Exploring Cycloids …

Using Geometer’s Sketchpad to Support Mathematical ThinkingAdpated from Algebra in Motion by Audrey Weekshttp://www.calculusinmotion.comCYCLOIDS(c)AWeeks1/03 www.calculusinmotion.comFor wheel size, drag W ⇒wheel radius = 46.000 pixelsFor pebble size, drag P ⇒PWWant More?Show Epi/HypocycloidAt any time, ESC erases traces.1. Move to starting position2. Select one ⇒3. Roll1 rev. 2 revs.4 revs. 5 revs.At any time, ESC erases traces.1 rev.3 revs.6 revs.2 revs.3 revs.WHAT IF ... a wheel has a pebble stuck in its rim.As the wheel rolls along a straight path,what path will the pebble travel?WHAT IF ... a farmer's cart has square wheels,but as he drives it down his road, he gets asmooth ride. What does the road look like?(c)AWeeks1/03 www.calculusinmotion.comcenter1. Conjecture shape of road.2. Press "Roll Wheel" button; adjust conjecture if needed.3. Put trace on quad. interior; roll again to see road.(To trace quad. interior, press "Reset". Click on interior to highlight it.From top pull-down menu "Display", release on "Trace Quad.")At any time, ESC erases traces.Roll WheelResetwheel radius = 4.692 cm2⋅π⋅wheel radius = 29.480 cmadjustwheel radiusShelly Berman p. 2 of 4 Jo Ann FrickerCycloids.doc

Using Geometer’s Sketchpad to Support Mathematical Thinkingand Hypocycloids …€The hypocycloid is the curve produced by fixed point P on the circumference ofa small circle of radius b rolling around the inside of a large circle of radiusa > b.Eric W. Weisstein. "Hypocycloid." From MathWorld--A Wolfram Web Resource.http://mathworld.wolfram.com/Hypocycloid.html€CYCLOIDS(c)AWeeks1/03 www.calculusinmotion.comFor wheel size, drag W ⇒wheel radius = 20.000 pixelsFor pebble size, drag P ⇒wheel radiusPFor rim size, drag R ⇒rim radius = 144.000 pixelsrim sizeRoll for Hypocycloidm rim sizem wheel radius = 7.200Shelly Berman p. 3 of 4 Jo Ann FrickerCycloids.doc

Using Geometer’s Sketchpad to Support Mathematical Thinkingand Epicycloids€An epicycloid is the path traced out by a point P on the edge of a circle ofradius b rolling on the outside of a circle of radius a. An epicycloids with onecusp is called a cardioid, one with two cusps is called a nephroid.Eric W. Weisstein. "Epicycloid." From MathWorld--A Wolfram Web Resource.http://mathworld.wolfram.com/Epicycloid.html€CYCLOIDS(c)AWeeks1/03 www.calculusinmotion.comFor wheel size, drag W ⇒wheel radius = 50.000 pixelsFor pebble size, drag P ⇒wheel radiusFor rim size, drag R ⇒rim radius = 120.000 pixelsrim sizeRoll for Epicycloidm rim sizem wheel radius = 2.400Shelly Berman p. 4 of 4 Jo Ann FrickerCycloids.doc