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3D Matrix Transformations

3D Matrix Transformations
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2018-02-06 21:31
VectorSolid GeometryMatrixTransformationGeometric Construction
This example demonstrates the effects of matrix transformations on 3D objects like cubes and pyramids in space. Key math concepts include matrix application, transformation, vectors, and solid geometry. GeoGebra commands such as SelectedIndex, Element, Vector, Circle, Line, Intersect, Segment, Point, Cube, ApplyMatrix, Polygon, Determinant, and Sequence are used to construct the transformations, with slider animations to visualize the changes. Suitable for high school and above students to learn about matrix transformations in solid geometry.

GeoGebra Commands & Parameters

  1. SelectedIndex(Examples) → index
  2. Element(MT, index) → T_0
  3. Vector((0, 0, 0), (x_0, y_0, z_0)) → u
  4. Vector((0, 0, 0), (x_1, y_1, z_1)) → v
  5. Vector((0, 0, 0), (x_2, y_2, z_2)) → w
  6. Circle(A, 1, xOyPlane) → c
  7. Line(A, xAxis) → f
  8. Intersect(f, c, 1) → B
  9. Segment(A, B) → g
  10. Point(Circle[B, Distance[A, B], Segment[A, B]]) → C
  11. Cube(A, B, C) → Cube, D, faceCDHG, faceEFGH, edgeAD, edgeAB, edgeBC, edgeCD, edgeAE, edgeDH, edgeEH, edgeEF, E, edgeBF, edgeFG, edgeCG, edgeGH, F, G, H, faceABCD, faceADHE, faceABFE, faceBCGF
  12. ApplyMatrix(T_a, A) → A'
  13. ApplyMatrix(T_a, B) → B'
  14. ApplyMatrix(T_a, C) → C'
  15. ApplyMatrix(T_a, D) → D'
  16. ApplyMatrix(T_a, E) → E'
  17. ApplyMatrix(T_a, F) → F'
  18. ApplyMatrix(T_a, G) → G'
  19. ApplyMatrix(T_a, H) → H'
  20. Polygon(A', B', C', D') → poly1, a', b', c', d'
  21. Polygon(E', A', B', F') → poly2, e', a'_1, b'_1, f'
  22. Polygon(B', C', G', F') → poly3, b'_2, c'_1, g', f'_1
  23. Polygon(A', E', H', D') → poly4, a'_2, e'_1, h', d'_1
  24. Polygon(H', D', C', G') → poly5, h'_1, d'_2, c'_2, g'_1
  25. Polygon(H', E', F', G') → poly6, h'_2, e'_2, f'_2, g'_2
  26. Determinant(T_0) → deter
  27. Sequence(Sequence[Sequence[(i, j, k), i, -4, 4, 8 / 3], j, -4, 4, 8 / 3], k, -4, 4, 8 / 3) → points3d
  28. Element(points3d, 1, 1) → KKA
  29. Sequence(OrthogonalLine[Element[KKA, i], z = 0], i, 1, 4) → list1
  30. Element(points3d, 1, 2) → KKA1
  31. Element(points3d, 1, 3) → KKA2
  32. Element(points3d, 1, 4) → KKA3
  33. Sequence(OrthogonalLine[Element[KKA1, i], z = 0], i, 1, 4) → list2
  34. Sequence(OrthogonalLine[Element[KKA2, i], z = 0], i, 1, 4) → list3
  35. Sequence(OrthogonalLine[Element[KKA3, i], z = 0], i, 1, 4) → list4
  36. Element(points3d, 2, 1) → KA
  37. Element(points3d, 3, 1) → KA1
  38. Element(points3d, 4, 1) → KA2
  39. Sequence(OrthogonalLine[Element[KKA, i], y = 0, space], i, 1, 4) → list5
  40. Sequence(OrthogonalLine[Element[KA1, i], y = 0, space], i, 1, 4) → list6
  41. Sequence(OrthogonalLine[Element[KA, i], y = 0, space], i, 1, 4) → list7
  42. Sequence(OrthogonalLine[Element[KA2, i], y = 0, space], i, 1, 4) → list8
  43. ApplyMatrix(T_a, list1) → list1'
  44. ApplyMatrix(T_a, list2) → list2'
  45. ApplyMatrix(T_a, list3) → list3'
  46. ApplyMatrix(T_a, list4) → list4'
  47. ApplyMatrix(T_a, list5) → list5'
  48. ApplyMatrix(T_a, list6) → list6'
  49. ApplyMatrix(T_a, list7) → list7'
  50. ApplyMatrix(T_a, list8) → list8'

AI Drawing Prompt Reference

Draw a 3D cube and its image after matrix transformation in space. First, create three vectors u, v, and w, then construct a circle c and a line f based on these points. Find the intersection point B of line f and circle c, connect point A and B to form segment g, and then find the point C on circle c closest to A. Use points A, B, and C to construct the cube. Apply matrix transformation T_a to each vertex of the cube to generate a new cube and its faces and edges. Finally, generate a series of points and draw their orthogonal lines on the x-axis and y-axis, and apply the same matrix transformation T_a to these lines.
The prompt above was compiled from the original drawing commands and is for reference only.