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Pasta Machine 2.0

Pasta Machine 2.0
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2020-06-25 16:52
Coordinate GeometrySolid GeometryMatrixGeometric ConstructionSlider Animation
This example demonstrates the dynamic construction of 3D shapes and their surfaces generated by a pasta machine. Core mathematical concepts include solids or 3D shapes, surface area, kites, median lines, matrices, distributions, centroids. It utilizes GeoGebra commands like Polygon, Translate, Surface, and Centroid with objects such as α, h, A-G, poly1-2, a-g, i-o, and centroid. Suitable for high school and above students studying solid geometry, coordinate geometry, and matrix applications.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C, D, E, F, G) → poly1, a, b, c, d, e, f, g
  2. Translate(Polygon[Rotate[A, α], Rotate[B, α], Rotate[C, α], Rotate[D, α], Rotate[E, α], Rotate[F, α], Rotate[G, α]], Vector[(0, 0, h)]) → poly2
  3. Surface(((x(A) + (t * (x(B) - x(A)))) * cos((α * s / h))) - ((y(A) + (t * (y(B) - y(A)))) * sin((α * s / h))), ((x(A) + (t * (x(B) - x(A)))) * sin((α * s / h))) + ((y(A) + (t * (y(B) - y(A)))) * cos((α * s / h))), s, t, 0, 1, s, 0, h) → i
  4. Surface(((x(B) + (t * (x(C) - x(B)))) * cos((α * s / h))) - ((y(B) + (t * (y(C) - y(B)))) * sin((α * s / h))), ((x(B) + (t * (x(C) - x(B)))) * sin((α * s / h))) + ((y(B) + (t * (y(C) - y(B)))) * cos((α * s / h))), s, t, 0, 1, s, 0, h) → j
  5. Surface(((x(C) + (t * (x(D) - x(C)))) * cos((α * s / h))) - ((y(D) + (t * (y(C) - y(D)))) * sin((α * s / h))), ((x(C) + (t * (x(D) - x(C)))) * sin((α * s / h))) + ((y(C) + (t * (y(D) - y(C)))) * cos((α * s / h))), s, t, 0, 1, s, 0, h) → k
  6. Surface(((x(E) + (t * (x(D) - x(E)))) * cos((α * s / h))) - ((y(E) + (t * (y(D) - y(E)))) * sin((α * s / h))), ((x(E) + (t * (x(D) - x(E)))) * sin((α * s / h))) + ((y(E) + (t * (y(D) - y(E)))) * cos((α * s / h))), s, t, 0, 1, s, 0, h) → l
  7. Surface(((x(E) + (t * (x(F) - x(E)))) * cos((α * s / h))) - ((y(E) + (t * (y(F) - y(E)))) * sin((α * s / h))), ((x(E) + (t * (x(F) - x(E)))) * sin((α * s / h))) + ((y(E) + (t * (y(F) - y(E)))) * cos((α * s / h))), s, t, 0, 1, s, 0, h) → m
  8. Surface(((x(G) + (t * (x(F) - x(G)))) * cos((α * s / h))) - ((y(G) + (t * (y(F) - y(G)))) * sin((α * s / h))), ((x(G) + (t * (x(F) - x(G)))) * sin((α * s / h))) + ((y(G) + (t * (y(F) - y(G)))) * cos((α * s / h))), s, t, 0, 1, s, 0, h) → n
  9. Surface(((x(G) + (t * (x(A) - x(G)))) * cos((α * s / h))) - ((y(G) + (t * (y(A) - y(G)))) * sin((α * s / h))), ((x(G) + (t * (x(A) - x(G)))) * sin((α * s / h))) + ((y(G) + (t * (y(A) - y(G)))) * cos((α * s / h))), s, t, 0, 1, s, 0, h) → o
  10. Centroid(poly1) → centroid

AI Drawing Prompt Reference

Draw a 3D solid with a heptagonal base and a rotated, elevated copy of the base as the top face. Connect corresponding vertices to form a twisted prism. Generate curved surfaces along each edge between the base and top, where each surface twists smoothly from the bottom edge to the top edge, with rotation angle increasing linearly with height. The overall shape resembles a spiral pasta machine output. Mark the centroid of the base polygon. Emphasize the helical twist and vertical extrusion.
The prompt above was compiled from the original drawing commands and is for reference only.