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Running Circles Around Circles: Part 2: Hypocycloid Astroid

1,247 views · Published 17 January 2017 · 14:21 · Indexed 20 September 2026

Channel: Math Easy Solutions · 2017 · Education

In this video I look at part 2 of the Laboratory Project titled “Running circles around circles”. In this part I use the amazing Desmos calculator to plot various hypocycloids from the parametric equations derived in part 1, but for various values of the a constant, while setting b equal to 1. I also take a closer look at the special case where a = 4, which is known as the “hypocycloid of four cusps” (or Astroid). When setting a = 4, I show how we can use the sin3x and cos3x trigonometric identities, which I derived earlier, to simplify the parametric equations to get x = 4cos^3(θ) and y = 4sin^3(θ). This is a very interesting video, not only because I further show the great graphing and animation capabilities of the Desmos calculator, but also show how to simplify the parametric equations using previous trig identities; so make sure to watch this video!

Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhuFI_qCMI-KHdba-Iw

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/running-circles-around-circles-part-2-hypocycloid-astroid

Related Videos:

Parametric Curves: Superellipses: https://youtu.be/JqkjQjtenPQ
Parametric Curves: Example 11: Conchoids of Nicomedes: https://youtu.be/Ir1tVdwGWec
Parametric Curves: Example 7: The Cycloid: Proof Part 1: https://youtu.be/XmHpIZqeRKk
Parametric Curves: Example 6: Graphing Devices: https://youtu.be/cwpEasc9mhQ
Parametric Curves: Example 5: Lissajous Figure: https://youtu.be/0OT1OnnAQ9I
Parametric Equations and Curves: https://youtu.be/Kd3XF4LZoFE
Parametric Equations and Polar Coordinates: https://youtu.be/usSors49Gdw
Trigonometry Identity: sin3x = 3sinx - 4sin^3(x): https://youtu.be/gsLYuQvdnDg
Trigonometry Identity: cos3x = 4cos^3(x) - 3cosx: https://youtu.be/ucfZgdryqq0 .

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Running Circles Around Circles: Part 2: Hypocycloid Astroid

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