the constructor for the type: T(). For instance, if the spline is interpolating Vector3D objects, then the default value will be <img src="task1_media/0017.png" style="height:20px" />.</li> <li>If there is only one knot in the spline, interpolation should always return the value of that knot (independent of the time). In other words, we simply have a constant interpolant.</li> <li>If the query time is less than or equal to the initial knot, return the initial knot’s value.</li> <li>If the query time is greater than or equal to the final knot, return the final knot’s value.</li> </ul> <p>Once we have two or more knots, interpolation can be handled using general-purpose code. In particular, we can adopt the following “mirroring” strategy to obtain the four knots used in our computation:</p> <ul> <li>Any query time between the first and last knot will have at least one knot “to the left” <img src="task1_media/0018.png" style="height:22px" /> and one “to the right” <img src="task1_media/0019.png" style="height:22px" />.</li> <li>Suppose we don’t have a knot “two to the left” <img src="task1_media/0020.png" style="height:22px" />. Then we will define a “virtual” knot <img src="task1_media/0021.png" style="height:22px" />. In other words, we will “mirror” the difference be observe between <img src="task1_media/0022.png" style="height:22px" /> and <img src="task1_media/0023.png" style="height:22px" /> to the other side of <img src="task1_media/0024.png" style="height:22px" />.</li> <li>Likewise, if we don’t have a knot “two to the right” <img src="task1_media/0025.png" style="height:22px" />), then we will “mirror” the difference to get a “virtual” knot <img src="task1_media/0026.png" style="height:22px" />.</li> <li>At this point, we have four valid knot values (whether “real” or “virtual”), and can compute our tangents and positions as usual.</li> <li>These values are then handed off to our subroutine that computes cubic interpolation over the unit interval.</li> </ul> <center><img src="task1_media/evaluate_catmull_rom_spline.png" style="height:300px" /> <br /></center> <p>An important thing to keep in mind is that <code class="language-plaintext highlighter-rouge">Spline::cubic_unit_spline()</code> assumes that the time value t is between 0 and 1, whereas the distance between two knots on our Catmull-Rom spline can be arbitrary. Therefore, when calling this subroutine you will have to normalize t such that it is between 0 and 1, i.e., you will have to divide by the length of the current interval over which you are interpolating. You should think very carefully about how this normalization affects the value computed by the subroutine, in comparison to the values we want to return. A transformation is necessary for both the tangents that you feed in to specify the unit spline.</p> <p>Internally, a Spline object stores its data in an STL map that maps knot times to knot values. A nice thing about an STL map is that it automatically keeps knots in sorted order. Therefore, we can quickly access the knot closest to a given time using the method <code class="language-plaintext highlighter-rouge">map::upper_bound()</code>, which returns an iterator to knot with the smallest time greater than the given query time (you can find out more about this method via online documentation for the Standard Template Library).</p> <h3 id="using-the-splines"> <a href="#using-the-splines" class="anchor-heading" aria-labelledby="using-the-splines"><svg viewBox="0 0 16 16" aria-hidden="true"><use xlink:href="#svg-link"></use></svg></a> Using the splines </h3> <p>Once you have implemented the functions in <code class="language-plaintext highlighter-rouge">spline.cpp</code>, you should be able to make simple animations by translating, rotating or scaling the mesh in the scene. The main idea is to:</p> <ul> <li>create an initial keyframe by clicking at a point on the white timeline at the bottom of the screen</li> <li>specify the initial location/orientation/scale of your mesh using the controls provided</li> <li>create more keyframes with different mesh locations/orientations/scales and watch the splines smoothly interpolate the movement of your mesh!</li> </ul> <center><img src="task1_media/animate_cow.gif" /></center> </div> </div> <div class="search-overlay"></div> </div> </body> </html>
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