digitalcourage.social

Khurram Wadee ✅Of course this was known to many famous <a href="https://mastodon.org.uk/tags/mathematicians" class="mention hashtag" rel="nofollow noopener" target="_blank">#mathematicians</a>, like <a href="https://mastodon.org.uk/tags/Euler" class="mention hashtag" rel="nofollow noopener" target="_blank">#Euler</a> and <a href="https://mastodon.org.uk/tags/Jacobi" class="mention hashtag" rel="nofollow noopener" target="_blank">#Jacobi</a> and they solved the equations in terms of <a href="https://mastodon.org.uk/tags/EllipticIntegrals" class="mention hashtag" rel="nofollow noopener" target="_blank">#EllipticIntegrals</a>. I find it satisfying to be able to visualize the <a href="https://mastodon.org.uk/tags/dynamics" class="mention hashtag" rel="nofollow noopener" target="_blank">#dynamics</a> of the this system, using <a href="https://mastodon.org.uk/tags/FreeSoftware" class="mention hashtag" rel="nofollow noopener" target="_blank">#FreeSoftware</a>, such as <a href="https://mastodon.org.uk/tags/WxMaxima" class="mention hashtag" rel="nofollow noopener" target="_blank">#WxMaxima</a>.

Khurram Wadee ✅We’re having a <a href="https://mastodon.org.uk/tags/Soliton" class="mention hashtag" rel="nofollow noopener" target="_blank">#Soliton</a> sale over here! Get yours quick! Seriously, though, this is a quick <a href="https://mastodon.org.uk/tags/animation" class="mention hashtag" rel="nofollow noopener" target="_blank">#animation</a> of the <a href="https://mastodon.org.uk/tags/SolitonInteraction" class="mention hashtag" rel="nofollow noopener" target="_blank">#SolitonInteraction</a> surface – the solution of the <a href="https://mastodon.org.uk/tags/Korteweg" class="mention hashtag" rel="nofollow noopener" target="_blank">#Korteweg</a>-deVries <a href="https://mastodon.org.uk/tags/Eauation" class="mention hashtag" rel="nofollow noopener" target="_blank">#Eauation</a> showing (x,t,u(x,t)) giving you an all-round view.<a href="https://mastodon.org.uk/tags/MyWork" class="mention hashtag" rel="nofollow noopener" target="_blank">#MyWork</a> <a href="https://mastodon.org.uk/tags/CCBYSA" class="mention hashtag" rel="nofollow noopener" target="_blank">#CCBYSA</a> <a href="https://mastodon.org.uk/tags/AppliedMathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#AppliedMathematics</a> <a href="https://mastodon.org.uk/tags/Mathematic" class="mention hashtag" rel="nofollow noopener" target="_blank">#Mathematic</a> <a href="https://mastodon.org.uk/tags/Phyiscs" class="mention hashtag" rel="nofollow noopener" target="_blank">#Phyiscs</a> <a href="https://mastodon.org.uk/tags/KdV" class="mention hashtag" rel="nofollow noopener" target="_blank">#KdV</a> <a href="https://mastodon.org.uk/tags/KdVEquation" class="mention hashtag" rel="nofollow noopener" target="_blank">#KdVEquation</a> <a href="https://mastodon.org.uk/tags/FreeSoftware" class="mention hashtag" rel="nofollow noopener" target="_blank">#FreeSoftware</a> <a href="https://mastodon.org.uk/tags/WxMaxima" class="mention hashtag" rel="nofollow noopener" target="_blank">#WxMaxima</a>

Khurram Wadee ✅I posted on <a href="https://mastodon.org.uk/tags/Soliton" class="mention hashtag" rel="nofollow noopener" target="_blank">#Soliton</a> interactions in the <a href="https://mastodon.org.uk/tags/Korteweg" class="mention hashtag" rel="nofollow noopener" target="_blank">#Korteweg</a>-deVries <a href="https://mastodon.org.uk/tags/Equation" class="mention hashtag" rel="nofollow noopener" target="_blank">#Equation</a> yesterday with an <a href="https://mastodon.org.uk/tags/Animation" class="mention hashtag" rel="nofollow noopener" target="_blank">#Animation</a> of how two <a href="https://mastodon.org.uk/tags/SolitaryWaves" class="mention hashtag" rel="nofollow noopener" target="_blank">#SolitaryWaves</a> interact. The <a href="https://mastodon.org.uk/tags/PhaseShift" class="mention hashtag" rel="nofollow noopener" target="_blank">#PhaseShift</a> experienced by both can also be seen in a static <a href="https://mastodon.org.uk/tags/3D" class="mention hashtag" rel="nofollow noopener" target="_blank">#3D</a> plot with time, t, plotted on one axis. You should be able to see a “dog-leg” in the trajectory of each wave after it interacts with the other.<a href="https://mastodon.org.uk/tags/MyWork" class="mention hashtag" rel="nofollow noopener" target="_blank">#MyWork</a> <a href="https://mastodon.org.uk/tags/CCBYSA" class="mention hashtag" rel="nofollow noopener" target="_blank">#CCBYSA</a> <a href="https://mastodon.org.uk/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Mathematics</a> <a href="https://mastodon.org.uk/tags/AppliedMathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#AppliedMathematics</a> <a href="https://mastodon.org.uk/tags/Physics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Physics</a> <a href="https://mastodon.org.uk/tags/FreeSoftware" class="mention hashtag" rel="nofollow noopener" target="_blank">#FreeSoftware</a> <a href="https://mastodon.org.uk/tags/WxMaxima" class="mention hashtag" rel="nofollow noopener" target="_blank">#WxMaxima</a>

Khurram Wadee ✅A subtlety probably difficult to spot in the animation is that the interaction of the two waves leaves them phase shifted, with the taller wave gaining position, while the shorter one loses it. This further animation shows the interactions again (purple) but I’ve also shown what would happen if each wave moved without interaction with the other (red and blue).<a href="https://mastodon.org.uk/tags/Animation" class="mention hashtag" rel="nofollow noopener" target="_blank">#Animation</a> <a href="https://mastodon.org.uk/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Mathematics</a> <a href="https://mastodon.org.uk/tags/AppliedMathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#AppliedMathematics</a> <a href="https://mastodon.org.uk/tags/Physics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Physics</a> <a href="https://mastodon.org.uk/tags/WxMaxima" class="mention hashtag" rel="nofollow noopener" target="_blank">#WxMaxima</a> <a href="https://mastodon.org.uk/tags/FreeSoftware" class="mention hashtag" rel="nofollow noopener" target="_blank">#FreeSoftware</a>

🚀 Несерьёзный Выдумщик 👨‍🔬Грядёт замена <a href="https://shitpost.poridge.club/tags/wxMaxima" rel="nofollow noopener" target="_blank">#wxMaxima</a> ? Оказывается в <a href="https://shitpost.poridge.club/tags/KDE" rel="nofollow noopener" target="_blank">#KDE</a> предлагается универсальный GUI для систем математики: <a href="https://shitpost.poridge.club/tags/Cantor" rel="nofollow noopener" target="_blank">#Cantor</a> — <a href="https://apps.kde.org/cantor/" rel="nofollow noopener" target="_blank">https://apps.kde.org/cantor/</a> Заявленный перечень environments включает: <a href="https://shitpost.poridge.club/tags/KAlgebra" rel="nofollow noopener" target="_blank">#KAlgebra</a>, <a href="https://shitpost.poridge.club/tags/Maxima" rel="nofollow noopener" target="_blank">#Maxima</a>, <a href="https://shitpost.poridge.club/tags/Sage" rel="nofollow noopener" target="_blank">#Sage</a>, <a href="https://shitpost.poridge.club/tags/Scilab" rel="nofollow noopener" target="_blank">#Scilab</a>, <a href="https://shitpost.poridge.club/tags/Qalculate" rel="nofollow noopener" target="_blank">#Qalculate</a> Т.е. полноценные, давно и хорошо известные системы computer algebra system (CAS) or symbolic algebra system (SAS). Хотя это скорее дополнительный вариант, тем кто на <a href="https://shitpost.poridge.club/tags/linux" rel="nofollow noopener" target="_blank">#linux</a> привык использовать <a href="https://shitpost.poridge.club/tags/wxMaxima" rel="nofollow noopener" target="_blank">#wxMaxima</a> — <a href="https://wxmaxima-developers.github.io/wxmaxima" rel="nofollow noopener" target="_blank">https://wxmaxima-developers.github.io/wxmaxima</a> Решает в алгебраическом виде уравнения и системы, численные вычисления, построение графиков. По возможностям близка к Maple и Mathematica. Из забавного позволяет видеть графики в динамичном виде: двумерные <a href="https://wxmaxima-developers.github.io/wxmaxima/images/wxmaxima-animations.mp4" rel="nofollow noopener" target="_blank">https://wxmaxima-developers.github.io/wxmaxima/images/wxmaxima-animations.mp4</a> трехмерные <a href="https://wxmaxima-developers.github.io/wxmaxima/images/wxmaxima-animation2.mp4" rel="nofollow noopener" target="_blank">https://wxmaxima-developers.github.io/wxmaxima/images/wxmaxima-animation2.mp4</a> И в целом всё то, что потом перешло в питоновский Jupyter как notebooks: <a href="https://shitpost.poridge.club/tags/математика" rel="nofollow noopener" target="_blank">#математика</a> <a href="https://shitpost.poridge.club/tags/машиннаяалгербра" rel="nofollow noopener" target="_blank">#машиннаяалгербра</a> <a href="https://shitpost.poridge.club/tags/mathematic" rel="nofollow noopener" target="_blank">#mathematic</a> <a href="https://shitpost.poridge.club/tags/CAS" rel="nofollow noopener" target="_blank">#CAS</a> <a href="https://shitpost.poridge.club/tags/ComputerAlgebra" rel="nofollow noopener" target="_blank">#ComputerAlgebra</a> <a href="https://shitpost.poridge.club/@russian_mastodon@mastodon.social" class="u-url mention" rel="nofollow noopener" target="_blank">@russian_mastodon@mastodon.social</a> <a href="https://shitpost.poridge.club/@ru@lor.sh" class="u-url mention" rel="nofollow noopener" target="_blank">@ru@lor.sh</a> <a href="https://shitpost.poridge.club/@Russia@3zi.ru" class="u-url mention" rel="nofollow noopener" target="_blank">@Russia@3zi.ru</a>

Khurram Wadee ✅So, the reason for this post is to share a dynamic way to see what the meaning of each of those Fmax curves is. They were produced using <a href="https://mastodon.org.uk/tags/WxMaxima" class="mention hashtag" rel="nofollow noopener" target="_blank">#WxMaxima</a> and various other <a href="https://mastodon.org.uk/tags/FOSS" class="mention hashtag" rel="nofollow noopener" target="_blank">#FOSS</a> utilities. I wish something like this was possible way back then, but even most handouts were hand-written in the 1980s.<a href="https://mastodon.org.uk/tags/MyWork" class="mention hashtag" rel="nofollow noopener" target="_blank">#MyWork</a> <a href="https://mastodon.org.uk/tags/CCBYSA" class="mention hashtag" rel="nofollow noopener" target="_blank">#CCBYSA</a> <a href="https://mastodon.org.uk/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Mathematics</a> <a href="https://mastodon.org.uk/tags/Physics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Physics</a> <a href="https://mastodon.org.uk/tags/Engineering" class="mention hashtag" rel="nofollow noopener" target="_blank">#Engineering</a> <a href="https://mastodon.org.uk/tags/Dynamics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Dynamics</a>

Khurram Wadee ✅It may be a bit soon but I’ve produced a <a href="https://mastodon.org.uk/tags/calendar" class="mention hashtag" rel="nofollow noopener" target="_blank">#calendar</a> for #2025. Unlike previous editions, which featured some of my photographs, this one contains <a href="https://mastodon.org.uk/tags/illustrations" class="mention hashtag" rel="nofollow noopener" target="_blank">#illustrations</a> to do with <a href="https://mastodon.org.uk/tags/mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#mathematics</a>, <a href="https://mastodon.org.uk/tags/geometry" class="mention hashtag" rel="nofollow noopener" target="_blank">#geometry</a>, <a href="https://mastodon.org.uk/tags/physics" class="mention hashtag" rel="nofollow noopener" target="_blank">#physics</a> and <a href="https://mastodon.org.uk/tags/engineering" class="mention hashtag" rel="nofollow noopener" target="_blank">#engineering</a> applications.<a href="https://commons.wikimedia.org/wiki/File:Calendar2025_v1.pdf" rel="nofollow noopener" translate="no" target="_blank">https://commons.wikimedia.org/wiki/File:Calendar2025_v1.pdf</a><a href="https://mastodon.org.uk/tags/MyWork" class="mention hashtag" rel="nofollow noopener" target="_blank">#MyWork</a> <a href="https://mastodon.org.uk/tags/CCBYSA" class="mention hashtag" rel="nofollow noopener" target="_blank">#CCBYSA</a> <a href="https://mastodon.org.uk/tags/PDF" class="mention hashtag" rel="nofollow noopener" target="_blank">#PDF</a> <a href="https://mastodon.org.uk/tags/png" class="mention hashtag" rel="nofollow noopener" target="_blank">#png</a> <a href="https://mastodon.org.uk/tags/Emacs" class="mention hashtag" rel="nofollow noopener" target="_blank">#Emacs</a> <a href="https://mastodon.org.uk/tags/FreeSoftware" class="mention hashtag" rel="nofollow noopener" target="_blank">#FreeSoftware</a> <a href="https://mastodon.org.uk/tags/WxMaxima" class="mention hashtag" rel="nofollow noopener" target="_blank">#WxMaxima</a>

Khurram Wadee ✅Yesterday, I posted an image of the <a href="https://mastodon.org.uk/tags/LorenzAttractor" class="mention hashtag" rel="nofollow noopener" target="_blank">#LorenzAttractor</a> showing the evolution of three trajectories (shown in red, green and blue) starting close together. Here, I’ve made it into a little animation to show how the paths initially stay close to each other but after about a quarter of the duration plotted, they <a href="https://mastodon.org.uk/tags/diverge" class="mention hashtag" rel="nofollow noopener" target="_blank">#diverge</a> from each other irrevocably (i.e. become uncorrelated) but remain part of the <a href="https://mastodon.org.uk/tags/ChaoticAttractor" class="mention hashtag" rel="nofollow noopener" target="_blank">#ChaoticAttractor</a>.<a href="https://mastodon.org.uk/tags/DynamicalSystems" class="mention hashtag" rel="nofollow noopener" target="_blank">#DynamicalSystems</a> <a href="https://mastodon.org.uk/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Mathematics</a> <a href="https://mastodon.org.uk/tags/AppliedMathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#AppliedMathematics</a> <a href="https://mastodon.org.uk/tags/CCBYSA" class="mention hashtag" rel="nofollow noopener" target="_blank">#CCBYSA</a> <a href="https://mastodon.org.uk/tags/FreeSoftware" class="mention hashtag" rel="nofollow noopener" target="_blank">#FreeSoftware</a> <a href="https://mastodon.org.uk/tags/WxMaxima" class="mention hashtag" rel="nofollow noopener" target="_blank">#WxMaxima</a>

Khurram Wadee ✅Just messing about a bit. Here is the famous <a href="https://mastodon.org.uk/tags/LorenzAttractor" class="mention hashtag" rel="nofollow noopener" target="_blank">#LorenzAttractor</a> plotted using <a href="https://mastodon.org.uk/tags/WxMaxima" class="mention hashtag" rel="nofollow noopener" target="_blank">#WxMaxima</a>. The three <a href="https://mastodon.org.uk/tags/trajectories" class="mention hashtag" rel="nofollow noopener" target="_blank">#trajectories</a>, shown in red, green and blue are for three fairly nearby <a href="https://mastodon.org.uk/tags/InitialConditions" class="mention hashtag" rel="nofollow noopener" target="_blank">#InitialConditions</a>.<a href="https://mastodon.org.uk/tags/DynamicalSystems" class="mention hashtag" rel="nofollow noopener" target="_blank">#DynamicalSystems</a> <a href="https://mastodon.org.uk/tags/ChaoticAttractors" class="mention hashtag" rel="nofollow noopener" target="_blank">#ChaoticAttractors</a> <a href="https://mastodon.org.uk/tags/StrangeAttractors" class="mention hashtag" rel="nofollow noopener" target="_blank">#StrangeAttractors</a> <a href="https://mastodon.org.uk/tags/NumericalSolutions" class="mention hashtag" rel="nofollow noopener" target="_blank">#NumericalSolutions</a> <a href="https://mastodon.org.uk/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Mathematics</a> <a href="https://mastodon.org.uk/tags/AppliedMathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#AppliedMathematics</a> <a href="https://mastodon.org.uk/tags/CCBYSA" class="mention hashtag" rel="nofollow noopener" target="_blank">#CCBYSA</a> <a href="https://mastodon.org.uk/tags/FreeSoftware" class="mention hashtag" rel="nofollow noopener" target="_blank">#FreeSoftware</a>

Khurram Wadee ✅Using the mapping w → z+1/z, you get something that looks remarkably like an <a href="https://mastodon.org.uk/tags/aerofoil" class="mention hashtag" rel="nofollow noopener" target="_blank">#aerofoil</a>.It is a <a href="https://mastodon.org.uk/tags/ConformalMapping" class="mention hashtag" rel="nofollow noopener" target="_blank">#ConformalMapping</a> meaning that angles are preserved during the mapping. In this animation, I’ve varied the imaginary part of the the eccentricity, while keeping the real part the same. With a zero <a href="https://mastodon.org.uk/tags/AngleOfAttack" class="mention hashtag" rel="nofollow noopener" target="_blank">#AngleOfAttack</a>, you can see the change in the airflow around the <a href="https://mastodon.org.uk/tags/wing" class="mention hashtag" rel="nofollow noopener" target="_blank">#wing</a> as its shape changes.<a href="https://mastodon.org.uk/tags/MyWork" class="mention hashtag" rel="nofollow noopener" target="_blank">#MyWork</a> <a href="https://mastodon.org.uk/tags/CCBYSA" class="mention hashtag" rel="nofollow noopener" target="_blank">#CCBYSA</a> <a href="https://mastodon.org.uk/tags/AppliedMathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#AppliedMathematics</a> <a href="https://mastodon.org.uk/tags/WxMaxima" class="mention hashtag" rel="nofollow noopener" target="_blank">#WxMaxima</a> <a href="https://mastodon.org.uk/tags/FreeSoftware" class="mention hashtag" rel="nofollow noopener" target="_blank">#FreeSoftware</a> <a href="https://mastodon.org.uk/tags/Aeronautics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Aeronautics</a> <a href="https://mastodon.org.uk/tags/Aerodynamics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Aerodynamics</a> <a href="https://mastodon.org.uk/tags/LaminarFlow" class="mention hashtag" rel="nofollow noopener" target="_blank">#LaminarFlow</a>

Khurram Wadee ✅The <a href="https://mastodon.org.uk/tags/Zhukovsky" class="mention hashtag" rel="nofollow noopener" target="_blank">#Zhukovsky</a> <a href="https://mastodon.org.uk/tags/Aerofoil" class="mention hashtag" rel="nofollow noopener" target="_blank">#Aerofoil</a> (sometimes transliterated as <a href="https://mastodon.org.uk/tags/Joukowsky" class="mention hashtag" rel="nofollow noopener" target="_blank">#Joukowsky</a> from <a href="https://mastodon.org.uk/tags/Russian" class="mention hashtag" rel="nofollow noopener" target="_blank">#Russian</a>), is a 2D model of <a href="https://mastodon.org.uk/tags/streamlined" class="mention hashtag" rel="nofollow noopener" target="_blank">#streamlined</a> <a href="https://mastodon.org.uk/tags/Airflow" class="mention hashtag" rel="nofollow noopener" target="_blank">#Airflow</a> past a <a href="https://mastodon.org.uk/tags/wing" class="mention hashtag" rel="nofollow noopener" target="_blank">#wing</a>. It uses <a href="https://mastodon.org.uk/tags/ComplexVariable" class="mention hashtag" rel="nofollow noopener" target="_blank">#ComplexVariable</a> and is an <a href="https://mastodon.org.uk/tags/AnalyticFunction" class="mention hashtag" rel="nofollow noopener" target="_blank">#AnalyticFunction</a> (i.e. <a href="https://mastodon.org.uk/tags/Differentiable" class="mention hashtag" rel="nofollow noopener" target="_blank">#Differentiable</a> everywhere, save at isolated <a href="https://mastodon.org.uk/tags/Singularities" class="mention hashtag" rel="nofollow noopener" target="_blank">#Singularities</a>). Take a circle in the <a href="https://mastodon.org.uk/tags/ComplexPlane" class="mention hashtag" rel="nofollow noopener" target="_blank">#ComplexPlane</a> which is not quite centred at the <a href="https://mastodon.org.uk/tags/origin" class="mention hashtag" rel="nofollow noopener" target="_blank">#origin</a> but passes through the <a href="https://mastodon.org.uk/tags/coordinate" class="mention hashtag" rel="nofollow noopener" target="_blank">#coordinate</a> (1,0) or (z=1+0i).<a href="https://mastodon.org.uk/tags/MyWork" class="mention hashtag" rel="nofollow noopener" target="_blank">#MyWork</a> <a href="https://mastodon.org.uk/tags/CCBYSA" class="mention hashtag" rel="nofollow noopener" target="_blank">#CCBYSA</a> <a href="https://mastodon.org.uk/tags/AppliedMathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#AppliedMathematics</a> <a href="https://mastodon.org.uk/tags/WxMaxima" class="mention hashtag" rel="nofollow noopener" target="_blank">#WxMaxima</a> <a href="https://mastodon.org.uk/tags/FreeSoftware" class="mention hashtag" rel="nofollow noopener" target="_blank">#FreeSoftware</a> <a href="https://mastodon.org.uk/tags/Aeronautics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Aeronautics</a> <a href="https://mastodon.org.uk/tags/Aerodynamics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Aerodynamics</a> <a href="https://mastodon.org.uk/tags/LaminarFlow" class="mention hashtag" rel="nofollow noopener" target="_blank">#LaminarFlow</a>

Khurram Wadee ✅This is why <a href="https://mastodon.org.uk/tags/Engineers" class="mention hashtag" rel="nofollow noopener" target="_blank">#Engineers</a> use <a href="https://mastodon.org.uk/tags/TransitionCurves" class="mention hashtag" rel="nofollow noopener" target="_blank">#TransitionCurves</a> in the form of <a href="https://mastodon.org.uk/tags/CornuSpirals" class="mention hashtag" rel="nofollow noopener" target="_blank">#CornuSpirals</a> or <a href="https://mastodon.org.uk/tags/Clothoids" class="mention hashtag" rel="nofollow noopener" target="_blank">#Clothoids</a> to connect from the straight segment onto a circular arc. It has the property that it starts with zero <a href="https://mastodon.org.uk/tags/curvature" class="mention hashtag" rel="nofollow noopener" target="_blank">#curvature</a> at the beginning and then the curvature increases linearly with distance travelled along the curve. At the appropriate curvature, it is connected to a circular arc with the same curvature.<a href="https://mastodon.org.uk/tags/MyWork" class="mention hashtag" rel="nofollow noopener" target="_blank">#MyWork</a> <a href="https://mastodon.org.uk/tags/WxMaxima" class="mention hashtag" rel="nofollow noopener" target="_blank">#WxMaxima</a> <a href="https://mastodon.org.uk/tags/CivilEngineering" class="mention hashtag" rel="nofollow noopener" target="_blank">#CivilEngineering</a> <a href="https://mastodon.org.uk/tags/HighwayEngineering" class="mention hashtag" rel="nofollow noopener" target="_blank">#HighwayEngineering</a> <a href="https://mastodon.org.uk/tags/CCBYSA" class="mention hashtag" rel="nofollow noopener" target="_blank">#CCBYSA</a>

Khurram Wadee ✅A high-speed road or railway track that is poorly designed. The vehicle moves at constant speed from the straight segment of road or track directly onto a CircularArc and thereby encounters an instantaneous application of centripetal acceleration. This is neither good for the driver and passengers who will feel a sudden force outwards, nor for the vehicle which my skid and impose large stress changes in the axles and bearings.<a href="https://mastodon.org.uk/tags/MyWork" class="mention hashtag" rel="nofollow noopener" target="_blank">#MyWork</a> <a href="https://mastodon.org.uk/tags/WxMaxima" class="mention hashtag" rel="nofollow noopener" target="_blank">#WxMaxima</a> <a href="https://mastodon.org.uk/tags/CivilEngineering" class="mention hashtag" rel="nofollow noopener" target="_blank">#CivilEngineering</a> <a href="https://mastodon.org.uk/tags/HighwayEngineering" class="mention hashtag" rel="nofollow noopener" target="_blank">#HighwayEngineering</a> <a href="https://mastodon.org.uk/tags/CCBYSA" class="mention hashtag" rel="nofollow noopener" target="_blank">#CCBYSA</a>

Khurram Wadee ✅Forgive the recent apparent obsession (I’d call it a fascination) with the <a href="https://mastodon.org.uk/tags/cycloid" class="mention hashtag" rel="nofollow noopener" target="_blank">#cycloid</a> but I’ve just discovered something I’d not heard of before. It is also called a <a href="https://mastodon.org.uk/tags/TautochroneCurve" class="mention hashtag" rel="nofollow noopener" target="_blank">#TautochroneCurve</a> or <a href="https://mastodon.org.uk/tags/Isochrone" class="mention hashtag" rel="nofollow noopener" target="_blank">#Isochrone</a> curve, which means that a particle starting from any location on the curve will get to the <a href="https://mastodon.org.uk/tags/MinimumPoint" class="mention hashtag" rel="nofollow noopener" target="_blank">#MinimumPoint</a> at precisely the same time as a particle starting at any other point.<a href="https://mastodon.org.uk/tags/Dynamics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Dynamics</a> <a href="https://mastodon.org.uk/tags/Kinematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Kinematics</a> <a href="https://mastodon.org.uk/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Mathematics</a> <a href="https://mastodon.org.uk/tags/AppliedMathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#AppliedMathematics</a> <a href="https://mastodon.org.uk/tags/Mechanics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Mechanics</a> <a href="https://mastodon.org.uk/tags/ClassicalMecanics" class="mention hashtag" rel="nofollow noopener" target="_blank">#ClassicalMecanics</a> <a href="https://mastodon.org.uk/tags/WxMaxima" class="mention hashtag" rel="nofollow noopener" target="_blank">#WxMaxima</a> <a href="https://mastodon.org.uk/tags/FreeSoftware" class="mention hashtag" rel="nofollow noopener" target="_blank">#FreeSoftware</a> <a href="https://mastodon.org.uk/tags/MyWork" class="mention hashtag" rel="nofollow noopener" target="_blank">#MyWork</a> <a href="https://mastodon.org.uk/tags/CCBYSA" class="mention hashtag" rel="nofollow noopener" target="_blank">#CCBYSA</a>

Khurram Wadee ✅Anyway, the <a href="https://mastodon.org.uk/tags/animation" class="mention hashtag" rel="nofollow noopener" target="_blank">#animation</a> took a bit of thought as it requires a bit of <a href="https://mastodon.org.uk/tags/Mechanics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Mechanics</a>, some <a href="https://mastodon.org.uk/tags/Integration" class="mention hashtag" rel="nofollow noopener" target="_blank">#Integration</a> and is made a bit more tricky as the curve is multi-valued and so you need to treat different branches separately. The <a href="https://mastodon.org.uk/tags/AnimatedGif" class="mention hashtag" rel="nofollow noopener" target="_blank">#AnimatedGif</a> was produce with <a href="https://mastodon.org.uk/tags/WxMaxima" class="mention hashtag" rel="nofollow noopener" target="_blank">#WxMaxima</a>.<a href="https://mastodon.org.uk/tags/MyWork" class="mention hashtag" rel="nofollow noopener" target="_blank">#MyWork</a> <a href="https://mastodon.org.uk/tags/CCBYSA" class="mention hashtag" rel="nofollow noopener" target="_blank">#CCBYSA</a> <a href="https://mastodon.org.uk/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Mathematics</a> <a href="https://mastodon.org.uk/tags/Maths" class="mention hashtag" rel="nofollow noopener" target="_blank">#Maths</a> <a href="https://mastodon.org.uk/tags/AppliedMathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#AppliedMathematics</a> <a href="https://mastodon.org.uk/tags/Physics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Physics</a> <a href="https://mastodon.org.uk/tags/Calculus" class="mention hashtag" rel="nofollow noopener" target="_blank">#Calculus</a>

Khurram Wadee ✅Imagine a circular wheel rolling, without skidding, on a flat, horizontal surface. The <a href="https://mastodon.org.uk/tags/locus" class="mention hashtag" rel="nofollow noopener" target="_blank">#locus</a> of any given point on its <a href="https://mastodon.org.uk/tags/circumference" class="mention hashtag" rel="nofollow noopener" target="_blank">#circumference</a> is called a <a href="https://mastodon.org.uk/tags/cycloid" class="mention hashtag" rel="nofollow noopener" target="_blank">#cycloid</a>. It is a <a href="https://mastodon.org.uk/tags/periodic" class="mention hashtag" rel="nofollow noopener" target="_blank">#periodic</a> <a href="https://mastodon.org.uk/tags/curve" class="mention hashtag" rel="nofollow noopener" target="_blank">#curve</a> with <a href="https://mastodon.org.uk/tags/period" class="mention hashtag" rel="nofollow noopener" target="_blank">#period</a> over the <a href="https://mastodon.org.uk/tags/circle" class="mention hashtag" rel="nofollow noopener" target="_blank">#circle</a>'s circumference and has <a href="https://mastodon.org.uk/tags/cusps" class="mention hashtag" rel="nofollow noopener" target="_blank">#cusps</a> whenever the point is in contact with the surface (the two sides of the curve are tangentially vertical at that point).<a href="https://mastodon.org.uk/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Mathematics</a> <a href="https://mastodon.org.uk/tags/Geometry" class="mention hashtag" rel="nofollow noopener" target="_blank">#Geometry</a> <a href="https://mastodon.org.uk/tags/Maths" class="mention hashtag" rel="nofollow noopener" target="_blank">#Maths</a> <a href="https://mastodon.org.uk/tags/AppliedMathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#AppliedMathematics</a> <a href="https://mastodon.org.uk/tags/Mechanics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Mechanics</a> <a href="https://mastodon.org.uk/tags/Kinematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Kinematics</a> <a href="https://mastodon.org.uk/tags/Dynamics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Dynamics</a> <a href="https://mastodon.org.uk/tags/Physics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Physics</a> <a href="https://mastodon.org.uk/tags/MyWork" class="mention hashtag" rel="nofollow noopener" target="_blank">#MyWork</a> <a href="https://mastodon.org.uk/tags/CCBYSA" class="mention hashtag" rel="nofollow noopener" target="_blank">#CCBYSA</a> <a href="https://mastodon.org.uk/tags/WxMaxima" class="mention hashtag" rel="nofollow noopener" target="_blank">#WxMaxima</a>

Khurram Wadee ✅Just a simple demonstration here of the <a href="https://mastodon.org.uk/tags/locus" class="mention hashtag" rel="nofollow noopener" target="_blank">#locus</a> of a <a href="https://mastodon.org.uk/tags/StraightLine" class="mention hashtag" rel="nofollow noopener" target="_blank">#StraightLine</a> in <a href="https://mastodon.org.uk/tags/ThreeDimensionalSpace" class="mention hashtag" rel="nofollow noopener" target="_blank">#ThreeDimensionalSpace</a> in <a href="https://mastodon.org.uk/tags/CartesianCoordinates" class="mention hashtag" rel="nofollow noopener" target="_blank">#CartesianCoordinates</a> passing through a known point with <a href="https://mastodon.org.uk/tags/PositionVector" class="mention hashtag" rel="nofollow noopener" target="_blank">#PositionVector</a> a and a direction d. A <a href="https://mastodon.org.uk/tags/Projection" class="mention hashtag" rel="nofollow noopener" target="_blank">#Projection</a> onto the x-y plane is also depicted.<a href="https://mastodon.org.uk/tags/AnimatedGif" class="mention hashtag" rel="nofollow noopener" target="_blank">#AnimatedGif</a> <a href="https://mastodon.org.uk/tags/FreeSoftware" class="mention hashtag" rel="nofollow noopener" target="_blank">#FreeSoftware</a> <a href="https://mastodon.org.uk/tags/WxMaxima" class="mention hashtag" rel="nofollow noopener" target="_blank">#WxMaxima</a> <a href="https://mastodon.org.uk/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Mathematics</a> <a href="https://mastodon.org.uk/tags/Maths" class="mention hashtag" rel="nofollow noopener" target="_blank">#Maths</a> <a href="https://mastodon.org.uk/tags/Geometry" class="mention hashtag" rel="nofollow noopener" target="_blank">#Geometry</a> <a href="https://mastodon.org.uk/tags/MyWork" class="mention hashtag" rel="nofollow noopener" target="_blank">#MyWork</a> <a href="https://mastodon.org.uk/tags/CCBYSA" class="mention hashtag" rel="nofollow noopener" target="_blank">#CCBYSA</a>

Khurram Wadee ✅A simple model for it might be to assume it is a <a href="https://mastodon.org.uk/tags/linear" class="mention hashtag" rel="nofollow noopener" target="_blank">#linear</a> <a href="https://mastodon.org.uk/tags/viscous" class="mention hashtag" rel="nofollow noopener" target="_blank">#viscous</a> fluid offering resistance proportional to the velocity of the projectile, i.e. R = -k.v, where v is the velocity vector and k is a constant which dictates how viscous the fluid is.Happily, this model has exact solutions so I didn’t need to do any numerical integration. Here are four different cases, including the ideal case k = 0. The others are k = 0.1, 1 and 10, with the k=1highlighted. Pt2...<a href="https://mastodon.org.uk/tags/MyWork" class="mention hashtag" rel="nofollow noopener" target="_blank">#MyWork</a> <a href="https://mastodon.org.uk/tags/CCBYSA" class="mention hashtag" rel="nofollow noopener" target="_blank">#CCBYSA</a> <a href="https://mastodon.org.uk/tags/Maths" class="mention hashtag" rel="nofollow noopener" target="_blank">#Maths</a> <a href="https://mastodon.org.uk/tags/WxMaxima" class="mention hashtag" rel="nofollow noopener" target="_blank">#WxMaxima</a>

AAA365Khurram Wadee ✅ (<a href="https://mastodon.org.uk/@mkwadee" class="u-url mention" rel="nofollow noopener" target="_blank">@mkwadee</a>) Engineer interested in nonlinear mechanics and science & maths in general. Also, politics & humour.The animation in my header is a demonstration of Kepler's Second Law of Planetary Motion (CC-BY-SA) -- see <a href="https://en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motion" rel="nofollow noopener" target="_blank">https://en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motion</a>You can check out others mathematical animations by me by searching for <a href="https://ohai.social/tags/WxMaxima" class="mention hashtag" rel="nofollow noopener" target="_blank">#WxMaxima</a>.Also, you can find my photographs by searching for <a href="https://ohai.social/tags/D7000" class="mention hashtag" rel="nofollow noopener" target="_blank">#D7000</a>. But please note that of course neither of these hashtags is unique to me.

Khurram Wadee ✅I was explaining to my wife how the <a href="https://mastodon.org.uk/tags/velocity" class="mention hashtag" rel="nofollow noopener" target="_blank">#velocity</a> <a href="https://mastodon.org.uk/tags/components" class="mention hashtag" rel="nofollow noopener" target="_blank">#components</a> of an idealized <a href="https://mastodon.org.uk/tags/projectile" class="mention hashtag" rel="nofollow noopener" target="_blank">#projectile</a> differ. She was having some problems understanding that only the <a href="https://mastodon.org.uk/tags/vertical" class="mention hashtag" rel="nofollow noopener" target="_blank">#vertical</a> component changes and that the <a href="https://mastodon.org.uk/tags/horizontal" class="mention hashtag" rel="nofollow noopener" target="_blank">#horizontal</a> component remains constant. In the end, I knocked up this little animation to tr and explain pictorially (the program in <a href="https://mastodon.org.uk/tags/WxMaxima" class="mention hashtag" rel="nofollow noopener" target="_blank">#WxMaxima</a> is actually interactive. so you can stop it, step through it or run it backwards).<a href="https://mastodon.org.uk/tags/Maths" class="mention hashtag" rel="nofollow noopener" target="_blank">#Maths</a> <a href="https://mastodon.org.uk/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Mathematics</a> <a href="https://mastodon.org.uk/tags/Mechanics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Mechanics</a> <a href="https://mastodon.org.uk/tags/ConstantAcceleration" class="mention hashtag" rel="nofollow noopener" target="_blank">#ConstantAcceleration</a> <a href="https://mastodon.org.uk/tags/Vectors" class="mention hashtag" rel="nofollow noopener" target="_blank">#Vectors</a> <a href="https://mastodon.org.uk/tags/Dynamics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Dynamics</a> <a href="https://mastodon.org.uk/tags/CCBYSA" class="mention hashtag" rel="nofollow noopener" target="_blank">#CCBYSA</a>

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