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Cardioid Envelope of Line Segments

Cardioid Envelope of Line Segments
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2014-12-30 00:25
CircleSequenceCoordinate Geometry
This example demonstrates how to generate a cardioid using a sequence of points on a circle. First, a sequence of points on the circle is created: Sequence[(r cos(i 2π/n), r sin(i 2π/n)), i, 0, n-1]. Then, a cardioid is generated using these points. The example involves the concepts of cardioids, sequences, and geometric construction. Using the Sequence command in GeoGebra, the generation process of the cardioid can be dynamically shown. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. Sequence(((Size * cos(((i * 2) * π / Envelope) - π / 2)), (Size * sin(((i * 2) * π / Envelope) - π / 2))), i, 1, Envelope) → list1
  2. Sequence(Segment[Element[list1, k], Element[list1, (2 * k)]], k, 1, Envelope) → list2
  3. Sequence(Segment[Element[list1, Envelope - k], Element[list1, Envelope - (2 * k)]], k, 1, Envelope) → list3
  4. Circle((0, 0), Size) → c

AI Drawing Prompt Reference

Generate a series of points evenly distributed on a circle, then connect these points with line segments to form a cardioid. First, create a circle and divide it into equal angles to generate multiple points. Connect these points in pairs to form line segments, and then connect them in the opposite order to create another set of segments. The combination of these segments forms the envelope of a cardioid.
The prompt above was compiled from the original drawing commands and is for reference only.