Updated cycloid notebook with animation and applications

tommytwoeyes · Nov 23, 2015 · c87156e · c87156e
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diff --git a/10_Parametric_Equations__Polar_Coordinates/.ipynb_checkpoints/Cycloid-checkpoint.ipynb b/10_Parametric_Equations__Polar_Coordinates/.ipynb_checkpoints/Cycloid-checkpoint.ipynb
@@ -12,7 +12,26 @@
     "\n",
     "A cycloid is the curve traced out by a point on the circumference of a circle as the circle rolls along on a straight line, in its own plane.\n",
     "\n",
-    "___"
+    "___\n",
+    "\n",
+    "### Parametric Equations of the (basic) Cycloid Curve\n",
+    "\n",
+    "$$\\mathbf{ x(t) = r(\\theta-\\sin\\theta) \\quad y(t) = r(1-\\cos\\theta) }$$\n",
+    "\n",
+    "$$\\text{ For } \\theta \\in \\mathbb{R}$$\n",
+    "\n",
+    "___\n",
+    "\n",
+    "## Applications\n",
+    "Cycloids have many applications, including solutions to the [Brachistochrone](https://en.wikipedia.org/wiki/Brachistochrone_curve) and [Tautochrone](https://en.wikipedia.org/wiki/Tautochrone_curve) problems.\n",
+    "\n",
+    "Brachistochrone &amp; Tautochrone curves, in turn, have applications in, for example:\n",
+    "\n",
+    "* Pendulums\n",
+    "* Clock-making (pendulums)\n",
+    "* Surfing &amp; Skateboard ramp design ([Google it](https://www.google.com/search?q=Brachistochrone+curve+skateboard+ramp&oq=Brachistochrone+curve+skateboard+ramp))\n",
+    "\n",
+    "<img src='https://upload.wikimedia.org/wikipedia/commons/b/bd/Tautochrone_curve.gif'/>"
    ]
   },
   {

diff --git a/10_Parametric_Equations__Polar_Coordinates/Cycloid.ipynb b/10_Parametric_Equations__Polar_Coordinates/Cycloid.ipynb
@@ -12,7 +12,26 @@
     "\n",
     "A cycloid is the curve traced out by a point on the circumference of a circle as the circle rolls along on a straight line, in its own plane.\n",
     "\n",
-    "___"
+    "___\n",
+    "\n",
+    "### Parametric Equations of the (basic) Cycloid Curve\n",
+    "\n",
+    "$$\\mathbf{ x(t) = r(\\theta-\\sin\\theta) \\quad y(t) = r(1-\\cos\\theta) }$$\n",
+    "\n",
+    "$$\\text{ For } \\theta \\in \\mathbb{R}$$\n",
+    "\n",
+    "___\n",
+    "\n",
+    "## Applications\n",
+    "Cycloids have many applications, including solutions to the [Brachistochrone](https://en.wikipedia.org/wiki/Brachistochrone_curve) and [Tautochrone](https://en.wikipedia.org/wiki/Tautochrone_curve) problems.\n",
+    "\n",
+    "Brachistochrone &amp; Tautochrone curves, in turn, have applications in, for example:\n",
+    "\n",
+    "* Pendulums\n",
+    "* Clock-making (pendulums)\n",
+    "* Surfing &amp; Skateboard ramp design ([Google it](https://www.google.com/search?q=Brachistochrone+curve+skateboard+ramp&oq=Brachistochrone+curve+skateboard+ramp))\n",
+    "\n",
+    "<img src='https://upload.wikimedia.org/wikipedia/commons/b/bd/Tautochrone_curve.gif'/>"
    ]
   },
   {