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In the Name of the Rose: Curves, Tangency, and Area Construction

In the Name of the Rose: Curves, Tangency, and Area Construction
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2020-09-12 00:17
CircleFunctionSequenceGeometric ConstructionGraphing
In collections Surfaces and flowers
This example demonstrates how combining circle geometry with tangency (via tangent-related functions/conditions) can generate a distinctive curve/surface configuration, together with measurable area or accumulated quantities. The main mathematical ideas are circle–tangent relationships, function-based modeling that encodes tangency behavior, parameter-driven construction of surfaces/trajectories, and area/increment accumulation (reflecting the use of Area and Addition concepts). Construction-wise, the GeoGebra command set and object list indicate: Sequence to build a parameter or stepwise set of elements, Surface to create the resulting surface geometry, and Segment to form the defining line segments. Additional derived objects such as circles (e.g., c), function-related elements (e.g., f/op/g), list containers (liste1), and intermediate labeled objects (e_1, l1, h, l2, etc.) are used to assemble the tangency-driven structure and to visualize the final image (image1), anchored by key points like O. Suitable for upper secondary to early university students, especially for lessons on geometric construction, graphing/parameterization, and the interplay between tangency constraints and area accumulation.

GeoGebra Commands & Parameters

  1. Sequence(Surface[(((1 + (c * (-1)^(i))) * cos((i * π / n) + (u * g(v)))) * f(v)), (((1 + (c * (-1)^(i))) * sin((i * π / n) + (u * g(v)))) * f(v)), (e * sin(v)), u, ((-π)) / n, π / n, v, -1.5708, 1.5708], i, 1, (2 * n)) → liste1
  2. Surface((((c + 1.05) * cos(u)) * cos(v)), (((c + 1.05) * sin(u)) * cos(v)), (e * sin(v)), u, -3.14159, 3.141592653589793, v, -1.5708, -0.5) → e_1
  3. Sequence(Surface[(((a + (c * (-1)^(i))) * cos(((i * 1.1) * π / n) + (u * g(v)))) * f(v)), (((a + (c * (-1)^(i))) * sin(((i * 1.1) * π / n) + (u * g(v)))) * f(v)), (e * sin(v)) - a / 5, u, ((-π)) / n, π / n, v, -1.5708, 1.5708], i, 1, (2 * n)) → l1
  4. Sequence(Surface[(((a + (c * (-1)^(i))) * cos(((i * 1.1) * π / n) + (u * g(v)))) * h(v)), (((a + (c * (-1)^(i))) * sin(((i * 1.1) * π / n) + (u * g(v)))) * h(v)), (e * sin(v)) + a / 5, u, ((-π)) / n, π / n, v, -1.5708, 1.5708], i, 1, (2 * n)) → l2
  5. Sequence(Surface[(((a / 2 + (c * (-1)^(i))) * cos(((i * 1.1) * π / n) + (u * g(v)))) * p(v)), (((a / 2 + (c * (-1)^(i))) * sin(((i * 1.1) * π / n) + (u * g(v)))) * p(v)), (e * sin(v)) + a / 1.5, u, ((-π)) / n, π / n, v, -1.5708, 1.5708], i, 1, (2 * n)) → l3
  6. Sequence(Surface[(((a / 1.25 + (c * (-1)^(i))) * cos((i * π / n) + (u * g(v)))) * q(v)), (((a / 1.25 + (c * (-1)^(i))) * sin((i * π / n) + (u * g(v)))) * q(v)), (e * sin(v)) + a / 2, u, ((-π)) / n, π / n, v, -1.5708, 1.5708], i, 1, (2 * n)) → l4
  7. Surface(x(O) + (((1.2 * j) * sin(i°)) * cos(((0.5 * m) * i°))), y(O) + (((1.2 * j) * cos(i°)) * cos(((0.5 * m) * i°))), z(O) + ((0.5 * (j - 1)) * cos(((0.5 * m) * i°))^(2)) + 0.25, i, 0, 720, j, 0, 1.8) → b
  8. Segment(O, O - (0, 0, 5)) → i

AI Drawing Prompt Reference

Create a complex geometric shape composed of multiple circles and curves, including a series of parameter-driven surfaces and segments. First, draw a series of parameterized surfaces by adjusting parameters like c, n, u, and v to form a series of ring-like structures. Then, draw a spiral surface around the origin. Finally, add a series of segments connecting key point O to form a 3D geometric structure.
The prompt above was compiled from the original drawing commands and is for reference only.