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Viviani Curve by wuguoning

Viviani Curve by wuguoning
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2020-03-15 11:29
CirclePolygonTrigonometryCoordinate GeometryDerivative
This example demonstrates the Viviani Curve, the locus of the feet of perpendiculars dropped from a point on the incircle of a triangle to its sides. It involves core concepts in coordinate geometry, incircle properties, derivatives, curve sketching, and polygon characteristics. The construction uses commands like Surface and CurveCartesian with parameters a, b, c, d, e, f. Suitable for high school and advanced learners to explore complex curves and parametric representations.

GeoGebra Commands & Parameters

  1. Surface(a / 2 + ((u * a / 2) * cos(v)), ((u * a / 2) * sin(v)), 0, u, 0, 1, v, 0, (2 * π)) → b
  2. Surface(a / 2 + ((u * a / 2) * cos(v)), ((u * a / 2) * sin(v)), sqrt(((3 * a^(2))) / 4 - (a^(2) / 4 * u^(2)) - ((a^(2) / 2 * u) * cos(v))), u, 0, 1, v, 0, (2 * π)) → c
  3. Surface(a / 2 + (a / 2 * cos(u)), (a / 2 * sin(u)), (v * sqrt(a^(2) - (a / 2 + (a / 2 * cos(u)))^(2) - ((a / 2 * sin(u)))^(2))), u, 0, (2 * π), v, 0, 1) → d
  4. CurveCartesian(a / 2 + (a / 2 * cos(u)), (a / 2 * sin(u)), sqrt(a^(2) - (a / 2 + (a / 2 * cos(u)))^(2) - ((a / 2 * sin(u)))^(2)), u, 0, (2 * π)) → e
  5. CurveCartesian(a / 2 + (a / 2 * cos(u)), (a / 2 * sin(u)), 0, u, 0, (2 * π)) → f

AI Drawing Prompt Reference

Draw a Viviani curve, which is the locus of the feet of perpendiculars dropped from a point on the incircle of a triangle to its sides. Start by drawing the incircle, then use parameters a, u, and v to calculate points on the surface, and finally use CurveCartesian to draw the curve.
The prompt above was compiled from the original drawing commands and is for reference only.