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Villarceau Circles

Villarceau Circles
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2014-11-19 00:37
CircleSequenceCoordinate Geometry
In collections surface 3D surface 3D
This example demonstrates Villarceau circles on a torus. Villarceau circles are special circles on a torus that are tangent to two intersecting circles on the torus. The construction process is shown using Circle, Sequence, and Intersect commands in GeoGebra. It involves concepts of circle, sequence, and coordinate geometry. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. Circle(zAxis, (R, 0, 0)) → c
  2. Sequence(Circle[Point[c, i], r, Tangent[Point[c, i], c]], i, 0, 1, 1 / n) → liste1
  3. Sequence(Intersect[pl, Element[liste1, i]], i, 1, Length[liste1]) → liste2
  4. Intersect(pl, zAxis) → A
  5. Sequence(Intersect[Line[Element[liste2, i], A], Element[liste1, i]], i, 1, Length[liste1]) → liste3

AI Drawing Prompt Reference

Draw a torus with a vertical central axis. Construct a series of small circles on the torus, evenly distributed along the main circle, each tangent to the main circle. Then, intersect these circles with a horizontal plane to obtain a set of points. From each point, draw a line to the central axis, and find the other intersection of this line with the corresponding small circle. Connect these points to form Villarceau circles, which are special circles tangent to two intersecting circles on the torus.
The prompt above was compiled from the original drawing commands and is for reference only.