Simple Envelopes

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2015-09-05 23:06

In collections My 3D Collection Ruled surfaces

This example demonstrates how to generate envelopes through geometric transformations. It involves concepts of circles and vectors. Using commands like Intersect, Point, Circle, Sequence, Vector, and Translate, a series of circles are constructed to generate the envelope. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

Intersect(xAxis, yAxis) → A
Point(yAxis) → B
Circle(A, B) → c
Sequence(Rotate[B, (π / 30 * n), A], n, 0, 60) → list1
Vector(A, C) → u
Translate(c, u) → c'
Translate(list1, u) → list1'
Sequence(Rotate[Element[list1, n], α, zAxis], n, 0, 120) → list3
Sequence(Segment[Element[list3, n], Element[list1', n]], n, 0, 120) → list2

AI Drawing Prompt Reference

Draw the intersection of the x‑ and y‑axes and a point on the y‑axis, then draw the circle through these two points. Create a series of points around the origin by rotating the point in small angular steps and draw circles centered at the origin passing through each of these points. Choose a point on the circle and use it with the origin to define a vector; translate the original circle and the whole set of points by this vector. Rotate the translated points around the origin through a range of angles to obtain a second set, and finally connect each point of the second set with its corresponding point from the translated set, producing a family of line segments that form an envelope curve.

The prompt above was compiled from the original drawing commands and is for reference only.