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Projections from 3D Object to 2D Plane

Projections from 3D Object to 2D Plane
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2019-05-01 03:08
Coordinate GeometrySolid GeometryTransformationGeometric ConstructionLocus
① This example demonstrates how to project a 3D object (such as a sphere, a cube, and points, lines, faces on it) onto a 2D plane via orthogonal projection, visualizing the mapping from 3D space to 2D. ② Core mathematical concepts include 3D geometry, orthogonal projection, intersections of solids and planes (e.g., a sphere intersecting a plane yields a conic section), intersections of lines and planes, and the spatial relationships among faces, edges, and vertices of a cube. ③ Construction details: using GeoGebra commands such as Sphere to create a sphere, PointIn to pick a point on the sphere, IntersectConic to find the conic intersection of the sphere and a plane, OrthogonalPlane to construct an orthogonal plane, Intersect to find intersection points, Cube to build a cube, Join to connect segments, Sequence to generate lists, SelectedIndex and Textfield for interactive control. Objects include size, c, a, A, e, g, B, i, j, h, k, C, C_2, l1a, l, D, E, F, G, H, the six faces of the cube (faceABCD, faceADHE, etc.) and twelve edges (edgeAD, edgeAB, etc.). ④ Suitable for high school or early college students learning solid geometry, spatial visualization, and projection transformations; also useful for math lab activities or teacher demonstrations of 3D-to-2D visualization.

GeoGebra Commands & Parameters

  1. Sphere(c, (size * sqrt(3) / 2)) → a
  2. PointIn(a) → A
  3. Sphere(A, size) → e
  4. IntersectConic(a, e) → g
  5. Point(g) → B
  6. Line(A, B) → i
  7. OrthogonalPlane(B, i) → j
  8. Sphere(B, A) → h
  9. IntersectConic(a, h) → k
  10. Intersect(j, k) → C, C_2
  11. Cube(A, B, C) → l, D, faceCDHG, faceEFGH, edgeAD, edgeAB, edgeBC, edgeCD, edgeAE, edgeDH, edgeEH, edgeEF, E, edgeBF, edgeFG, edgeCG, edgeGH, F, G, H, faceABCD, faceADHE, faceABFE, faceBCGF
  12. Line(c, A) → f
  13. Join({Sequence[l1a(i), i, 1, 3], l1b}) → l1
  14. Sequence(Line[Element[l1, i], vz], i, 1, Length[l1]) → l2
  15. Sequence(Intersect[Element[l2, i], xOyPlane], i, 1, Length[l2]) → l3
  16. Sequence(Segment[Element[l3, Element[l4, (2 * i) - 1]], Element[l3, Element[l4, (2 * i)]]], i, 1, Length[l4] / 2) → l5
  17. SelectedIndex(l6) → index
  18. Point(zAxis) → eye
  19. Sequence(Ray[eye, Element[l1, i]], i, 1, Length[l1]) → l7
  20. Sequence(Intersect[Element[l7, i], xOyPlane], i, 1, Length[l2]) → l8
  21. Sequence(Segment[Element[l8, Element[l4, (2 * i) - 1]], Element[l8, Element[l4, (2 * i)]]], i, 1, Length[l4] / 2) → l9
  22. Point(zAxis) → ec
  23. Textfield(eyeDistance) → InputBox1
  24. Sequence(Ray[left, Element[l1, i]], i, 1, Length[l1]) → l10
  25. Sequence(Intersect[Element[l10, i], xOyPlane], i, 1, Length[l2]) → l11
  26. Sequence(Segment[Element[l11, Element[l4, (2 * i) - 1]], Element[l11, Element[l4, (2 * i)]]], i, 1, Length[l4] / 2) → l12
  27. Sequence(Ray[right, Element[l1, i]], i, 1, Length[l1]) → l13
  28. Sequence(Intersect[Element[l13, i], xOyPlane], i, 1, Length[l13]) → l14
  29. Sequence(Segment[Element[l14, Element[l4, (2 * i) - 1]], Element[l14, Element[l4, (2 * i)]]], i, 1, Length[l4] / 2) → l15
  30. Textfield(decX) → InputBox2
  31. Textfield(decY) → InputBox3
  32. Sequence(Line[Element[l1, i], vOblique], i, 1, Length[l1]) → l16
  33. Sequence(Intersect[Element[l16, i], xOyPlane], i, 1, Length[l16]) → l17
  34. Sequence(Segment[Element[l17, Element[l4, (2 * i) - 1]], Element[l17, Element[l4, (2 * i)]]], i, 1, Length[l4] / 2) → l18

AI Drawing Prompt Reference

Draw a sphere, then select a point A on the sphere and draw a circle e with A as the center and the same radius as the sphere. The sphere and circle e intersect to form an ellipse g, and select a point B on the ellipse. Draw a line i through points A and B, and construct a plane j perpendicular to line i. The sphere with center B and radius AB intersects with plane j to form another ellipse h, and the ellipses g and h intersect to form two lines C and C_2. Then, construct a cube with points A and B as vertices, and connect the faces and edges of the cube. Finally, draw a series of lines parallel to the z-axis through points A and B, observe their projections on the xOy plane, and adjust the viewing angle to see the cube's projection changes from different perspectives.
The prompt above was compiled from the original drawing commands and is for reference only.