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Cone & Cylinder by Revolution: Dynamic Solid Geometry

Cone & Cylinder by Revolution: Dynamic Solid Geometry
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2017-06-07 09:26
FunctionSequenceSolid GeometryTransformationParametric Equation
① This example demonstrates how rotating plane figures (such as a triangle and a rectangle) generates cones and cylinders, visually illustrating the concept of solids of revolution. ② Core mathematical concepts include solids of revolution, geometric properties of cones and cylinders, 3D coordinate transformations, and parametric equations. ③ Construction highlights: using GeoGebra commands such as Vector, Polygon, Sequence, and CurveCartesian to build the axis of rotation and generating profiles, with multiple objects (e.g., h, E_z, E_y, E_x, c, R_z, etc.) controlling rotation parameters and 3D coordinates, resulting in an interactive dynamic solid. ④ Suitable for high school or early college students learning solid geometry, introductory calculus (volume of revolution), and dynamic mathematics; also useful for teachers in classroom demonstrations or for students to explore the generation of solids independently.

GeoGebra Commands & Parameters

  1. Vector(1.1 * d * W_x) → u
  2. Vector(1.1 * d * W_y) → v
  3. Vector(1.1 * d * W_z) → w
  4. Polygon(1.1 * d * (W_x + W_y), 1.1 * d * (-W_x + W_y), 1.1 * d * (-W_x - W_y), 1.1 * d * (W_x - W_y)) → poly1, g, i, j, l
  5. Sequence(r * cos(0.125 * i) * W_x + r * sin(0.125 * i) * W_y, i, 0, p) → list1
  6. Sequence(r * cos(0.125 * i) * W_x + r * sin(0.125 * i) * W_y + k * W_z, i, 0, p) → list3
  7. CurveCartesian(x(r * cos(0.125 * i) * W_x + r * sin(0.125 * i) * W_y), y(r * cos(0.125 * i) * W_x + r * sin(0.125 * i) * W_y), i, 0, p) → e_1
  8. CurveCartesian(x(r * cos(0.125 * t) * W_x + r * sin(0.125 * t) * W_y + k * W_z), y(r * cos(0.125 * t) * W_x + r * sin(0.125 * t) * W_y + k * W_z), t, 0, p) → e_2
  9. Sequence(Segment[A, Element[list1, i]], i, 1, p + 1) → list2
  10. Sequence(Segment[Element[list1, i], Element[list3, i]], i, 1, p + 1) → list4
  11. Sequence(Segment[(0, 0) + k * W_z, Element[list3, i]], i, 1, p + 1) → list5

AI Drawing Prompt Reference

Create a 3D dynamic solid geometry scene showing plane figures rotating around an axis to generate a cone and a cylinder. The scene includes a vertical rotation axis, a triangle, and a rectangle as generating profiles. The triangle rotates to form a cone, and the rectangle rotates to form a cylinder. The solids are represented by discrete line segments and curves, with radial lines from the axis to points on circular cross-sections, and vertical lines connecting corresponding points at different heights. The circles are made of equally spaced points generated parametrically. Use a wireframe or semi-transparent style to emphasize the rotational symmetry and 3D structure.
The prompt above was compiled from the original drawing commands and is for reference only.